Proof. Use the split integral group with the given root datum and the strictly henselian trait \(R=\mathbf Z_{(p)}^{\mathrm{sh}}\). Its closed geometric point is \(\overline{\mathbf F}_p\); its geometric generic point has characteristic zero. First forget the left parahoric equivariance. On each relative partial affine flag variety \(LG_R/Q_R\), use its Iwahori-cell stratification, whose strata are affine spaces over \(R\), and let \(\mathscr C_R\) be the thick category generated by their relative standard constant complexes, with finite Schubert support. A parahoric orbit is a union of these cells, and its standard constant complex belongs to \(\mathscr C_R\) by localization. We do not regard parahoric orbits themselves as affine spaces.

Here are the base-change hypotheses being used. Start the Tate lifts over \(S_0=\operatorname{Spec}\mathbf Z[1/\ell]\), which is an admissible base in (Richarz and Scholbach 2021, Notation 2.1). In that paper’s Section 2.4 the target \(T\) of a base change \(T\to S_0\) need only be Noetherian of finite Krull dimension. Thus its Lemma 2.12 applies to \(T=\operatorname{Spec}R\) and to both geometric fibers. It says that \(*\)-extension and \(!\)-restriction of a stratum commute with these pullbacks on the Tate lifts. Its proof first treats Iwahori cells and then obtains the parahoric version by localization. Realize these comparison maps in ordinary etale complexes. Transitivity gives the maps for restriction from \(R\) to either fiber; closing their equivalence loci under cones and retracts gives the assertion on \(\mathscr C_R\). The same argument shows that restrictions and corestrictions to a fine cell lie in the thick category generated by its constant complex. No motivic \(t\)-structure or new Beilinson–Soulé assertion over \(R\) is needed.

On an affine stratum we have \[R\Gamma(\mathbf A_R^n,\mathbf k)=\mathbf k =R\Gamma(\mathbf A_{\bar\eta}^n,\mathbf k) =R\Gamma(\mathbf A_{\bar s}^n,\mathbf k).\] Consequently fiber restriction identifies mapping complexes between objects generated by its constant complex. Induct on a finite closed union of strata containing the supports of \(A,B\in\mathscr C_R\). For a closed stratum \(i:Z\hookrightarrow Y\) with complementary open \(j:U\hookrightarrow Y\), the fiber sequence \[R\operatorname{Hom}_Z(i^*A,i^!B) \longrightarrow R\operatorname{Hom}_Y(A,B) \longrightarrow R\operatorname{Hom}_U(j^*A,j^*B)\] and the boundary comparison prove full faithfulness on mapping complexes. Fiber restriction is essentially surjective on the specified categories because it contains their standard generators and preserves cones and retracts. This also proves compatibility with composition: the comparisons are induced by one enhanced restriction functor.

Here is the equivariant version of this argument. On a bounded support choose a finite-type smooth quotient \(K_R\) of the acting parahoric; its discarded kernel is split pro-unipotent and has trivial positive equivariant cohomology. Compatible quotients can be used for inclusions of parahorics and for \(I^u\subset L^+G\) (Richarz and Scholbach 2021, Lemma 2.18). Each \(K_R\) is an extension of a split reductive group by split unipotent groups. The cohomology of \(K_R^a\) commutes with restriction to either geometric fiber: this follows from the Bruhat decomposition of its reductive quotient, homotopy invariance for the unipotent factors, and localization and duality for products of affine spaces and split tori. Compute equivariant mapping complexes by the bar presentation \[Y_R,\quad K_R\times Y_R,\quad K_R^2\times Y_R,\quad\ldots.\] At each simplicial degree use the strata \(K_R^a\times\mathbf A_R^n\) coming from the fine Iwahori cells of \(Y_R\). The underlying coefficients in the bar calculation are pulled back from \(Y_R\). The preceding recollement argument therefore applies, with the single-stratum calculation replaced by that for \(K_R^a\). It gives an equivalence of the mapping complexes at that degree. Totalizing these equivalences gives the equivariant comparison. No exchange of a scalar extension with an infinite totalization is used. The common relative parahoric standard complexes are equivariant objects; their fiber restrictions generate the specified equivariant categories. Full faithfulness on the bar mapping complexes and closure under cones and retracts therefore give essential surjectivity on those categories as well. The maps of bar presentations for subgroup inclusions give the claimed compatibility with restriction of equivariance.

Convolution is twisted exterior product followed by proper direct image on bounded supports. Smooth descent for the torsor defining the twisted product and proper base change therefore commute with fiber restriction. The comparison for all iterated products uses the same iterated convolution diagram and respects all associativity compatibilities. This is also the realized convolution comparison of (Richarz and Scholbach 2021, Proposition 3.14 and Lemma 3.15). The argument applies to the smooth proper projections between the partial affine flags. The stratumwise perverse conditions agree on the two fibers: the restriction and corestriction degrees agree, and the relative cell dimensions are the same. The equivalence is therefore perverse \(t\)-exact and preserves intermediate extensions, hence intersection complexes. Wakimoto complexes are finite convolutions of standard and costandard complexes, so they are preserved as well.

An algebraic closure of the generic field embeds in \(\mathbf C\). On each finite support and each finite bar term, algebraically closed extension in characteristic zero is fully faithful on bounded constructible \(\overline{\mathbb Q}_\ell\)-complexes (Javanpeykar et al. 2025, Lemma A.1). The etale–Betti comparison is likewise fully faithful (Sun 2017, Corollary 3.4.3, Proposition 3.5.9, and Remark 3.5.12). These comparisons use the derived pullback and site functors; their isomorphisms on Hom into every shift give the mapping-complex equivalences. Full faithfulness suffices here: the specified categories are generated from the matched standard complexes by finite cones and retracts. Comparing the same bar presentations at fixed coefficients before totalization gives the asserted characteristic-zero comparison. All constructions above use only \(p\ne\ell\). ◻

Proof. The geometric input for reductive groups is the monoidal dg equivalence of (Ben-Zvi et al. 2024, Theorem 6.6.1), which strengthens the monoidal triangulated calculation of (Bezrukavnikov and Finkelberg 2008, Theorem 5). We use the corrected version of the latter paper; its purity and multiplicative formality construction is in (Bezrukavnikov and Finkelberg 2008, secs. 6.5–6.6). The Galois-compatible formulation in (Ben-Zvi et al. 2024, Theorem 6.7.5) uses an arithmetic shearing. We verify below the split Weil normalization and dilation required here.

The regular Satake Ext algebra is the graded equivariant algebra \(\operatorname{Sym}(\widehat{\mathfrak g}^*)\), with its linear generators in degree \(2\); the Ext spaces between the other Satake objects are \[\operatorname{Ext}^{\bullet} (\operatorname{Sat}(V),\operatorname{Sat}(W)) =(S\otimes V^*\otimes W)^{\widehat G}.\] These identifications respect tensor products and composition. Lemma 3 carries the complete ordinary diagram of these calculations to every residue characteristic.

For the Weil calculation we need pointwise purity. Over every finite field, the intersection complex of a split parahoric Schubert variety is very pure and Tate with semisimple Frobenius (Cataldo et al. 2018, sec. 1.2 and Theorem 2.2.1). That theorem is stated for arbitrary split connected reductive groups and all \(p\ne\ell\). Writing \(\operatorname{IC}\) here for that paper’s unshifted complex, our normalization is \(\operatorname{IC}[d](d/2)\). It says that a stalk cohomology group in degree \(a\) has Frobenius scalar \(q^{a/2}\). Normalized Verdier self-duality gives the same assertion for point costalks. On a smooth orbit of dimension \(d_O\), the identity \(x^!j^!B=(j^!B)_x[-2d_O](-d_O)\) then gives that scalar also on the degree-\(b\) cohomology of the ordinary fiber of \(j^!B\).

The equivariant orbit filtration of a morphism complex has local terms computed from these restrictions and the cohomology of the split orbit stabilizer \(H\). Its reductive quotient is a split Levi, and the remaining unipotent factors are cohomologically trivial; \(H^c(BH)\) is Tate with scalar \(q^{c/2}\). A local term \[\operatorname{Hom}(H^a(A_x),H^b((j^!B)_x))\otimes H^c(BH)\] therefore has degree \(b-a+c\) and scalar \(q^{(b-a+c)/2}\). All spectral-sequence differentials vanish by their distinct weights. This proves purity and the semisimplified scalar on the Ext groups; it alone does not exclude extensions of equal-weight Frobenius modules. On a degree-two adjoint generator the scalar is nevertheless exactly \(q\): its semisimplification is \(q\), and it commutes with \(\widehat G\); each simple adjoint summand occurs once. On the central summand the generators are the first Chern classes of the split torus, on which Frobenius is also exactly \(q\). This determines Frobenius on the generated algebra. The corresponding goodness statement for convolution (Cataldo et al. 2018, Corollary 2.2.3), together with the perversity of spherical convolution, makes its normalized tensor multiplicity spaces Tate of weight zero with trivial Frobenius. Thus the canonical top-cell Weil structures give the split Satake tensor identifications used above.

For clarity, formality can be applied to this entire diagram at once. For a mixed morphism complex \(C=\bigoplus_w C_w\), use the multiplicative intermediate complex \[\bigoplus_w\tau_{\leq w}C_w.\] Its inclusion into \(C\) and its projection to \(\bigoplus_w H^w(C_w)\) are quasi-isomorphisms. Here \(C_w\) denotes the generalized weight summand; no semisimplicity of an arbitrary pure Weil complex is being inferred. Composition and tensor product add weights and degrees, so these quasi-isomorphisms are compatible with every composition and tensor operation. They are Frobenius-equivariant. Closing this enhanced diagram under finite cones and retracts gives the assertion. Only its associative monoidal structure is required below. ◻

Proof. The central construction and the Wakimoto filtration for every split reductive group over the base in question follow from (Cass et al. 2025, Theorem 1.4 and Theorem 5.24), by realization. The scope of its centrality assertion is exactly the homotopy category. More precisely, its Theorem 4.38 constructs the two natural maps in its Equation (4.7) from the enhanced nearby-cycles and convolution functors, using Lemma 4.20 and Theorem 4.36(3). For a fixed central label these give the stated natural equivalence of exact functors in the other kernel variable, so it applies to an actual cofiber diagram. Theorem 4.38 supplies the homotopy hexagon, and Theorem 4.41 the homotopy tensor compatibility. These finite identities are the ones needed for products of character operators on cohomology below. The maps are defined over \(\mathbf F_q\) and carry their actual Weil structures, after fixing the nearby-cycles Frobenius lift.

We record explicitly the graded-interchange argument, to include all root data. The Wakimoto graded category is the category of \(\Lambda\)-graded vector spaces, and the graded central functor has the weights of restriction from \(\widehat G\) to \(\widehat T\). For every dominant character \(\lambda\) of the actual torus there is its highest-weight quotient to the line labelled by \(J_\lambda\). On this quotient the central interchange is the ordinary interchange: restrict the fusion diagram to the open Cartan-position cell, where the sheaves are normalized constants and the quotient is the highest-line restriction. The two tensor generators are interchanged there, with exactly the fixed Satake parity correction. Naturality and the central tensor identity now imply the same interchange for an arbitrary central label with \(J_\lambda\). Indeed the two opposite central interchanges have product one, and passage to this quotient identifies one of them with the ordinary interchange, so identifies the other with its inverse. This is the argument of (Arkhipov and Bezrukavnikov 2009, Lemma 18 and Section 3.6.6, Equation (21)). Dominant characters generate \(\Lambda\) as a group. Tensor compatibility and invertibility of the \(J_\lambda\) extend the assertion to every \(\lambda\). The argument uses the actual character lattice, and applies to products and central torus factors as stated. Its top-cell maps are defined over \(\mathbf F_q\), so it preserves the specified normalization and Weil maps. Projection compatibility follows from proper base change for the nearby-cycles construction: the projection of flag levels is a smooth proper flag fibration, and the commutation before taking nearby cycles is the commutation of modifications at disjoint points. Using the same multiple-point diagram makes these compatibilities simultaneous for all the parahorics and all products.

The split Weil multiplicities can also be checked without any choice of splitting of the filtration. The trace of the central object is the Bernstein central character, over each finite extension. The Wakimoto trace functions are linearly independent. Consequently all power traces on \(V_a\) equal \(\dim V_a\); their semisimplified eigenvalues are one. This statement permits unipotent extensions, which do not affect the eigenvalue separations used later. ◻

Proof. We give the algebra-to-module calculation, including its root-datum and equivariance conventions. Let \(\mathcal R\) be the regular Satake ind-object. Its second regular action gives the \(\widehat G\)-action on the following algebras and modules. The regular calculations are \[\operatorname{Ext}^{\bullet}_{L^+G}(\delta_0,\mathcal R)=S, \qquad \operatorname{Ext}^{\bullet}_{I^u}(\delta_0,\mathcal R) =\mathbf k[\mathcal N]_{\rm sh},\] and the Wakimoto calculation for nef weights, including its products, is \[ \bigoplus_{\eta\ \text{nef}} \operatorname{Ext}^{\bullet}_{I^u} (\delta_0,J_\eta\star\mathcal R) \simeq \left(\bigoplus_{\eta\ \text{nef}} \Gamma(\widetilde{\mathcal N},\mathcal O(\eta))\right)_{\rm sh}. \tag{6}\] The algebra calculation is (Arkhipov et al. 2004, Theorems 7.3.1 and 7.6.1); the section algebra and its product maps are (Arkhipov et al. 2004, Theorem 8.5.2 and Equations (8.7.2)–(8.7.8)). We explain why using the actual lattice \(\Lambda\) entails no isogeny restriction here. Wakimoto inversion identifies the left side for \(\eta\) with the appropriate fixed-point costalk of \(\mathcal R\). Its cohomological filtration is the principal-nilpotent filtration on the \(-\eta\) weight space of \(\mathbf k[\widehat G]\). The fixed-point and filtration assertions of (Ginzburg 2000, Propositions 5.5.2 and 5.6.2) are stated for the full cocharacter lattice of an arbitrary semisimple group. The section-filtration identification used in (Arkhipov et al. 2004, Equations (8.6.2)–(8.6.3)) applies to every dominant character of the actual dual torus. Multiplication of the regular representation, followed by the fixed-point-to-convolution base-change maps in (Arkhipov et al. 2004, Equations (8.7.6)–(8.7.8)), identifies the products in Equation (6). This proves the required calculation for every semisimple isogeny type, without an assertion about all objects in a generalized ABG equivalence.

For a torus the calculation is immediate: its Grassmannian is \(\Lambda\), its pro-unipotent group is cohomologically trivial, and the algebra map is \(\operatorname{Sym}(\operatorname{Lie}(\widehat T)^*[-2])\to\mathbf k\). For a general reductive group, perform the characteristic-zero calculation using its central isogeny to the product of its adjoint group and its abelianization. Each Grassmannian component identifies with the corresponding component in that product; finite central kernels have trivial positive cohomology with these coefficients. On the dual side this restricts the regular representation and its products to the prescribed central-character lattice. It therefore restricts precisely the preceding section-algebra calculation. Central translations have length zero, and central linear variables map to zero. Finally Lemma 3 transports this ordinary enhanced diagram, including the forgetful map of algebras and the module action, to every \(p\ne\ell\). The regular ind-object causes no problem: mapping out of the point in these Ind-completions commutes with its filtered union.

We now identify the polynomial map. The equivariant-cohomology spectral sequence for the regular internal Hom has second page \[H^{\bullet}(BG)\otimes \operatorname{Ext}^{\bullet}_{I^u}(\delta_0,\mathcal R).\] Both factors are even, so its edge map induces \[\mathbf k\otimes_{H^{\bullet}(BG)} \operatorname{Ext}^{\bullet}_{L^+G}(\delta_0,\mathcal R) \simeq \operatorname{Ext}^{\bullet}_{I^u}(\delta_0,\mathcal R).\] Freeness over \(H^{\bullet}(BG)\) follows by lifting a homogeneous basis on this second factor and comparing its associated graded. Taking \(\widehat G\)-invariants identifies \(H^{\bullet}(BG)\) with \(S^{\widehat G}\), since the invariant part of \(\mathcal R\) is the unit. The map thus has kernel \(S(S^{\widehat G})_{>0}\) and is restriction to the nilpotent cone. On degree-two simple adjoint factors its identification differs from the usual one at most by nonzero scalars, by equivariance and surjectivity. Absorb these constant scalars in the choice of coordinates; they commute with the Frobenius dilation. This also identifies the action on each module in Equation (6).

The comparisons just used are comparisons of dg modules. In characteristic zero the simultaneous purity construction of (Arkhipov et al. 2004, Proposition 9.5.2 and Lemma 9.5.3), or the multiplicative weight truncation in Theorem 4, formalizes the algebra, its map, and its positive modules together. The ordinary enhanced specialization carries that diagram to the desired characteristic. No identification of an arbitrary Weil structure on a Springer test is needed for the present assertion.

For every finite-dimensional \(V\), adjunction now gives \[\begin{align*} R\operatorname{Hom}(\operatorname{Sat}(V),\operatorname{Av}_*W_\eta) &\simeq R\operatorname{Hom}(\operatorname{For}\operatorname{Sat}(V),W_\eta)\\ &\simeq \bigl(R\Gamma(\widetilde{\mathcal N}, V^*\otimes\mathcal O(\eta))\bigr)^{\widehat G}_{\rm sh}\\ &\simeq R\operatorname{Hom}_{S,\widehat G} (S\otimes V,(R\pi_*\mathcal O(\eta))_{\rm sh}). \end{align*}\] For every nef \(\eta\), including walls, the higher line-bundle cohomology vanishes by (Broer 1993, Theorem 2.4 and Remark 2.7(2)). Thus the section-module computation indeed computes the derived pushforward appearing here. This use of vanishing is entirely on the characteristic-zero dual side. The polynomial computation proves naturality in all morphisms between the free labels, which compactly generate. Yoneda proves the asserted equivalence. The pushforward is coherent on the smooth affine space \(\widehat{\mathfrak g}\); an equivariant graded finite free resolution, followed by shearing, shows that it is a perfect \(S\)-module.

Finally our forgetful functor preserves the underlying sheaf. If one instead uses \(\operatorname{For}[c](c/2)\), its right adjoint is \(\operatorname{Av}_*[-c](-c/2)\). The integer \(c\) is the dimension fixed by that change of level, independently of \(\eta\). ◻

Proof. For two kernels, the path description is \[(P,Q;\ U(P,Q),\ V(Q,{}^{\mathrm{Fr}}P)).\] Rotating the first step sends \((P,Q)\) to \((Q,{}^{\mathrm{Fr}}P)\). To obtain pullback in that direction use its inverse, which sends \((P,Q)\) to \(({}^{\mathrm{Fr}^{-1}}Q,P)\). The two coefficients on this rotated path are \(\Phi(V)({}^{\mathrm{Fr}^{-1}}Q,P)\) and \(U(P,Q)\). This is precisely Equation (7). In this description \(\mathrm{Fr}^{-1}\) may be interpreted on perfections. Etale constructible sheaves and their compact direct images are invariant under the resulting universal homeomorphisms, so the construction also defines the same map before perfection.

For any finite list of kernels use the stack of all its intermediate bundles. Multiplication forgets an intermediate bundle, a unit repeats one, and rotation moves an end bundle through Frobenius. The identities between these operations are identities of the corresponding path diagrams. Applying compact direct image, base change, and the projection formula in the enhanced six-operation formalism preserves these identities coherently. The required precise finite-type foundations are the enhanced adic operations and their base-change and projection formula compatibilities in (Liu and Zheng 2024, sec. 7.1 and Theorem 7.2.7).

One can express the coherence as a balanced bar construction. In simplicial degree \(n\) retain all intermediate kernels in the path; inner faces compose adjacent kernels, degeneracies insert units, and the end face is the rotation just described. Their common refinements are again the same path stack, which supplies every simplicial identity and its higher compatibility. This verifies coherence in the enhancement, rather than only pairwise isomorphisms in its homotopy category.

The same reasoning applies to the locally finite type stack. A fixed finite diagram uses finitely many bounded kernels. The image and inverse image of a finite-type bounded open under the relevant bounded-position maps are contained in some larger finite-type bounded open. Thus that diagram is computed on a finite-type stage, after enlarging stages when necessary. The finite-type maps are compatible under extension by zero, so their filtered colimit preserves the same identities. Central-lattice quotients are treated by the corresponding locally finite sums of these diagrams.

When \(U=\mathbf1\), inverse rotation of the path is \(\mathrm{Fr}^{-1}\). Its compact direct image is pullback by \(\mathrm{Fr}\). Composing with the Weil structure is therefore exactly cohomological Frobenius. Moving disjoint groups of labels gives commuting rotations of the common path diagram, with product the complete rotation. ◻

Proof. The kernels in use are rigid. Standard simple kernels are invertible with inverse the corresponding costandard kernel; length-zero kernels are invertible as well. Finite standard filtrations, cones, and retracts preserve dualizability. The projection kernels have the pullback and proper-direct-image adjunctions of the smooth proper flag projection.

The character formula is multiplicative under convolution: the coevaluation for a product is the composite of its two coevaluations, and the cyclic and interchange identities move the two factors successively. Applying the triangle identities then gives the product of their characters on cohomology. This uses only the homotopy hexagon and tensor identities specified in Theorem 5. For additivity, fix the central label \(L\) and apply its chosen natural equivalence of exact enhanced functors to the cofiber diagram of the test kernels. This supplies a morphism of that cofiber diagram, not merely unrelated maps on its three objects. On a mapping-cone model the interchange is triangular with respect to the two summands, while coevaluation and evaluation pair equal summands. The two diagonal contributions are the character of the second term and the negative of the character of the first term. This also explains why shifts give their usual alternating signs.

For the quadratic relation use the unshifted closed kernel \(C_s=\mathbf k_{P_s/I}\), where \(P_s/I\simeq\mathbf P^1\). Convolution through the partial level gives \[C_s\star C_s\simeq R\Gamma(\mathbf P^1,\mathbf k)\otimes C_s.\] The open–closed triangle gives \(\chi_{C_s}=H_s+1\). The Weil trace of \(R\Gamma(\mathbf P^1,\mathbf k)\) at \(x\) is \(1+q_x\). Thus \((H_s+1)^2=(1+q_x)(H_s+1)\), the asserted relation. Length-additive convolution of open cells is an isomorphism, giving the length and braid relations. For a degree-\(d_x\) place, all kernels are external products on its geometric legs with cyclic Frobenius. The trace of a graded cycle on \(C^{\otimes d_x}\) is the trace of the composite Frobenius on \(C\); hence the parameter is \(q^{d_x}\). The same identity shows that a shifted simple or translation kernel contributes its single-factor sign \((-1)^{\ell(t^a)}\), not a spurious sign depending on the number of geometric legs. This proves Equation (9). Length parity is a character of the translation lattice, so multiplicativity is preserved.

At \(L=\mathbf1\) the path stack is the Frobenius-fixed bundle groupoid. On it, the unit, counit, and Weil structure in Equation (8) sum the stalk traces over the Hecke correspondence, with its stack multiplicities. This is the ordinary action on compactly supported functions on the rational bundle groupoid. The construction for other central labels uses the same diagrams and their central interchange, so the verified relations apply there as well.

For general \(P\) repeat the closed-kernel computation with the full flag \(P/I\). Connected transition groups act trivially on its cohomology, so its cohomology is the constant Tate complex with Frobenius trace \(c_P=\sum_{w\in W_P}q_x^{\ell(w)}\). The composite of the two transfers at the \(P\)-level is multiplication by \(c_P\); the opposite composite is \(\sum_{w\in W_P}H_w\). Since \(c_P\ne0\), the two normalized transfers give the stated idempotent identification. Theorem 5 identifies the projected central kernel with Satake at a hyperspecial level. Trace additivity on its Wakimoto filtration then gives the usual Bernstein central character, which proves compatibility of the spherical action.

We spell out the assertion about parallel walls, which is needed for a geometric correspondence later. For a translation \(t^a\) parallel to the \(s\) wall, its character on the rank-one root subgroup has valuation zero. Thus the exact partial relative position identifies the rank-one refinement flag at the two ends. On the Frobenius path stack the allowed flags are the fixed flags of that semilinear identification. They form a finite etale family \(Z\to Y\) of degree \(q_x+1\). The closed kernel factors through the partial level, and its character factors through the two transfer maps. Those maps are the pullback and sum for \(Z\to Y\). Indeed, smooth proper base change reduces the unit/counit calculation to the diagonal of \(\mathbf P^1\) and the graph of its twisted Frobenius. Their intersections are transverse: Frobenius has zero tangent map and the diagonal contributes the identity. Every fixed flag therefore has coefficient one. Etale locally on \(Y\), where \(Z\) is a disjoint union of sections, the resulting maps are the diagonal map and the sum map between constant coefficient complexes. They agree on overlaps and hence descend. This calculation is unchanged after tensoring with the coefficient pulled back from the partial path stack, or after retaining automorphisms of the bundles. Subtracting the diagonal, by the open–closed triangle, gives the other-flags operator \(H_s\).

Finally, naturality of full rotation makes Frobenius commute with every Weil Hecke action. A slide at an auxiliary disjoint place commutes with this action by its common disjoint-leg diagram. Dividing the full slide by the other, invertible slides proves the same assertion for the selected-leg slide. Projection transfers use those same diagrams. The graded translation assertion follows from the map compatibility in Theorem 5. More explicitly, those filtered interchanges and rotation maps commute with the inclusion, quotient, and connecting maps of each successive filtration triangle. Applying cohomology gives the asserted equivariant long exact sequences. This is all the filtration compatibility needed here; it does not require an action on a derived completion of the Iwahori trace. ◻

Proof. Put \(\mathcal C=\operatorname{Mod}_S (D(\operatorname{Rep}_{\rm rat}(\widehat G)))\) and let \(\Phi_r\) be extension of scalars by \(\alpha\mapsto r\alpha\). The free objects \(S\otimes V\) are compact, dualizable generators: rational invariants of a reductive group in characteristic zero are exact and commute with direct sums (Milne 2017, Remark 15.13 and Proposition 15.16). Thus this category is the Ind-completion of its perfect objects.

We first calculate the universal trace for one factor. Let \(\mathcal C^e=\mathcal C\otimes\mathcal C^{\rm rev}\). Use the regular right action \(L\cdot(P,Q)=Q\otimes L\otimes P\) and the twisted left action \((P,Q)\cdot M=\Phi_r(P)\otimes M\otimes Q\). The pairing \((L,M)\mapsto\mathsf K(M\otimes L)\) is coherently balanced: the balancing map is precisely \[\mathsf K(M\otimes Q\otimes L\otimes P) \xrightarrow{\rho} \mathsf K(\Phi_r(P)\otimes M\otimes Q\otimes L).\] Proposition 7 verifies all the bar identities. Consequently \(\mathsf K\) extends to a continuous functional on the relative tensor product of these two module categories.

Here is an explicit algebra calculation of that relative tensor product. It is the derived intersection of the diagonal and the graph \((\Phi_r,\operatorname{id})\) in the product presentation of \([\operatorname{Spec}S/\widehat G]\). This language abbreviates a module-algebra computation: (Ben-Zvi et al. 2010, Proposition 4.1) identifies the tensor product of module categories with modules over the derived tensor product of their defining algebra objects. It applies in the stable presentable category of equivariant modules, without a connectivity condition.

An object of this intersection has a diagonal coordinate \(x\), a graph coordinate \(y\), and two group identifications \[a:x\longrightarrow r y, \qquad b:x\longrightarrow y.\] Resolve the equations \(\operatorname{Ad}(a)x-r y=0\) and \(\operatorname{Ad}(b)x-y=0\) by their degree-one Koszul generators. Their algebra, before group descent, is the polynomial algebra on \(x,y\) and the two group variables with those Koszul generators. These resolutions are \(K\)-flat: their exterior-degree filtrations have free graded polynomial quotients. Eliminate \(y\) and its Koszul generator using the second equation. This is an invertible linear substitution followed by removal of contractible Koszul pairs. Trivialize \(b\) by the product group action, put \(e=y\) and \(s=a b^{-1}\), and transport the remaining generator by the same change of basis. The residual group is the diagonal \(\widehat G\), its action is conjugation, and the remaining differential is \[d\eta_\alpha= \langle\alpha,\operatorname{Ad}(s)e-r e\rangle.\] This is exactly \(B_r\). In particular the specified null-homotopies of the equation have been retained.

The universal trace sends a label \(P\) to \(B_r\otimes_S P\). The comparison from the second-coordinate representation to the first-coordinate representation in the preceding diagram is \(a b^{-1}:y\to r y\). It is the last-to-first slide of Equation (7). Hence its action on a representation is \(V(s)\), with the asserted orientation. Coevaluation and evaluation then give \(\operatorname{Tr}_V(s)\) for a rigid character transfer.

For a cyclic degree-\(d\) leg, retain \(d\) coordinates and the \(d\) twisted identifications. Trivializing all but one of the group variables eliminates the intermediate coordinates and their Koszul pairs. The remaining transport is their ordered product \(s\), and its equation is \(\operatorname{Ad}(s)e=q^d e\). Carrying a representation label around this full cycle acts by \(V(s)\). For an external tensor label every factor has this same full transport; a single partial turn also performs the appropriate graded cyclic permutation. This explains both the \(d\)-fold slide and the map from the original \(S^{\otimes d}\) to the loop algebra in Equation (10).

It remains to justify that the continuous functional is pairing with a module, including when it is not coherent. Let \(\mathcal T=\operatorname{Mod}_{B_r} (D(\operatorname{Rep}_{\rm rat}(\widehat G)))\) and write the functional as \(L:\mathcal T\to D(\mathbf k)\). Again \(B_r\otimes V\) are compact dualizable generators. The exact contravariant functor on \(\mathcal T^c\) given by \(P\mapsto L(P^\vee)\) is represented by an object \(\mathcal A\in\operatorname{Ind}(\mathcal T^c) =\mathcal T\). Tensor duality gives, for compact \(M\), \[L(M)\simeq \operatorname{Hom}_{\mathcal T}(M^\vee,\mathcal A) \simeq(\mathcal A\otimes_{B_r}M)^{\widehat G}.\] Both sides preserve colimits, proving the formula for all \(M\), and hence Equation (10).

The character calculation is natural in labels and their evaluation and coevaluation maps. The projection transfer of Proposition 8 is therefore the same character operation on cohomology in this model. Finally, for a linear degree-two morphism \(u:P\to Q[2]\), cyclic naturality and Theorem 4 give \[\sigma_Q[2]\,\mathsf K(u) =q^d\,\mathsf K(u)\,\sigma_P,\] where \(\sigma\) is the full selected-leg slide. This proves the stated change in eigenvalue. The degree is even, so no additional Koszul sign occurs. ◻

Proof. We recall precisely the principal-bundle input. In arbitrary characteristic the canonical parabolic reduction exists and is unique; its Levi bundle is semistable. Semistable Levi bundles in a fixed component form a quasicompact stack. The reduced Harder–Narasimhan strata are of finite type, and the stack of canonical reductions maps finitely and radicially onto its stratum. These are the assertions of (Schieder 2015, sec. 2.3.1, Theorem 2.3.3(a),(c), and § 2.4.1). We use the finite radicial description, not an assertion that this map is an isomorphism in every characteristic.

For each of the finitely many standard Levis \(M\), central cocharacters cover the free degree lattice of \(M\) up to finite index; see (Schieder 2015, sec. 2.1.3). After tensoring by central bundles, its semistable bundles consequently belong to finitely many bounded families. Degree-zero central bundles are included in these families, since the corresponding Picard stacks are of finite type. Fix all the representations of these Levis that occur in the root filtrations and in highest-weight realizations of the finitely many maximal flag varieties. On the bounded families, the maximal and minimal slopes of these associated vector bundles are bounded. Restoring a central translation changes each central weight quotient by its prescribed scalar degree. It follows that a quotient of central weight \(\beta\) has all slopes in \[ [\langle\beta,\nu\rangle-C_0, \langle\beta,\nu\rangle+C_0], \tag{13}\] where \(\nu\) is the canonical type and \(C_0\) is independent of \(\nu\). There are only finitely many representations here, so one constant suffices. This argument uses boundedness of families; it does not use preservation of semistability by a representation in positive characteristic.

Choose a fixed large number \(A>C_0+2g_X-2\), enlarged for the finitely many weight filtrations if necessary. Coarsen the canonical parabolic by retaining exactly the simple gaps greater than \(A\). Write this parabolic as \(P=LU\). In its induced Levi bundle the remaining semisimple gaps are bounded by \(A\). It therefore belongs, modulo central translation in \(L\), to a bounded family: its canonical Levi is one of the semistable Levis considered above, and its remaining degree gaps range over a finite set of rational lattice points.

Use the relative-root height filtration of \(U\), whose successive quotients are vector groups with linear \(L\)-action. This parabolic filtration and its commutator inclusions hold in every characteristic; see (Conrad 2014, Theorem 5.4.3). Filtering them further by the weights of the original canonical Levi, every resulting root has a positive coefficient on a retained gap. Equation (13) therefore gives minimal slope greater than \(2g_X-2\). Their \(H^1\) vanishes by Serre duality. The \(H^1\) of every successive vector-group quotient vanishes as well, and the nonabelian cohomology sequence, applied successively, shows that the \(P\)-bundle is induced from its \(L\)-bundle. The cutoff was chosen from the original semistable-Levi families before coarsening, so there is no circular choice of a bound depending on \(A\).

It follows that every bundle is induced from one of finitely many bounded families for a Levi, followed by a central translation in that Levi. The translations required in an HN ball are lattice points in a ball of radius \(O(1+b)\) and are polynomially many. These translations can be chosen over the fixed finite field after passing to a fixed finite-index degree lattice: choose a divisor of positive degree over that field and use its line bundle. The remaining degree cosets are absorbed into the bounded families. The bounded principal-bundle families have finitely many rational classes. To see this directly, fix a faithful representation and a twist for which all the associated vector bundles are globally generated and have vanishing \(H^1\). A basis of global sections rigidifies each bundle into one of finitely many finite-type framed algebraic spaces. Every rational bundle admits such a basis over the finite field, and each framed space has finitely many rational points. The canonical reduction of a rational bundle descends by uniqueness, and the vanishing of \(H^1(U)\) above holds over the finite field itself. Thus every rational class is obtained from the same finite list of rational Levi families. This proves (a).

Choose finite-type scheme atlases for these bounded families, with dimensions bounded by \(C\). Each central translate of each atlas maps to the bundle stack. A geometric fiber of such a map over an object with automorphism dimension \(r\) has dimension at least \(r\): it parametrizes isomorphisms from a family member to that object. The dimension formula bounds the image stratum by \(C-r\). Within each fixed HN stratum only finitely many translating choices occur, and the HN stratification is locally finite. Refining the images and the stabilizer dimensions thus proves (b) with a locally finite stratification. Universal homeomorphisms in the canonical-reduction description do not change these dimensions or the constructible-sheaf calculation.

For (c), pass to a subsequence on which a simple gap tends to infinity relative to \(1+b_n\), and fix its maximal parabolic \(P_i\). Choose an actual \(G\)-equivariant projective embedding \(G/P_i\hookrightarrow\mathbb P(V_i)\) using a sufficiently ample linearized line bundle with weight a divisible multiple of the fundamental weight. In particular the scheme-theoretic stabilizer of the base-point line is \(P_i\); no assertion about the stabilizer of an arbitrary simple highest-weight module in small characteristic is required. The line defining the coarsened canonical reduction is the unique highest-weight line. Every other weight differs from this weight by a positive combination of simple roots involving the selected root. Equation (13) shows that the degree of this line exceeds every slope of the quotient by the selected gap minus a fixed constant. A modification of size \(b_n\) in this fixed representation has poles bounded by a divisor of degree \(O(b_n)\). The resulting homomorphism from the highest line to the target quotient therefore vanishes when the gap is sufficiently large. The meromorphic isomorphism preserves the reduction line and hence its maximal-parabolic reduction. For point Frobenius, the target canonical type and degrees are the same: point Frobenius acts on the parameter field, not by Frobenius pullback along \(X\). Uniqueness of the canonical reduction gives the descent and iteration assertions.

Finally, a full level structure has a space of choices of fixed finite type and fixed dimension; over the finite field its possible trivializations form a torsor under the fixed finite group \(G(\mathcal O_D)\). Parahoric flags have the same boundedness property. They change all the preceding constants by fixed amounts. No argument uses reducedness of \(D\). ◻

Proof. We use middle perversity with smooth descent, as in (Laszlo and Olsson 2009, Lemma 4.1 and Theorems 4.2, 5.1), so that \(\mathbf k[-r]\) is perverse on the classifying stack of a smooth group of dimension \(r\). Stratify further by stabilizer dimension and by the isomorphism relation. Over a pair of isomorphic bundles with stabilizer dimension \(r\), the stalk of \(\Delta_!\mathbf k\) is the compact cohomology of their isomorphism scheme and has upper degree \(2r\). The corresponding locus of isomorphic pairs has dimension at most \(C-2r\), by Proposition 10(b). The support criterion for the upper perverse truncation gives a bound independent of \(r\).

At a point of the Frobenius graph the stabilizer dimension in \(\mathcal B\times\mathcal B\) is \(2r\). Every stratum through it has dimension at least \(-2r\). Thus the stalk upper bound of \(\mathcal F\) is at most \(N+2r\). On the graph the relevant stratum has dimension at most \(C-r\). Compact cohomology of a sheaf on a finite-type algebraic stack of dimension \(a\) vanishes in degrees above \(2a\); see (Sun 2012, Theorem 1.4). The resulting bound is \[(N+2r)+2(C-r)=N+2C.\] Stratification and the spectral sequence for cohomology sheaves give the same estimate for complexes. Apply the calculation on finite-type opens and then take the defining filtered colimit. The bounds are independent of the open, so they persist in that colimit. The central-degree convention is treated componentwise. ◻

Proof. Compute the trace by applying the Hecke transforms to one factor of \(\Delta_!\mathbf k\) and then applying Lemma 11. Every fixed bounded kernel has finite upper perverse amplitude. A costandard kernel has a bound independent of its length, as follows.

Let \(\mathcal H_w\) be its exact-position Hecke stratum, with source and target maps \(p,q\). The map \(p\) is smooth of relative dimension \(\ell(w)\) and \(q\) is affine; these facts are checked after smooth local trivialization, where their fibers are the affine Schubert cell. In the closure correspondence the target map is proper. Here is the coefficient comparison for an arbitrary locally bounded constructible input. Choose a sufficiently deep jet subgroup acting trivially on the chosen Schubert closure, and pass to its smooth torsor of trivializations. The open and closed Hecke spaces then become products over the source. For bounded constructible complexes the formula \[R(\operatorname{id}\times j)_*(A\boxtimes B) \simeq A\boxtimes Rj_*B\] follows by Verdier duality from the corresponding extension-by-zero formula. It does not require \(A\) to be lisse. Thus smooth base change (Laszlo and Olsson 2008, sec. 12.1) and descent identify the costandard coefficient with \(Rj_*p^*(-)[\ell(w)]\). Proper pushforward from the closure identifies the costandard transform with \[Rq_*p^*(-)[\ell(w)].\] Smooth pullback with this shift preserves perversity, and affine Artin vanishing says that \(Rq_*\) preserves an upper perverse bound; see (Beilinson 1987, sec. 3.2, paragraph preceding Theorem 3.2). Both assertions descend from the smooth scheme charts, since the maps of the correspondence are representable. The resulting upper bound is independent of \(w\).

Passage between the fixed parahoric levels costs only a fixed amplitude, because their fibers are fixed flag varieties. Forgetting and then right-averaging torus equivariance has a representable torus-torsor fiber. Its direct image has a finite Postnikov filtration with constant cohomology pieces \(H^i(T,\mathbf k)\): translations act trivially on torus cohomology. This is cohomology of \(T\), not of \(BT\), and no splitting of the filtration is needed. It adds only fixed shifts. This is the averaging convention in Proposition 6. By the same proposition, nef Springer line bundles correspond to costandard Wakimoto translations, with shifts independent of their weights. A bounded additional translation can be composed with a nef translation at a fixed cost. Convolution of a fixed number of these transforms preserves the uniform upper bound just proved. The spherical polynomial action and the identification of their products with derived tensor products are those of Theorem 4 and Proposition 6. ◻

Proof. By Proposition 9, the holonomy satisfies the derived linear equation \(\operatorname{Ad}(s)e=Qe\). The fiber of the adjoint quotient over \([t]\) consists of elements conjugate to \(tu\), with \(u\) unipotent in \(H\). Indeed the adjoint quotient is the Weyl quotient of a maximal torus (Lee 2015, Theorem 4.1(2)), and representation characters are unchanged by the commuting unipotent factor in Jordan decomposition. Here is an algebraic slice description, with no completion of the coefficient module. Let \(W\) be the sum of the nonresonant \(\operatorname{Ad}(t)\)-eigenspaces in \(\widehat{\mathfrak g}\). Choose an \(H\)-stable affine open \(U\subset H^\circ\) containing the unipotent locus on which \(1-\operatorname{Ad}(tu)\) on \(\widehat{\mathfrak g}/\operatorname{Lie}H\) and \(\operatorname{Ad}(tu)-Q\) on \(W\) are invertible. Both determinants are nonzero at every unipotent \(u\). There are only finitely many \(H\)-classes of semisimple \(u\) for which \(tu\) is conjugate to \(t\), since their torus representatives come from the finite Weyl orbit of \(t\). Invariant functions allow us also to remove all these classes except \(u=1\). The conjugation map \[\widehat G\mathbin{\times}^{H}(tU)\longrightarrow\widehat G\] is etale: its transverse differential is the first displayed invertible operator. After the removal of the other sheets, uniqueness of the Jordan semisimple part identifies every conjugating ambiguity over the class fiber with \(H\). The restriction to the schematic class fiber is therefore etale and universally bijective, hence an isomorphism. Being etale, it is an isomorphism over every infinitesimal thickening of that fiber as well. Flat pullback and descent therefore identify the supported quasi-coherent complexes on these neighborhoods, equivariantly. This assertion applies to arbitrary complexes: on affine charts it holds for modules killed by each power of the fiber ideal, hence for their filtered unions; flatness computes pullback on cohomology in every degree. The finite Koszul specialization has cohomology annihilated by the fiber ideal, so it is covered by this argument. In particular, all equivariance remains ordinary rational \(H\)-equivariance.

Over \(\mathcal O(U)\), the invertible nonresonant block of \(\operatorname{Ad}(tu)-Q\) gives a change of odd Koszul generators for which their differentials are precisely the coordinates of \(W\). Eliminating these contractible pairs is an actual dg-algebra quasi-isomorphism over \(\operatorname{Sym}(E_t^*[-2])\). Solve successively for the transported variables at the other geometric factors in the same way. Push forward the unipotent and additional derived variables along their affine projection. They remain in \(M\), including all invariant Koszul generators and the residual equation \(\operatorname{Ad}(u)e=e\). The polynomial-action assertion follows by solving the equations starting at the chosen geometric factor. The description is \(H\)-equivariant.

Tensoring with the finite Koszul complex changes upper bounds by only a fixed amount. For a fixed finite-dimensional representation of a reductive subgroup of \(\widehat G\), every irreducible summand occurs in a restriction from \(\widehat G\). One proof uses the surjection \(\mathbf k[\widehat G]\to\mathbf k[H]\), takes a finite-dimensional \(\widehat G\)-subrepresentation containing lifts of its matrix coefficients, and then uses complete reducibility. Thus all fixed \(H\)-tests are available. Induction from \(H^\circ\) to \(H\) is finite, so it also allows \(H^\circ\)-invariants and fixed \(H^\circ\)-tests. These retracts are taken after specialization on the slice; their maps need not extend equivariantly before specialization.

Here is the perfect-generation detail for the flag factors. On the smooth projective flag variety choose an ample equivariant line bundle \(A\). Equivariant Serre evaluation gives surjections from finite sums of \(V\otimes A^{-n}\) onto any equivariant coherent sheaf. For an explicit finiteness argument, let the flag variety have dimension \(a\). More than \(2a\) successive evaluations give an extension whose class vanishes, since the local-to-global Ext spectral sequence and exact reductive invariants give \(\operatorname{Ext}^{n}_{H^\circ}(F,K)=0\) for coherent \(F,K\) and \(n>2a\). The truncated evaluation complex therefore contains \(F\) as a retract. Applying finite cohomology filtrations shows that every perfect complex is in the finite thick closure of these bundles. Apply this statement to its dual and dualize: the same perfect complex is in the finite thick closure of bundles \(V\otimes A^n\), \(n\geq0\). All representations, twists, cones, and shifts in this construction are fixed once the perfect complex is fixed. Tensoring a variable nef line with these positive twists keeps it nef. The structure sheaf of a fixed closed smooth \(H^\circ\)-flag orbit is perfect on the ambient flag variety, so the assertion applies to its restriction test. This proves the last claim with constants independent of the variable nef weights. ◻

Proof. Choose Borels containing a reference torus with \(t,\gamma\). First order their roots by \(\gamma\) and \(-\gamma\), respectively; within \(L\), order them positively on \(\lambda\) and complete the order arbitrarily on the zero-weight roots. Their \(H^\circ\)-orbits are complete flag varieties. Use two Springer tests restricted to these orbits, as permitted by Lemma 13.

Let \(P_\gamma,P_{-\gamma}\) be the two opposite parabolics. Choose an integral character \(\chi\) strictly dominant for the first partial flag variety and zero on the coroots of \(L\). The extremal pairing in mutually dual representations gives an invariant section of the product of two nef line bundles on the partial flags. Project the selected complete-flag orbits to these partial-flag varieties. On the product of the two resulting \(H^\circ\)-orbits, the section’s nonvanishing locus is the locus of opposite partial flags, which is \(H^\circ/H_L\). Indeed the ambient big-cell Gauss factorization is unique and is preserved by conjugation by \(t\). If an element of \(H^\circ\) admits that factorization, its unipotent and Levi factors centralize \(t\) as well and lie in the corresponding parabolics of \(H^\circ\). Thus ambient oppositeness restricts to exactly the opposition locus for \(H^\circ\). Localizing by that section is the filtered colimit of simultaneous positive twists by the two nef lines. The section and its localization maps have degree zero.

The pairs of complete flags form a fibration over this opposition locus. Over the reference opposite partial-flag pair, its fiber is the product of two complete flag varieties for \(H_L\). The resonant parts of the two ambient nilradicals are \[E_{\gamma\geq0}\quad\text{and}\quad E_{\gamma\leq0}.\] Indeed the zero \(\gamma\)-part \(E_L\) has positive \(\lambda\)-weight and belongs to both nilradicals, independently of the remaining flags; the roots of \(H_L\) have \(\lambda\)-weight zero. The displayed subspaces span \(E_t\) and intersect in \(E_L\), so \[\mathbf k[E_{\gamma\geq0}] \otimes^{\mathbf L}_{\mathbf k[E_t]} \mathbf k[E_{\gamma\leq0}] \simeq \mathbf k[E_L].\] Thus their derived intersection over \(E_t\) has no excess.

For completeness, the other torus eigenspaces do not obstruct splitting off this intersection. Decompose the ambient Lie algebra as \(E_t\oplus W\) by \(t\)-weights. The projectors are polynomials in \(\operatorname{Ad}(t)\), so this also splits the universal nilradical subbundles globally and equivariantly. Its Koszul base change to \(E_t\) has zero differential on the factors from \(W\), so these factors are finite exterior algebras of equivariant vector bundles. Their exterior-degree-zero summand is canonical; no splitting of the varying nonresonant subbundle inside \(W\) is required. Taking this summand in both tests leaves exactly \(\mathbf k[E_L]\) on the opposite-pair slice. Tensoring with \(M\) therefore gives \(M_L\). The remaining complete flags contribute \(R\Gamma(\mathcal O)=\mathbf k\). By equivariant descent from \(H^\circ/H_L\), the resulting invariant test is the expression in Equation (14).

We check the uniformity in \(\eta\). Give the first partial-flag line fiber character \(\eta\). After the \(n\)th localization twist the two fiber characters are, with the opposite-Borel conventions, \[\eta-n\chi,\qquad n\chi.\] Both lines are nef for every sufficiently large \(n\), since \(\chi\) is strictly positive on precisely the omitted simple coroots and \(\eta\) vanishes on the Levi coroots. The starting value of \(n\) may depend on \(\eta\). It changes neither the localization colimit nor the bound: every term in this cofinal tail is in the same uniformly bounded nef family of Proposition 12. The fixed resolutions in Lemma 13 introduce only fixed additional positive twists and fixed shifts. The character \(\eta\) and the localization characters are trivial on the derived subgroup of \(L\), so the residual complete flags still push their structure sheaves to constants. If \(L=\widehat G\), the character line is underlying-trivial and already nef; this also covers the case with no partial flags.

A fixed \(U\) is obtained from a fixed representation test by the restriction and summand argument in Lemma 13. Finally, global sections on these fixed flag varieties have finite cohomological dimension, reductive invariants are exact, and filtered colimits of vector spaces are exact. All the operations and the direct-summand projection above are therefore valid for arbitrary dg \(M\). They preserve a uniform upper bound, proving the proposition. ◻

Proof. Choose a faithful algebraic representation \(\rho:L\hookrightarrow\operatorname{GL}(V)\). The relation \(\operatorname{Ad}(t)e=Qe\) says that \(d\rho(e)\) is conjugate to \(Q\,d\rho(e)\). Its finite multiset of eigenvalues is therefore preserved by multiplication by \(Q\). Since \(Q>1\), every eigenvalue is zero. Thus \(d\rho(e)\), and hence \(e\), is nilpotent. Start with the Jacobson–Morozov theorem in characteristic zero (Bertoloni Meli et al. 2024, Theorems 3.1 and 3.2). On the torsor of triples with first member \(e\), let \((t,Q)\) act by \[(e,h,f)\longmapsto (Q^{-1}\operatorname{Ad}(t)e,\operatorname{Ad}(t)h, Q\operatorname{Ad}(t)f).\] The closure of the subgroup generated by \((t,Q)\) is diagonalizable. The torsor is under the unipotent radical of the centralizer of \(e\) (Bertoloni Meli et al. 2024, Proposition 3.3); successive vector-group quotients have vanishing first cohomology for a diagonalizable group in characteristic zero. It therefore has a fixed point, which is the required compatible triple.

Put \(c=tQ^{-h_e/2}\). It commutes with the triple. Decompose \(\operatorname{Lie}L\) into simultaneous \(c\)-eigenspaces and irreducible \(\mathfrak{sl}_2\)-modules \(V_n\). A vector of \(h_e\)-weight \(k\) belongs to \(E_L\) precisely when its \(c\)-eigenvalue is \(Q^{1-k/2}\). Its predecessor of weight \(k-2\), when present, has \(t\)-eigenvalue one and lies in \(\operatorname{Lie}H_L\). The raising map is onto this vector unless \(k=-n\) is the lowest weight. Hence every normal weight is of the form \(-n\leq0\).

For finiteness, conjugate \(h_e\) into the fixed torus by \(H_L\). Its central projection in \(L\) is zero, because the triple comes from \(\mathrm{SL}_2\). Its root pairings are integral and bounded in absolute value by \(\dim\operatorname{Lie}L\): they are weights of an \(\mathfrak{sl}_2\)-representation of that dimension. There are consequently only finitely many possibilities in the fixed cocharacter lattice. Fix one such \(h\). On the \(c=1\) part of the same calculation the raising map gives a surjection \[\operatorname{Lie}Z_{H_L}(h) \longrightarrow E_L\cap(\operatorname{Lie}L)_2(h).\] Thus every \(e\) admitting this \(h\) has an open \(Z_{H_L}(h)\)-orbit in that vector space. An irreducible vector space has at most one open orbit. The finite list of \(h\)’s therefore gives finitely many \(H_L\)-orbits. ◻

Proof. Deshear as above. In an invariant open subset where the orbit is closed it is a smooth regular immersion into the smooth space \(E_L\). If \(\mathcal I\) is its ideal, local cohomology is \[R\Gamma_{H_Le}(M_L) =\operatorname*{colim}_{p} R\mathcal Hom(\mathcal O/\mathcal I^p,M_L).\] Each quotient is perfect on the regular ambient space. Its finite filtration has graded pieces the symmetric powers of the conormal bundle. Regular-immersion duality therefore expresses the corresponding support terms as \[i_*\bigl(Li^*M_L\otimes\det N\otimes \operatorname{Sym}^{a}N\bigr)[-c], \qquad c=\operatorname{codim}(H_Le),\] where \(M_L\) is read in the desheared grading and \(i\) denotes the regular immersion. The shift \([-c]\) raises degrees by \(c\); the normal symmetric powers, rather than the conormal ones, are the factors relevant to the upper bound.

The action of \(\rho_e\) on a normal vector of \(h_e\)-weight \(k\) has weight \(k-2\). By Lemma 15 this is at most \(-2\). Consequently arbitrary normal powers cannot increase the corrected upper bound. The determinant and the codimension shift cost a constant depending only on the orbit. This establishes a bound uniform in \(p\) before passage to the filtered colimit.

Equivariant sections on the orbit are rational group cohomology of \(S_e\). Its unipotent radical is computed by the finite exterior Lie-algebra cochain complex, and invariants for a reductive Levi are exact. The triple centralizer is a Levi in the centralizer of \(e\) (Bertoloni Meli et al. 2024, Proposition 3.3); taking the fixed points of the compatible diagonalizable action gives the same decomposition for \(S_e\). The cocharacter \(\rho_e\) lies in, and is central in, this triple-centralizer Levi. If \(\mathfrak u\) is the Lie algebra of the unipotent radical, a possible additional cost is \[\max_{0\leq a\leq\dim\mathfrak u} \left(a+\max\operatorname{wt}_{\rho_e} \bigwedge^a\mathfrak u^*\right).\] The exterior complex has finitely many columns, so it is valid for unbounded coefficients without an infinite-totalization issue. Finite component invariants are exact in characteristic zero. The test \(V\) is desheared too: its \(a\)-weight-\(\alpha\) space is placed in numerical degree \(\alpha\). Its numerical degree plus its \(\rho_e\)-weight is therefore its \(h_e\)-weight. Taking invariants of the reductive triple-centralizer Levi forces total \(\rho_e\)-weight zero. On the original \(H_L\)-invariants the total \(a\)-weight is zero as well, so the resulting numerical degree is the original cohomological degree. This gives Equation (16).

No finite-generation hypothesis has been used. Tensoring with the perfect quotients and their finite filtrations is legitimate for unbounded dg modules. The operation displayed above is a filtered colimit, not an inverse limit. Exactness of filtered colimits preserves the upper bound uniform in \(p\). Finally there are finitely many orbits, so their support triangles give a finite devissage. ◻

Proof. Induct on the semisimple rank of \(L\), using Proposition 14 for all its smaller allowed Levis. If the centralizer of the triple in \(H_L\) has a torus noncentral in \(L\), choose a cocharacter in that torus noncentral in \(L\). Its centralizer is a proper Levi \(L'\) of \(L\) containing the triple. It is still allowed: combine this cocharacter with \(\gamma\), using a sufficiently large multiple to specify the same successive centralizers. Derived restriction is transitive, so the derived fiber obtained from \(M_{L'}\) is the same as that from \(M_L\). Induction proves the assertion for this orbit.

Consider an orbit for which no such torus exists. The direction \(\lambda-h_e/2\) centralizes the triple, so it is central in \(L\). Since \(\lambda\) is central in \(H_L\), this implies that \(H_L\) centralizes \(h_e\). In the centralizer of \(h_e\), an element centralizing \(e\) also centralizes \(f\), by uniqueness of the compatible triple with \(e,h_e\) fixed. Thus \(S_e\) equals the triple centralizer in \(H_L\). It is reductive and finite modulo \(Z(L)^\circ\). Since \(\lambda-h_e/2\) is central in \(L\), every vector in \(E_L\) has \(h_e\)-weight two. Lemma 15 gives only nonpositive weights in the normal space, so that space is zero: \(H_Le\) is open in \(E_L\). The irreducible space \(E_L\) has only one open orbit.

Every boundary orbit has already been handled by the proper-Levi case. Apply Lemma 16 to those finitely many orbits with tests \(U\otimes\eta\). Every neutral cocharacter of a triple in \(L\) annihilates \(\eta\in X^*(L)\). Their boundary contributions therefore have upper bounds independent of \(\eta\). Together with Equation (14), the localization triangle shows that the open orbit alone satisfies the same type of uniform bound for each fixed \(U\).

We explain why these invariant bounds detect the entire corrected fiber. The group \(S_e\) is reductive, possibly disconnected, and contains the central torus \(Z(L)^\circ\) with finite quotient. Restrictions of characters of \(L\) form a finite-index sublattice of \(X^*(Z(L)^\circ)\). Modulo tensoring by these restrictions, there are finitely many irreducible rational representations of \(S_e\): there are finitely many central-character cosets, and for each fixed central character the irreducibles are representations of a finite-dimensional twisted group algebra of the finite group \(S_e/Z(L)^\circ\). Choose finitely many fixed \(H_L\)-representations \(U_1,\ldots,U_r\) whose restrictions detect the duals of representatives of these classes. Such representations exist by the matrix-coefficient and complete-reducibility argument used above.

Arbitrary rational representations of a reductive group in characteristic zero are direct sums of finite-dimensional irreducibles (Milne 2017, Remark 12.53(a), Theorem 22.42, and Corollary 22.43). Invariants are exact, so every nonzero cohomology class of the corrected fiber is detected by one of the tests \(U_j\otimes\eta\). The correction on that test has only the \(h_e\)-weights of \(U_j\), since \(h_e\) annihilates \(\eta\). More explicitly, an irreducible constituent of \(H^j(W_e)\) of \(\rho_e\)-weight \(b\) has a nonzero invariant pairing with one of these tests. If the chosen test component has \(h_e\)-weight \(a\), that pairing has ordinary degree \(j+b+a\): the total \(\rho_e\)-weight is zero, and the test’s numerical degree plus its \(\rho_e\)-weight is its \(h_e\)-weight. Hence \[j+b\leq C'(U_j)-\min\operatorname{wt}_{h_e}(U_j).\] The maximum over the finite list gives the bound in Equation (15). This argument includes rank zero, where \(E_L=0\) and all representation types are characters. It starts the induction and completes the proof. ◻

Proof. In the loop description, the representation label becomes \(V_\mu^{\otimes d}\) and the \(d\)-fold slide acts by its holonomy; this is the slide convention of Proposition 9. Among the weights of \(V_\mu\), pairing with \(\lambda\) is uniquely maximal at \(\mu\). Indeed a weight lies in the convex hull of the Weyl orbit of \(\mu\), and the linear functional defined by \(\mu\) attains its maximum on that orbit only at \(\mu\) itself. This argument also applies on chamber walls. Thus the extreme part is the line of character \(d\mu\). It is \(H\)-stable; its character is trivial on the derived subgroup of \(H\), and the unipotent part of holonomy acts trivially on it. Nilpotent thickenings in the Koszul specialization do not change this semisimple eigenvalue separation.

Apply Proposition 17 and Lemma 16 with this one-dimensional representation. There are finitely many orbits by Lemma 15. The maximum of their fixed costs is a single constant \(C\), and the representation contribution on the orbit of \(e\) is \(d\langle\mu,h_e\rangle\). This proves Equation (17).

Compatibility of the triple implies that \(\lambda-h_e/2\) centralizes it. The invariant metric extends, on the derived Lie algebra, to an invariant bilinear form; therefore \[(h_e,\lambda-h_e/2) =([e,f],\lambda-h_e/2)=0.\] Consequently \[ 2d\langle\lambda,\mu\rangle -d\langle\mu,h_e\rangle =2dm\|\lambda-h_e/2\|^2. \tag{18}\] All directions here are projected away from the connected center. If the right side vanishes, the semisimple element \(tQ^{-h_e/2}\) centralizes the triple and has absolute values one in the remaining directions. It is conjugate into a maximal compact subgroup after removing the absolute-value central part (Adams and Taibi 2016, Theorems 3.6 and 3.7). This is exactly the required compact-times-Jacobson–Morozov form. Conversely that form gives equality for its triple. If equality occurs for no orbit, the finite list of squared norms in Equation (18) has a strictly positive minimum. This is the asserted uniform linear gap.

Finally, replacing \(t\) by \(tz\) leaves \(H\), \(E_t\), their orbits, the allowed Levis, and every flag test unchanged. It also leaves the semisimple direction \(\lambda\), the rational deshearing, and all corrected-fiber and representation-test gradings unchanged, even when \(\log_Q|z|\) is irrational. The only change in the finite invariant Koszul test is in the scalar values subtracted from a fixed list of invariant generators. Its length, shifts, and the preceding geometric bounds are unchanged. All constants in the slice, flag, support, and finite-representation arguments can therefore be chosen uniformly on this central family. The label \(\mu\) is central-trivial, so its extreme line is unchanged as well. For finitely many hyperspecial variants take the maximum of their constants. Kernels at other places were fixed throughout Proposition 12 and do not affect this uniformity. ◻

Proof. Suppose the instability in \(Y_u\) were unbounded. Proposition 10 would give a maximal-parabolic canonical reduction preserved by \(\phi\), where preservation means that the reduction is carried to its Frobenius image. Work first at a geometric point with all its data defined over \(\mathbb F_{q^b}\), enlarging \(b\) to be divisible by every \(d_j\). Iterating \(\phi\) \(b\) times gives an ordinary meromorphic automorphism \(\gamma\) of the bundle. Equation (22) puts all its powers, at every geometric point above \(x_j\), in the Iwahori cells of translations \(mbu_j\).

Apply the preceding Newton calculation to the preserved reduction. The valuation of its fundamental character at one geometric leg is \(b\langle\varpi,w_ju_j\rangle\) for some \(w_j\). The original Frobenius path intertwines the consecutive local automorphisms and their reductions. Since the character of the split parabolic is also preserved, its valuation is the same at all \(d_j\) geometric points above \(x_j\). The product formula for the meromorphic character therefore gives \[b\sum_jd_j\langle\varpi,w_ju_j\rangle=0,\] contrary to Equation (19). Proposition 10 now bounds the instability, so boundedness of bundles and of the Hecke positions gives finite type. This argument includes wall labels, which merely allow more Weyl choices.

The shtuka condition makes inertia unramified. Infinitesimally an automorphism compatible with \(\phi\) equals its transported Frobenius pullback; the latter has zero differential, so its tangent vector is zero. Automorphism schemes of a bundle with bounded level are affine of finite type. Thus the inertia is quasi-finite, and \(Y_u\) is Deligne–Mumford.

For separation, let two objects over a discrete valuation ring \(R\) be isomorphic over its fraction field. Choose a faithful representation of \(G\). At the generic point of the special fiber of \(X_R\), both path maps are integral isomorphisms: the legs are horizontal. If \(m\) is the smallest valuation of a matrix entry of the generic isomorphism, compatibility with the paths gives \(m=qm\), since base Frobenius multiplies vertical valuations by \(q\) and multiplication by an integral invertible matrix preserves the smallest valuation. Hence \(m=0\). The same argument applies to the inverse. The isomorphism consequently extends at that vertical generic point. All horizontal codimension-one points are already covered by the generic-fiber bundle isomorphism. Normality of the smooth surface \(X_R\) and affineness of the isomorphism torsor extend it across codimension two. The path identities extend by density. So do the prescribed level identities: \(D_R\) is flat over \(R\), including when \(D\) is nonreduced. The same argument treats fixed parahoric flags. This proves the valuative criterion for the quasi-finite diagonal, which is therefore proper.

For completeness, finite-dimensional compact cohomology can be reduced to the corresponding assertion for algebraic spaces. On this bounded family, choose a sufficiently large effective divisor \(N\) disjoint from the legs and the existing level such that an automorphism equal to the identity on \(N\) is the identity. A faithful representation reduces this to the uniform vanishing of sections of a bounded family of endomorphism bundles twisted by \(-N\). Frobenius-compatible trivializations on \(N\) form a finite étale torsor under \[H_N=\bigl(\operatorname{Res}_{N/\mathbb F_q}(G_N)\bigr)(\mathbb F_q).\] Indeed the jet group \(\operatorname{Res}_{N/\mathbb F_q}(G_N)\) is smooth and connected, and its Lang map is finite étale and surjective. The resulting rigidified shtuka is a separated algebraic space of finite type with a finite-group quotient equal to \(Y_u\). Compact cohomology is therefore bounded and finite-dimensional; taking \(H_N\)-invariants is exact over \(\mathbf k\). The same reduction applies to weights. A finite-type algebraic space over a field admits a finite stratification by schemes, obtained by successively choosing a dense open subspace that is a scheme. Excision transfers the scheme weight bounds to this algebraic space, and exact finite-group invariants retain them. Thus the pure constant coefficient in Equation (20) has the usual integral-weight compact cohomology bounds, without any appeal to a weight theorem for arbitrary Artin stacks.

Finally, \(a=nu+v\) is regular positive, and for each fixed pairing in Equation (19) the corresponding pairing for \(a\) is a nonconstant affine function of \(n\). Thus \(a\) satisfies the same nonvanishing condition for all sufficiently large \(n\). A rational ordinary loop of type \(a\) with excessive instability has, by Proposition 10, a preserved maximal-parabolic reduction. The same character and product-formula argument excludes it. The quantitative bound is \(O(1+n)\). There are finitely many rational bundles in that range, and each admits only finitely many rational meromorphic automorphisms with these pole bounds: in a faithful representation they lie in a fixed finite-dimensional space of sections over a finite field. The automorphism group of every such rational loop has finitely many rational points. This proves the last assertion. ◻

Proof. We first work at one geometric leg. Take a reduced gallery for \(t^v\). After a prefix \(b\), cyclic rotation changes the shtuka label to the translation \(\tau=b^{-1}t^ub\). Since \(u\) and \(v\) lie in one closed chamber, \[\ell(t^ub)=\ell(t^u)+\ell(b).\] For the next simple reflection \(s\), one has \(\ell(\tau s)=\ell(\tau)+1\). A translation has opposite left and right length changes at a wall to which its vector is not parallel. Consequently either \(s\) is an initial letter of \(\tau\), or \(\tau\) is parallel to the \(s\)-wall and commutes with \(s\).

In the first case write \(\tau=sw\). Unique open subdivision identifies the step with the cyclic rotation from the path \(sw\) to the path \(ws\). It is an isomorphism on perfections. The cancellation of the inverse simple kernel against that initial factor in the categorical action is precisely the cyclic isomorphism of Proposition 7. Thus its unnormalized action is pullback along this geometric step.

For a closed leg of degree \(d\), perform this rotation on the whole block of \(d\) geometric factors. Cyclicity applied to \(\Delta_s^{\boxtimes d}\star\Delta_w^{\boxtimes d}\) changes the repeated label \(\tau=sw\) to the repeated label \(ws=s\tau s\); unique reduced subdivisions identify both ends with the stated path stacks.

In the parallel case, pass from the alcove to its \(s\)-panel. The partial relative position identifies the two rank-one flag varieties: on the root groups for that panel, translation conjugation has exponent zero. A refinement of the partial shtuka is therefore a flag fixed by the resulting Frobenius-semilinear identification. Over a degree-\(d\) closed leg, these refinements form a finite étale family of \(q^d+1\) flags. To check this over a general base, trivialize the rank-one torsor étale locally and use the Lang map for its connected rank-one group; the fixed flags then become \(\mathbb P^1(\mathbb F_{q^d})\). If \(\pi\) denotes the map forgetting this refinement, the closed simple correspondence acts by \(\pi^*\pi_!\), and the open simple correspondence acts by \[ \pi^*\pi_!-\operatorname{id}. \tag{24}\]

Here Equation (24) is also an equality with the categorical transfer, rather than only an equality of trace functions. Use the unshifted closed rank-one kernel, which factors through the panel projection. Its unit and counit give the diagonal and projection maps for the proper flag variety. After restriction to the twisted trace, the Frobenius graph meets the diagonal transversely: its linearized fixed-point equation has derivative \(-1\). Every fixed flag therefore has coefficient one in the unit/counit transfer. The projection formula identifies the resulting operator with sum and pullback over the finite family of fixed flags. This is the explicit rank-one calculation in Proposition 8. Subtracting the identity kernel by the standard open/closed triangle gives Equation (24). In particular no compact cohomology shift of the flag variety remains in this finite transfer.

At a closed leg the same calculation is made in the product of its \(d\) geometric flag varieties, with Frobenius cyclically permuting the factors. A fixed tuple is determined by one \(\mathbb F_{q^d}\)-flag. The same transverse intersection proves the coefficient-one assertion and gives the parameter \(q^d\). Graded cyclic permutation produces the single-factor parity in the pure standard kernel. The correction defining \(T_v\) removes that parity even when \(v\) is odd. Thus normalization gives the factor \(q^{-L_v/2}\) in Equation (23).

Length-zero gallery steps identify the relevant paths. Composing the two kinds of simple steps proves finiteness on perfections and the operator assertion. Unique subdivision also proves compatibility of composition and of commuting-translation shuffles. When \(v=u\), the two initial steps in the commuting square are subdivisions of the same path of position \(2u\) and hence coincide. The square is therefore \(h=\phi\), \(\mathcal E'=\mathrm{Fr}(\mathcal E)\), and \(\phi'=\mathrm{Fr}(\phi)\). Its action is \(\mathrm{Fr}^*\) on \(C(u)\); after the normalization in Equation (20) it is Frobenius on \(M(u)\). ◻

Proof. With the direction \(p_!q^*\) used above, the fixed-point identification is \(p=\mathrm{Fr}^nq\). Thus its data consist of a shtuka \((\mathcal E,\phi)\) and a closing step \(h:\mathrm{Fr}^n\mathcal E\to\mathcal E\) of type \(v\), compatible with \(\phi\). Concatenate \[\mathcal E\xrightarrow{\phi}\mathrm{Fr}\mathcal E \longrightarrow\cdots\longrightarrow \mathrm{Fr}^n\mathcal E\xrightarrow{h}\mathcal E.\] The total position is \(nu+v\) by Equation (22). Compatibility with \(h\) identifies rotation of the initial \(u\)-step with point Frobenius.

Conversely, subdivide a loop of position \(nu+v\) into \(n\) successive \(u\)-steps and one \(v\)-step. The given identification of the first rotation with Frobenius turns the first step into \(\phi:\mathcal E\to\mathrm{Fr}\mathcal E\); subsequent steps are its Frobenius translates. If \(\gamma\) is the total loop, the last step is \(h=\gamma(\phi^{(n)})^{-1}\). The identity \(\phi\gamma=\mathrm{Fr}(\gamma)\phi\) gives the required commuting square for \(h\). Unique open subdivision makes these constructions inverse on objects, intermediate isomorphisms, and arrows. The trace formula just proved gives Equation (26). ◻

Proof. A noncentral invariant cocharacter can be placed in the closure of a Weyl chamber. Its stabilizer is the Weyl group of a proper standard Levi. Choose a fundamental character for a simple root missing from that Levi, multiplied to be integral. It is fixed by the arithmetic monodromy of \(S_c\), and so defines a rational character \(\chi\) of that torus. The value \(\chi(\gamma_c)\) is a rational function on \(X\). Away from the legs it is a unit: any realization of the loop is an integral automorphism there, as is its inverse, and hence all its torus-character valuations vanish. At a leg its valuation is \(\langle\varpi,w_ja_j\rangle\) for a Weyl choice \(w_j\). The product formula contradicts the assumed nonvanishing. This argument uses the tautological torus element and therefore applies even before a rational bundle realizing \(c\) has been chosen. ◻

Proof. We give the comparison separately for each rational invariant \(c\). First we construct a connected group acting on suitable full-level subgroupoids of loops. We then realize rotation as a rational point of that group and use Lang’s theorem to compare the two descent conditions. Let \(Y\to X\) be the normalization in a connected arithmetic Galois splitting field of \(S_c\). Its group \(\Gamma\) is a subgroup of \(W\); write \(N=|\Gamma|\). This cover need not be geometrically connected. It is split and unramified at the regular legs. Let \(\mathsf M=X_*(T)\) be the abstract split cocharacter lattice used on the cover. There is a norm homomorphism of generic tori \[ \nu:\operatorname{Res}_{F(Y)/F} (\mathbb G_m\otimes_{\mathbb Z}\mathsf M)\longrightarrow S_c, \tag{28}\] which multiplies Galois conjugates with their descent action on \(\mathsf M\). We will use a connected source for this norm. Neither surjectivity nor separability of the norm itself is required.

Choose a finite rational set \(Z\subset X\) disjoint from the legs and containing the support of \(D\), the ramification points, the points where \(\gamma_c\) is not regular integral, and all other prescribed level points. Add one auxiliary closed point outside the legs. For an effective rational modulus \(B\) supported on the full inverse image of \(Z\), with positive multiplicity at every such point, put \[ Q_B=\operatorname{Pic}^{\underline 0}(Y,B) \otimes_{\mathbb Z}\mathsf M. \tag{29}\] The superscript means degree zero on each geometric component. The modulus meets every component, so a line bundle with its specified \(B\)-trivialization has no scalar automorphisms. Consequently \(Q_B\) is an actual smooth connected commutative algebraic group. One can see this from the generalized-Picard exact sequence: over the algebraic closure it is an extension of the product of the component Jacobians by the products of the local truncated unit groups, modulo one constant multiplicative group per component. A truncated unit group is smooth and connected, including for nonreduced moduli in positive characteristic. This is also the precise representability and connectedness statement in (Jordan et al. 2024, sec. 1.4, Proposition 1.6 and the following paragraph). The auxiliary point rigidifies this source only; it adds no level to the original loop problem.

For the descent comparison, it suffices to construct a coherent action of \(Q_B(\overline{\mathbb F}_q)\) on geometric objects and arrows, compatible with Frobenius. We now impose the integral conditions that retain the full level. Let \(\mathcal L_{a,c,B}\) be the full subgroupoid on objects for which norms of split congruence units at \(B\) preserve the local lattices, all flags, and the complete \(D\)-trivialization. Every object belongs to such a subgroupoid after increasing the multiplicities of \(B\). Indeed the generic torus embedding and the norm are continuous for the local valuation topologies; sufficiently deep units map into the prescribed principal congruence stabilizer. Away from \(Z\) and the legs the regular integral centralizer is an unramified torus, and its integral units preserve the lattice. At the legs this follows from the centralizer-apartment description above.

Represent an element of \(Q_B(\overline{\mathbb F}_q)\) by an \(\mathsf M\)-valued divisor disjoint from \(B\). Such a representative exists by choosing meromorphic sections agreeing with its trivializations to the specified finite orders. Modify the bundle at the image of its support by the norm cocharacters in Equation (28): change the local trivializations at those points and glue them to the unchanged torsor away from the support. This uses formal gluing for smooth affine torsors on a curve, applied at finitely many points; see (Pappas and Zhu 2013, arXiv version, Section 5.a, Lemma 5.1). At \(Z\) keep the original lattice and its level. All modifications commute with \(\gamma_c\), so the result remains a loop with the same invariant and the same relative positions at the legs. If two representatives differ by the divisor of a rational split-torus function \(h\) with \(h|_B=1\), the meromorphic automorphism \(\nu(h)\) identifies the two results. At \(Z\) it extends and preserves the full nonreduced level by the defining stabilizer condition; elsewhere it is the usual change of lattice trivialization. The normalized function \(h\) is unique: the ratio of two choices is constant on each geometric component and equals one along \(B\). The ratios of three representatives multiply, so these isomorphisms satisfy their cocycle identities. Tensor product therefore gives a coherent action of \(Q_B(\overline{\mathbb F}_q)\) on the groupoid. An arrow between loops identifies their canonical centralizers and carries each norm modification to the corresponding one. Thus the action is natural in all arrows, including every target automorphism.

The action preserves \(\mathcal L_{a,c,B}\). Namely a torus modification does not change which elements of the commuting norm-unit subgroup stabilize a lattice. The same observation proves stability under \(R_u\). All these constructions are Frobenius-compatible, since the cover and the modulus are rational.

We next exhibit rotation in the connected source Equation (29). At a geometric point \(y\) of \(Y\) over the leg \(x_j\), let \(\nu_y\in\mathsf M\otimes\mathbb Q\) be the valuation vector of the tautological element \(\gamma_c\) in the split coordinates on \(Y\). It belongs to the Weyl orbit of \(a_j\). Since \(a_j\) is regular, there is a unique \(w_y\in W\) with \(\nu_y=w_ya_j\). Define the \(\mathsf M\)-valued divisor \[\Delta=\sum_j\ \sum_{y\mapsto x_j}w_y(u_j/N)[y],\] where the inner sum runs over all geometric points above \(x_j\). These cocharacters are integral because \(u_j\in2|W|\Lambda_0\) and \(N\) divides \(|W|\). The valuation vectors are transported by the deck action and preserved by Frobenius on the abstract constant lattice \(\mathsf M\). Uniqueness of \(w_y\) therefore makes \(\Delta\) both rational and \(\Gamma\)-equivariant with its descent action on the labels. Under the ordered local identification, every lift contributes \(u_j/N\) to its norm. There are \(N\) lifts over each geometric leg, so the norm is exactly the rotation modification by \(u_j\). The support of \(\Delta\) avoids \(B\), giving its canonical trivialization along the full modulus.

We verify its multidegree. Let \(\Gamma_{\mathrm{geom}}\) be the geometric monodromy subgroup, and write \(e_C\) for the vector degree of \(\Delta\) on a geometric component \(C\). Deck equivariance gives \[e_{\gamma C}=\gamma e_C,\] so \(\Gamma_{\mathrm{geom}}\) fixes \(e_C\). Frobenius preserves the labels of the abstract split torus, and rationality gives \(e_{\mathrm{Fr}C}=e_C\). Choose \(\sigma\in\Gamma\) inducing the same permutation of geometric components as Frobenius. Then \[\sigma e_C=e_{\sigma C}=e_{\mathrm{Fr}C}=e_C.\] The coset of \(\sigma\) generates the cyclic group \(\Gamma/\Gamma_{\mathrm{geom}}\). Together with invariance under \(\Gamma_{\mathrm{geom}}\), this proves \[ e_C\in(\mathsf M\otimes\mathbb Q)^\Gamma. \tag{30}\] By Lemma 23, this vector is central. Every coefficient of \(\Delta\) lies in the rational subspace annihilating \(X^*(G)\), and so does \(e_C\). That derived subspace meets the rational central cocharacter subspace in zero. Thus \(e_C=0\) on every component. This is the point at which arithmetic, rather than merely geometric, monodromy is needed. We have obtained an element \(a_B\in Q_B(\mathbb F_q)\) whose action is \(R_u\).

We can now compare descent. Lang’s theorem applies to the smooth connected source \(Q_B\); see (Lang 1956, Theorem 1 and its corollary). In this commutative case its surjectivity is especially direct: \(\mathrm{Fr}-1\) has derivative \(-1\), hence has open image, and an open subgroup of a connected algebraic group is the entire group. Choose \(b\in Q_B(\overline{\mathbb F}_q)\) such that \[\mathrm{Fr}(b)-b=a_B.\] If \(\mathsf t_b\) denotes its action, Frobenius compatibility supplies a natural identification \[R_u^{-1}\mathrm{Fr}\,\mathsf t_b \simeq\mathsf t_b\,\mathrm{Fr}.\] Hence \(\mathsf t_b\), with inverse \(\mathsf t_{-b}\), gives an equivalence of descent groupoids \[ \operatorname{Fix}(\mathrm{Fr},\mathcal L_{a,c,B}) \simeq \operatorname{Fix}(R_u^{-1}\mathrm{Fr},\mathcal L_{a,c,B}). \tag{31}\] The identity in the source is an isomorphism of rigidified line bundles, with no scalar ambiguity. Equation (31) therefore includes the Frobenius isomorphisms and their compatible arrows. It preserves automorphism groups, not only geometric isomorphism classes.

We also spell out effectivity of the descent used here. An object and a Frobenius isomorphism are defined over a finite extension. After iterating by that extension degree, the resulting automorphism belongs to the rational points of an affine finite-type group over a finite field, hence has finite order. A further iterate is the identity. The datum is thus finite Galois descent, which is effective for bundles, levels, and their meromorphic isomorphisms. Ordinary Frobenius descent is precisely the rational-point groupoid in Equation (27). The shifted version can either be read from Equation (31) or from the explicit concatenation construction of Lemma 22.

The subgroupoids \(\mathcal L_{a,c,B}\) exhaust the geometric loop groupoid as the multiplicities grow. Both fixed-point groupoids have finitely many isomorphism classes by hypothesis. Choose \(B\) large enough to contain representatives of both. The equivalence then proves equality of their entire weighted counts for \(c\). Summing over the finitely many contributing rational invariants proves Equation (27).

Finally, a norm modification arising from componentwise degree-zero source line bundles changes no character degree of \(G\). Its action commutes with central modifications and therefore descends to the quotient by \(\Xi\). The initial degree-lifting observation identifies its closed paths with the ones just considered. This proves the central-quotient assertion as well. ◻

Proof of Theorem 19. With \(U_v=p_!q^*\) on the unnormalized complex, the identity to prove is equivalently \[ \sum_{[\mathcal E]}(T_{nu+v})_{\mathcal E,\mathcal E} =q^{-(nL_u+L_v)/2} \operatorname{Tr}_{\mathrm{alt}} (\mathrm{Fr}^{n*}U_v\mid C(u)). \tag{32}\] Set \(a=nu+v\). For \(n\) large it is regular and satisfies the nonvanishing condition. Lemma 20 makes the ordinary loop count finite, and the large-Frobenius formula makes the shifted count finite. For a fixed rational bundle \(\mathcal E\), closing a Hecke modification whose target is isomorphic to \(\mathcal E\) amounts to choosing such an isomorphism, modulo its automorphisms. Consequently its diagonal matrix entry is the corresponding loop mass. Summing gives \[\begin{aligned} \sum_{[\mathcal E]}(T_a)_{\mathcal E,\mathcal E} &=q^{-L_a/2}|\mathcal L_a(\mathbb F_q)|\\ &=q^{-L_a/2}|\mathcal L_a^{\mathrm{sh}}|\\ &=q^{-L_a/2}\operatorname{Tr}_{\mathrm{alt}} (\mathrm{Fr}^{n*}U_v\mid C(u)). \end{aligned}\] The first equality includes the stack multiplicities; no extra reciprocal automorphism factor is to be inserted in the matrix trace. The second equality is Lemma 24, and the third is Lemma 22. Since lengths add in the positive chamber, \(L_a=nL_u+L_v\), proving Equation (32). Equation (23) and the even shift in Equation (20) give Equation (21). The operator \(T_v\) here is function-normalized; the parity correction for an arbitrary \(v\) was kept in Lemma 21. ◻

Proof. We first treat the full semisimple coweight lattice, then descend through a finite lattice extension, and finally pass from simple modules to idempotent ideals.

The full lattice. Suppose first that \(L\) is the direct sum of the full coweight lattice of the semisimple root system and a central lattice. The latter contributes a central Laurent polynomial algebra and may be specialized at its character. It therefore suffices to consider the semisimple factors. For a character \(r\) put \[I(r)=\mathcal H\otimes_A\mathbf k_r.\] Its spherical line is spanned by \(\mathbf 1_r=\sum_{w\in W}\mathsf T_w\otimes1\). We use the following precise principal-series theorem: for the full weight lattice of a root system, the spherical vector generates the principal-series module if and only if \[ \prod_{\alpha>0}\bigl(1-Qr(\alpha^\vee)\bigr)\ne0. \tag{38}\] This is (Ram 2003, Proposition 2.11(b)), applied to the dual root system. The parameter denoted by \(q\) there is \(Q^{1/2}\) here, and its finite generator is \(Q^{-1/2}\mathsf T_s\). Thus the exceptional root value in that proposition is \(Q^{-1}\), as in Equation (38). The result includes singular characters. For a product of root systems it follows by tensor product. Its characteristic-zero formulation over \(\mathbf k\) follows by extension of scalars: the finite matrices determining cyclicity are defined over the field generated by the finitely many character values and \(Q^{1/2}\), which admits a complex realization.

If \(r\) is dominant, every positive-coroot value has modulus at least one, so Equation (38) holds. Consequently \(I(r)\) is generated by \(\mathbf 1_r\). A simple module \(S\) containing the generalized weight \(r\) contains an ordinary eigenvector of that weight: a finite-dimensional module over a commutative algebra has a common eigenvector in each nonzero generalized weight space. Inducing such a vector gives a surjection \(I(r)\to S\). The generating spherical vector cannot map to zero, so \(eS\ne0\).

Conversely, there is exactly one spherical simple at each central character. For completeness, this follows directly from the spherical Satake identity \(e\mathcal He=Ze\) (Haines et al. 2009, sec. 4.1 and 4.6, Equation (4.6.1)). At a central character \(c'\), the module \[P=\mathcal He\otimes_Z\mathbf k_{c'}\] is finite-dimensional and satisfies \(eP=\mathbf k e\). Every proper submodule of \(P\) has zero \(e\)-projection, since any submodule containing \(eP\) contains the generator of \(P\). The sum of its proper submodules is therefore proper and is its unique maximal submodule. Any simple module \(S\) with \(eS\ne0\) and central character \(c'\) is a quotient of \(P\): for \(0\ne u\in eS\), simplicity gives \(S=\mathcal H u\) and \(eS=(e\mathcal He)u=\mathbf k u\). This proves uniqueness.

At \(c'=Wr\), every simple quotient of \(I(r)\) is spherical by the preceding paragraph. Hence its unique spherical simple is such a quotient and contains \(r\), since the inducing vector generates \(I(r)\) and has a nonzero image in every nonzero quotient. We have proved, for the full lattice and every dominant \(r\), \[ S[r]\ne0\quad\Longleftrightarrow\quad eS\ne0, \tag{39}\] where \(S[r]\) denotes the generalized weight space.

A finite enlargement for a general reductive lattice. Let \(V=L\otimes_{\mathbf Z}\mathbf Q\). Write \[V=V_{\mathrm{ss}}\oplus V_{\mathrm c},\] where \(V_{\mathrm{ss}}\) is the rational span of the coroots and \(V_{\mathrm c}\) is the common kernel of the roots. Let \(Q^\vee\) and \(P^\vee\) be the coroot and full coweight lattices in \(V_{\mathrm{ss}}\), and put \[L_{\mathrm{ss}}=L\cap V_{\mathrm{ss}},\qquad L_{\mathrm c}=\operatorname{pr}_{\mathrm c}(L),\qquad L'=P^\vee\oplus L_{\mathrm c}.\] The projection is rational, so \(L_{\mathrm c}\) is a lattice. Root integrality gives \(\operatorname{pr}_{\mathrm{ss}}(L)\subset P^\vee\); hence \(L\subset L'\). The inclusion has finite index, and \[ \Gamma=L'/L\simeq P^\vee/L_{\mathrm{ss}}. \tag{40}\] Indeed the map from \(P^\vee\) to \(L'/L\) is surjective because \(L\to L_{\mathrm c}\) is surjective, and its kernel is \(L_{\mathrm{ss}}\). In particular \(Q^\vee\subset L_{\mathrm{ss}}\).

Set \(\mathcal H'=\mathcal H(L',Q)\) and \(A'=\mathbf k[L']\). The Weyl group acts trivially on \(\Gamma\), since \(w a'-a'\in Q^\vee\) for \(a'\in L'\). The Bernstein presentation consequently grades \(\mathcal H'\) by \(\Gamma\), with degree-zero subalgebra \(\mathcal H\). Each component has an invertible homogeneous element \(\theta_{a'}\), which commutes with \(A\). Thus \[ \mathcal H'=\bigoplus_{\gamma\in\Gamma} \theta_{a'_{\gamma}}\mathcal H \tag{41}\] is a strong finite grading. No splitting of \(\Gamma\) into a subgroup of units is required.

We shall also use a different set of homogeneous units. The length-zero stabilizer \(\Omega'_{\mathrm{ss}}\) of the semisimple base alcove is isomorphic to \(P^\vee/Q^\vee\), and its map onto \(\Gamma\) is surjective by Equation (40). Choose \(\omega_\gamma\in\Omega'_{\mathrm{ss}}\) in each degree and put \(U_\gamma=\mathsf T_{\omega_\gamma}\). Then \[ e_\gamma=U_\gamma^{-1}eU_\gamma\in\mathcal H \tag{42}\] is the idempotent of a hyperspecial vertex. To see the inclusion in \(\mathcal H\), one may either use its degree zero or note that conjugation by \(\omega_\gamma\) takes the finite Weyl subgroup at the base vertex to the finite parahoric Weyl subgroup at that hyperspecial vertex. Central translations fix the reduced building and give no further idempotents. Taking all of \(\mathcal V\), rather than just the vertices arising from these representatives, is therefore harmless.

Descent of the simple-module detector. Let \(S\) be a simple \(\mathcal H\)-module at the central character \(c\). It is finite-dimensional, since \(\mathcal H\) is finite over \(Z\). Choose a simple quotient \(S'\) of the finite-dimensional induced module \(\mathcal H'\otimes_{\mathcal H}S\). Adjunction gives a nonzero map \(S\to S'\), hence an embedding on restriction. We identify \(S\) with its image. Simplicity of \(S'\) and the two choices of homogeneous units give \[ S'=\sum_{\gamma\in\Gamma}\theta_{a'_\gamma}S =\sum_{\gamma\in\Gamma}U_\gamma S. \tag{43}\] Each summand is a simple \(\mathcal H\)-module, because each homogeneous unit normalizes the degree-zero algebra. This also proves directly the needed Clifford-theoretic assertion that the restriction is a semisimple sum of conjugates. More particularly, \(\theta_{a'_\gamma}\) commutes with \(A\), so all summands in the first sum have the same \(A\)-weight support. Consequently \[ S[t]\ne0\quad\Longleftrightarrow\quad S'[t]\ne0, \tag{44}\] with the right-hand side interpreted as an \(A\)-weight space. The second sum gives the complementary assertion \[ eS'\ne0 \quad\Longleftrightarrow\quad e_\gamma S\ne0\text{ for some }\gamma\in\Gamma. \tag{45}\]

Suppose first that \(S[t]\ne0\). The nonzero \(A\)-weight space \(S'[t]\) contains an \(A'\)-weight \(t'\) restricting to \(t\). Since \(L\) has finite index in \(L'\), restriction determines the real absolute-value exponent uniquely; thus \(t'\) is dominant. Applying Equation (39) to \(\mathcal H'\) gives \(eS'\ne0\). Equation (45) now gives a nonzero hyperspecial projection on \(S\).

Conversely, suppose that \(e_vS\ne0\) for a hyperspecial vertex \(v\). The idempotent \(e_v\) is conjugate to \(e\) by a length-zero unit of \(\mathcal H'\), so \(S'\) is spherical. Let \(Wr'\) be its \(A'^W\)-central character. Restriction of this central character to \(Z\) is \(Wt\); hence, after replacing \(r'\) by a Weyl conjugate, its restriction to \(L\) is exactly \(t\). This representative \(r'\) is dominant. The full-lattice detector says that the spherical simple \(S'\) contains \(r'\), and hence contains \(t\) on restriction to \(A\). Equation (44) gives \(S[t]\ne0\). We have proved the asserted equivalence on all simple modules at \(c\).

The completed ideals. The algebra \(B\) is finite over the complete local ring \(R\). Every simple \(B\)-module is annihilated by the maximal ideal \(\mathfrak m_R\): it is finite over \(R\), and centrality and simplicity make \(\mathfrak m_R S\) either zero or \(S\), with the second possibility excluded by Nakayama’s lemma. The same assertion holds for every quotient of \(B\). Thus the simple \(B\)-modules are precisely the simple modules of the residue fiber to which the preceding detector applies.

Put \(J=\sum_{v\in\mathcal V}Be_vB\). In \(B/J\), the image of \(p_t\) annihilates every simple module by that detector. It belongs to the Jacobson radical and is an idempotent, so it is zero. Therefore \(Bp_tB\subset J\). Conversely, in \(B/Bp_tB\) every image of \(e_v\) annihilates every simple module. These images are again idempotents in the Jacobson radical and are zero. This proves the reverse inclusion and Equation (37). ◻

Proof. Equation (37) supplies, for each \(v\), a finite expression \(e_v=\sum_j a_jp_tb_j\) with \(a_j,b_j\in B\). Hence \(e_vN\subset\sum_j a_j(p_tN)\), proving finite-dimensionality. Conversely, write \(p_t\) as a finite sum of elements of the ideals \(Be_vB\). If all \(e_vN^i\) were zero, this expression would give \(p_tN^i=0\). Finally, the image of a projector commuting with \(B\) is a \(B\)-submodule, so the same argument applies to it. ◻

Proof. We use the following precise form of Langlands’ spectral theorem. At a fixed compact open level, the discrete pairs \((M,\sigma)\) with nonzero fixed-level induced vectors form finitely many classes under Weyl association and unramified twisting. Their compact induced models are finite-dimensional, their unitary twisting spaces are compact real tori modulo finite stabilizers, and Eisenstein wave packets on these spaces give the Hecke-equivariant Hilbert spectral decomposition. This formulation is stated in (Lafforgue 2013, Theorem III.7 and the preceding definitions, pp. 33–36), where it is identified with (Mœglin and Waldspurger 1995, sec. VI.2.1); the function-field treatment in the English edition is (Mœglin and Waldspurger 1995, sec. VI.2.7). The statement includes discrete inducing representations. If the given level is not contained in the chosen standard maximal compact subgroup, intersect it with that subgroup first and then take invariants under the original compact level. This replaces the full compact models by closed invariant subspaces and preserves all the asserted finiteness and norm bounds.

We explain the effect of \(\Xi\). Unramified characters are characters of a free degree lattice. On the central degree lattice of \(G\), the condition that \(\sigma\otimes\chi\) be trivial on \(\Xi\) fixes finitely many characters because the image of \(\Xi\) has finite index. For \(M=G\) this leaves finitely many points of each twisting torus. There are finitely many pairs by the quoted theorem and their fixed-level models have finite dimension, so \(\mathcal D_K\) is finite. For a proper Levi \(M\), the condition cuts its unitary parameter torus into finitely many translates of a closed subtorus, hence leaves a compact parameter space. The wave-packet construction allows finite Weyl identifications between parameters; equivalently it realizes the corresponding Hilbert space inside finitely many full induced compact models. This proves (i) in precisely the form needed here.

Boundedness of fixed-level induction ranks can also be read directly from the compact picture. The quotient of a maximal compact subgroup by its intersection with the inducing parabolic and the prescribed open level is finite. Its finitely many representatives replace the level on \(\sigma\) by finitely many compact open subgroups of \(M\). Their invariant spaces are finite-dimensional. Unitary unramified twists are trivial on these compact subgroups, so the same dimension bounds hold throughout each torus.

For (ii), use the exact constant-term assertion on deep function-field Siegel sectors (Morris 1982, I, Section 5.9, p. 107); see also (Mœglin and Waldspurger 1995, sec. I.2.7). Once the root coordinates outside a Levi are sufficiently large, a function at the prescribed compact level equals its constant term for that parabolic. The threshold is uniform at a fixed level and on the bounded transverse data. The constant-term exponent theorem says that translating an automorphic form’s normalized constant term in the split-center degree variables gives a finite sum of characters times polynomials (Bernstein and Lapid 2024, sec. 6.2, pp. 200–201, Equations (6.1)–(6.3)); see also (Mœglin and Waldspurger 1995, secs. I.3.1–I.3.2 and I.4.2). We need the strict decay consequence, and give its elementary proof to include all boundary directions.

Partition a Siegel domain according to which simple-root coordinates are above the constant-term threshold. The remaining roots define the Levi; their bounded coordinates range over bounded data modulo its center. After passing to finite-index degree lattices, finitely many transverse values and translates of positive orthants suffice. Take the image of each sufficiently deep sector in \(P(F)\backslash G(\mathbb A)/(K\Xi)\), with its quotient Iwasawa measure \(\delta_P(m)^{-1}\,du\,dm\,dk\). The compact unipotent quotient \(U(F)\backslash U(\mathbb A)\) has volume one, and the function equals its constant term \(f_P\) on this sector. The deep-sector separation statement (Morris 1982, I, Section 4.2, p. 101) puts every additional rational identification within a sector in \(P(F)\), which has already been quotiented out. The finitely many bounded transverse charts of (Morris 1982, I, Section 4.6, pp. 102–103) therefore give bounded residual multiplicity for the map to the automorphic quotient. The sector’s weighted squared norm is consequently at most a fixed multiple of the global squared norm. This statement uses the parabolic quotient and its measure; the integral-unipotent multiplicity in a raw subset of \(G(\mathbb A)\) need not be bounded.

Absorb the finitely many transverse measure factors into a norm on a finite-dimensional space \(V\). Multiplication by \(\delta_P^{-1/2}\) then normalizes the constant term to an exponential-polynomial sequence \[F\colon\mathbf N^r\longrightarrow V, \qquad \dim V<\infty,\] and the preceding norm comparison gives \(F\in\ell^2(\mathbf N^r,V)\).

Let \(E\) be the span of all forward coordinate shifts of \(F\). It is finite-dimensional because \(F\) is exponential-polynomial. Each coordinate shift \(S_j\) preserves \(E\) and is a contraction for its inherited \(\ell^2\) norm. It has no eigenvalue of modulus one: if \(S_jh=zh\), \(|z|=1\), and \(h\ne0\), a nonzero slice in the other coordinates gives a coordinate ray of constant nonzero norm, contradicting square summability. Thus choose \(\rho<1\) greater than the moduli of every eigenvalue of every \(S_j|_E\). Finite-dimensional Jordan estimates and commutation of the shifts give \[\|S_1^{n_1}\cdots S_r^{n_r}\|\le C\rho^{n_1+\cdots+n_r}.\] Evaluation at the origin is bounded in the \(\ell^2\) norm, so this proves exponential decay of \(F\). Summing over the polynomially many lattice points of a given radius gives an exponential squared norm tail. Bounded sectors contribute none for large radius. The reduction height and the HN radius are comparable up to fixed constants, by the degree description of the canonical reductions. This proves Equation (46) for each discrete form. A basis of the finite-dimensional space \(\mathcal D_K\) makes the constants uniform on that space.

Finally, compatibility of normalized parabolic induction with Iwahori invariants identifies the Iwahori invariants of the local induced representation with the corresponding induced Hecke module (Prasad 2005, sec. 5.8, Theorem 8). We use left modules and the positive-translation convention of Equation (34); the conversion from the source’s right-module convention is described in its Section 7. The Bernstein presentation now proves (iii). The ambient Hecke algebra is free as a right module over the Levi Hecke algebra, with basis indexed by the representatives \(w\) for which \(w\Phi_M^+\subset\Phi_G^+\). Commuting a Bernstein element past these basis elements by Equation (34) is triangular in Bruhat order, and its diagonal characters are exactly Equation (47). The real exponent of a unitary \(\sigma\) is zero in the split central directions of \(M\). The finite list of inducing data and the bounded fiber dimensions from (i) give the last assertions of (iii). ◻

Proof. On the circle \(|z|=r\), the determinant of \(z-A\) has absolute value at least \((r-r_0)^{\dim A}\). In an orthonormal basis, its adjugate has a bound depending only on \(B,r,R_0\). The resolvent is therefore uniformly bounded on that circle. The contour formula for \(A^n\) gives the stated estimate, after increasing \(C\) to cover the zero-dimensional case. ◻

Proof. Complexify the twisting torus of each family in Proposition 28. Its ordered \(x\)-weights form a coset of an algebraic subtorus. There are finitely many such cosets. A coset not containing the target on \(L^0\) can be annihilated by a Laurent polynomial nonzero at the target. Raise that polynomial to a power exceeding the uniformly bounded generalized-weight length, and multiply over these cosets. The resulting \(P_0\) kills their actions, including generalized weights.

Consider a remaining family, induced from \(M\) and shuffled by \(w\). Let \(C_M\) be its real twisting-center subspace after that shuffle, and let \(\nu\) be the real exponent of its unitary contour. Unitarity of the inducing central character gives \(\nu\perp C_M\). Membership of the target in its complexified coset gives \(\lambda_0-\nu\in C_M\). The possible ambiguity from working on \(L^0\) is a connected-central factor of \(G\) and is already contained in the twisting center of \(M\). Therefore \[ \nu=\operatorname{pr}_{C_M^\perp}(\lambda_0),\qquad (\lambda_0,\nu)=\|\lambda_0\|^2 -\|\operatorname{pr}_{C_M}(\lambda_0)\|^2. \tag{48}\] If the second subtracted term were zero, the twisting factor taking the inducing parameter to \(t_0\) would be unitary. Moreover, \(w\Phi_M^+\subset\Phi_G^+\) and dominance of \(\lambda_0\) would make the corresponding inducing weight dominant for \(M\). The induction hypothesis would give Equation (33) for that weight, and hence for its unitary central twist and Weyl conjugate \(t_0\). This contradicts the choice of \(t_0\). Thus every remaining family has \[ (\lambda_0,\nu)<\|\lambda_0\|^2. \tag{49}\] The finite list of strict inequalities persists in a small open cone of directions adjacent to \(\lambda_0\) and inside the regular dominant chamber. The \(y\)-exponents run through a finite list on the unitary contours. Their pairings with the fixed \(\zeta\) are bounded constants, whereas the strict \(x\)-gap grows linearly with \(\mu\). This proves the assertion for all sufficiently large \(\mu\) in a slightly smaller cone. ◻

Proof. The complex \(M(u)\) has finite-dimensional total cohomology by Lemma 20. If the target character did not occur, a Laurent polynomial in the two degree-zero Bernstein algebras would annihilate its entire cohomology and remain nonzero at the target. Multiply it by \(P_0\) and by polynomial factors killing every other discrete joint character. Powers larger than the finite generalized weight lengths kill the corresponding generalized spaces. Denote the resulting polynomial by \(P\). It is nonzero at the target, kills \(M(u)\), and on the discrete spectrum retains only the target joint generalized weight space \(\mathcal D_0\).

Multiply \(P\) by a Bernstein monomial with sufficiently large positive labels at both places. This changes none of these properties, since all Bernstein monomials are invertible. We can consequently write \[P=\sum_{v}c_vT_v\] with a finite set of regular positive degree-zero labels \(v\). For the remainder of this argument \(u\) and \(P\) are fixed and only the positive integer \(n\) varies. Apply Theorem 19 to each term. The normalized version of that identity gives, for every sufficiently large \(n\), \[ \sum_b(T_u^nP)_{b,b} =\operatorname{Tr}_{\mathrm{alt}} (\mathrm{Fr}^nP\mid M(u))=0. \tag{51}\] Here \(b\) indexes the normalized point basis of \(\mathcal L_K\); normalization by the square root of its measure does not change diagonal matrix entries. The translation \(u\) is even; each insertion \(T_v\) is in the function normalization of that theorem, including its known parity correction to the perverse trace. No evenness condition is imposed on \(v\). Only this ordinary diagonal sum is used; the operator is not asserted to be trace class on all of \(\mathcal L_K\).

The support bound in Theorem 19 puts every contributing diagonal entry in an HN ball of radius \(Bn\) for a fixed \(B\). Increase \(B\) if necessary and write \(Q_n=Q_{Bn}\). The left side of Equation (51) is therefore \(\operatorname{Tr}(Q_nT_u^nP Q_n)\). By Proposition 10, \(\operatorname{rank}Q_n\) is polynomial in \(n\).

Put \[a=\mu(t_0)\zeta(t_{0,y}),\qquad R=|a|>0.\] Lemma 30 gives an \(r_0<R\) bounding the moduli of all continuous eigenvalues of \(T_u\) on the image of \(P\). Fixed Hecke operators are uniformly bounded on all unitary fibers by their \(L^1\) norms. The image of \(P\) in each fiber is invariant under \(T_u\), has bounded dimension, and inherits its Hilbert norm. Choose \(r_0<r<R\) and apply Lemma 29 on this image. We obtain \[\|T_u^nP\|_{\mathrm{cont}}\le C r^n.\] This bound is uniform even where the dimension of the image changes. Hence the continuous contribution to the compressed trace is at most \(\operatorname{rank}(Q_n)Cr^n=o(R^n)\).

On \(\mathcal D_0\), simultaneous triangularization of the commuting Bernstein operators gives \[ \operatorname{Tr}_{\mathcal D_K}(T_u^nP) =\dim(\mathcal D_0)P(t_0,t_{0,y})a^n. \tag{52}\] The coefficient is nonzero; generalized-weight nilpotents do not change the diagonal of these matrices. The norm of \(T_u^nP\) on this finite-dimensional space is at most \(C(1+n)^N R^n\) for fixed \(C,N\). Equation (46) therefore makes the difference between its full trace and its \(Q_n\)-compressed trace \(o(R^n)\). Decomposing the Hilbert space into its orthogonal discrete and continuous summands, Equations (51) and (52) would imply \[0=\dim(\mathcal D_0)P(t_0,t_{0,y})a^n+o(R^n),\] an impossibility. Thus the target character occurs in \(M(u)\).

Since the full Bernstein torus also acts on this finite-dimensional space, the target character on \((L^0)^2\) has at least one full ordered extension \((t_x,t_y)\) in its support. The cyclic-slide identity of Proposition 7 identifies normalized Frobenius on that generalized weight space with the translation \(T_u\), up to the already harmless convention sign. Every one of its Frobenius eigenvalues has modulus \[|\mu(t_x)\zeta(t_y)| =q^{d_x\langle\lambda_0,\mu\rangle +\log_q|\zeta(t_{0,y})|}.\] Recall the weighted translation length \(L_u=d_x\ell(t^\mu)+d_y\ell(t^\zeta)\) and the normalization in Equation (20). The reduction in the proof of Lemma 20 applies the compact-support weight theorem (Deligne 1980, Corollary 3.3.4) to \(Y_u\): the weights of the constant sheaf’s \(H_c^j\) are integers at most \(j\). Since \[H^i(M(u))=H_c^{i+L_u}(Y_u,\mathbf k)(L_u/2),\] the weights of this normalized group are integers at most \(i\). Comparing with the displayed Frobenius modulus proves Equation (50). ◻

Proof. The Frobenius weights of \(M(u)\) are integers by the weight theorem and normalization just used in Proposition 31. Set \(b=2\log_q|\zeta(t_{0,y})|\). The occurrence in Proposition 31, not merely its degree inequality, gives \[ 2d_x\langle\lambda_0,\mu\rangle+b\in\mathbf Z \tag{53}\] for every sufficiently large allowed label in the open cone. Let \(L_1\subset L^0\) be the fixed finite-index lattice imposing the divisibility conditions. Given \(a\in L_1\), choose \(\mu\) sufficiently deep in that cone that both \(\mu\) and \(\mu+a\) belong to it, and avoid the finitely many affine hyperplanes excluded by Equation (19) for both labels. Such labels exist: a full-dimensional cone has lattice points arbitrarily far from any prescribed finite collection of proper affine hyperplanes. Subtracting the two instances of Equation (53) gives \(2d_x\langle\lambda_0,a\rangle\in\mathbf Z\). These pairings on the full-rank lattice \(L_1\) prove the assertion. ◻

Proof. Every weight has the form \(\mu-\sum_\alpha n_\alpha\alpha\) with \(n_\alpha\ge0\) in the positive simple roots of the dual group. Equality of its pairing with that of \(\mu\) forces every root appearing in this difference to be orthogonal to \(\lambda_0\). The highest weight \(\mu\) has zero pairing with the coroots of that Levi root system. Its highest line therefore generates the trivial representation of the corresponding derived Levi. Applying only its lowering operators produces no different weight. The maximizing face consists of the highest line alone. Additivity of the pairing proves the assertion about tuples. ◻

Proof. Use the finite Wakimoto filtration of Theorem 5 and its cohomology spectral sequence. Its first page has labels \(\mathbf a=(a_1,\ldots,a_d)\) of weights of \(V_\mu\), with their weight-space multiplicities. The Bernstein algebra and \(S_x\) act on this spectral sequence. The commutation statements of Propositions 7 and 8 also make \(S_x\) commute with the full Hecke action on its abutment. Tensoring the pages and the abutment with the flat algebra \(R_x\) gives a convergent spectral sequence with its finite filtration. Write \(\Theta_b\) for the Bernstein element of a label \(b\). Taking the \(d\)-fold slide first returns each tuple \(\mathbf a\) to its original ordering. For \(b=a_1+\cdots+a_d\), the Wakimoto interchange formula of Theorem 5 identifies \(S_x\) on this graded term with \[\Theta_b^\kappa\otimes U_{\mathbf a},\] where \(U_{\mathbf a}\) acts on the finite tensor product of weight-multiplicity spaces. The chosen power removes the parity signs, and the trivial semisimplified Weil structures make \(U_{\mathbf a}\) unipotent. The two factors commute. After specializing the Bernstein algebra at an ordering \(r\), its only possible eigenvalue is \[ \left(\prod_{j=1}^{d}a_j(r)\right)^\kappa. \tag{55}\]

Here is an annihilator construction entirely on cohomology. The completed Bernstein algebra is finite over \(R_x\). For each of the finitely many labels \(\mathbf a\), put \[D_{\mathbf a}(z)=\prod_{w\in W} \bigl(z-\Theta_{wb}^\kappa\bigr).\] Its coefficients are invariant, and hence lie in \(R_x\). One factor gives \(D_{\mathbf a}(\Theta_b^\kappa)=0\). Therefore \(D_{\mathbf a}(S_x)\) lies in the ideal generated by \(1\otimes(U_{\mathbf a}-1)\) in the commutative algebra of these operators. Choose \(n_{\mathbf a}\) with \((U_{\mathbf a}-1)^{n_{\mathbf a}}=0\). The monic polynomial \(Q_{\mathbf a}=D_{\mathbf a}^{n_{\mathbf a}}\in R_x[z]\) annihilates the entire first-page term, regardless of the size of its cohomology. Its residual roots are exactly the values in Equation (55) at the orderings of \(t_x\). Put \(Q_0=\prod_{\mathbf a}Q_{\mathbf a}\). This annihilates the entire first page, hence every later page and the associated graded of the abutment. If the filtration has \(b_\mu\) steps, then \(Q=Q_0^{b_\mu}\) annihilates \(\widehat H_\mu\): each application of \(Q_0(S_x)\) lowers its filtration by at least one step. The same \(Q\) also annihilates every page.

The maximal modulus of these residual roots is independent of the ordering of \(t_x\), by Weyl invariance of the weight polytope. Hensel factorization over \(R_x\) gives \(Q=Q_+Q_-\), with \(Q_+\) collecting all roots of that maximal modulus and \(Q_-\) all remaining roots. The factors are relatively prime. Their Bezout identity defines an idempotent \(\varepsilon_{\mathrm{ext}}\), polynomial in \(S_x\), on every page, the filtered cohomology, and its abutment. On the abutment it commutes with the full Hecke algebra, since \(S_x\) does and its polynomial coefficients are central.

Apply also the ordered idempotent \(p_{t_x}\) from Lemma 26. It commutes with the spectral-sequence maps and with \(\varepsilon_{\mathrm{ext}}\). Lemma 33 shows that exactly one label tuple survives on the projected first page: every entry is \(\mu\). Its multiplicity is one. The Weil normalization and interchange compatibility in Theorem 5 identify this term with the \(t_x\)-ordered part of \(H^*(M(\mu,\zeta))\). Central factors cause no additional eigenvalues, since \(\mu\in L^0\) and \(V_\mu\) is trivial on \(Z(\widehat G)^\circ\). There is no other surviving filtration step to support a differential. It follows that \(p_{t_x}\varepsilon_{\mathrm{ext}}\widehat H_\mu\) is finite-dimensional and contains the nonzero high-degree piece of Proposition 31. Flat completion preserves this finite-dimensional generalized central-character piece.

Apply Corollary 27 to the graded module \(\varepsilon_{\mathrm{ext}}\widehat H_\mu\). Each \(N_v\) is finite-dimensional, and some \(N_v\) retains a nonzero piece in the same high degree. The cohomology transfer in Proposition 8 identifies \(e_v\widehat H_\mu\) with \(H^*(C_{\mu,v}^{\mathrm{sph}})\otimes_{Z_x}R_x\) and intertwines \(S_x\) and the center. This proves the stated identification. The finite set of hyperspecial variants is among those allowed in Theorem 18. ◻

Proof. Use Proposition 9. On the finite representation bundle \(V_\mu^{\otimes d}\), the \(d\)-fold slide is the degree-zero closed matrix \[\rho_\mu(s)=V_\mu(s)^{\otimes d}\] over \(\mathcal O(\widehat G)\). Therefore \[P_\mu(z)=\det\bigl(z-\rho_\mu(s)^\kappa\bigr) \in\mathcal O(\widehat G)^{\widehat G}[z]=Z_x[z]\] satisfies \(P_\mu(S_x)=0\) as a literal identity of chain maps. Indeed Cayley–Hamilton holds on the finite bundle before tensoring with the arbitrary dg module \(\mathcal A\) in that proposition; tensoring and then taking exact rational invariants preserve the identity. In particular this assertion requires no finiteness assumption on \(\mathcal A\) or on the whole spherical cohomology.

Now the completed complex \[\widehat C_{\mu,v}^{\mathrm{sph}} =C_{\mu,v}^{\mathrm{sph}}\otimes_{Z_x}R_x\] is defined using that actual dg module structure. The residual roots of \(P_\mu\) are the products of weight values in Equation (55) at \(t_x\), with their tensor multiplicities. Their set is Weyl invariant, so it is the same root set used for \(Q\). Factor \(P_\mu=P_+P_-\) over \(R_x\) by Hensel’s lemma, separating the same maximal-modulus residual roots from the others. For a Bezout identity \(uP_++vP_-=1\), the chain map \[e_+=v(S_x)P_-(S_x)\] is idempotent: \(e_+^2-e_+\) is a multiple of \(P_\mu(S_x)=0\). Its image \(E_{\mu,v}\) is an actual dg direct summand. The coefficients lie in \(R_x\), so this projector commutes with tensoring by any finite Koszul complex over \(R_x\).

We verify that its cohomology is the transferred extreme part; polynomial annihilators on cohomology alone would not have sufficed for the preceding chain construction. On \(H^*(\widehat C_{\mu,v}^{\mathrm{sph}})\) both \(Q(S_x)\) and \(P_\mu(S_x)\) vanish, and the two polynomial projectors commute. On the intersection of the \(Q_+\) part with the \(P_-\) part, \(Q_+(S_x)\) and \(P_-(S_x)\) vanish. Their residual roots have different moduli, so these two polynomials are relatively prime over \(R_x\); their Bezout identity makes this intersection zero. The same argument interchanging \(+\) and \(-\) kills the other cross intersection. Thus \(e_+\) and the transferred \(\varepsilon_{\mathrm{ext}}\) act identically on cohomology. This argument applies to arbitrary modules, without finite generation. Consequently \(H^*(E_{\mu,v})=N_v\), as claimed. ◻

Proof. The bounded Koszul complex is a complex of finite free modules, so its tensor product has a convergent spectral sequence \[E_2^{p,q}=H^p\bigl(K_R(f;H^q(C))\bigr) \ \Longrightarrow\ H^{p+q}(C\otimes_R K_R(f)).\] Its term at \((p,q)=(0,N)\) is \(H^N(C)/\mathfrak mH^N(C)\), which is nonzero by Nakayama’s lemma. No later differential enters this term, since its source would have second index greater than \(N\); none leaves it, since its target would have positive Koszul degree. The term survives in total degree \(N\). ◻

Proof of Theorem 25. For semisimple rank zero, the group is a split torus. Its unitary local characters have zero absolute-value exponent, so the assertion holds with trivial \(\phi\). Assume the result in lower semisimple rank and choose a violating weight \(t_0\) as above. All the preceding constructions are then available for arbitrarily large \(m\) in Equation (54).

Use Proposition 34 to choose \(v\), and then Proposition 35 to obtain the dg summand \(E_{\mu,v}\). Let \(N\) be its highest nonzero cohomological degree. It exists because \(H^*(E_{\mu,v})=N_v\) has finite-dimensional total graded space, and it satisfies \[ N\ge 2dm\|\lambda_0\|^2+b, \qquad b=2\log_q|\zeta(t_{0,y})|. \tag{56}\] Use the invariant algebra generators \(a_1,\ldots,a_r\) fixed before Lemma 13, and put \(f_j=a_j-a_j(t_x)\). These functions generate the maximal ideal of \(Wt_x\) in \(Z_x\), and their images generate the maximal ideal of \(R_x\). Apply their finite Koszul complex to the actual \(R_x\)-dg summand \(E_{\mu,v}\). Lemma 36 preserves a nonzero class in degree \(N\). The chain projector of Proposition 35 commutes with this tensor product, so the tested summand is a direct summand of the tested completed spherical complex.

Completion can be removed from the latter test. Indeed it is flat, and each \(f_j\) acts null-homotopically on its Koszul complex. The tested cohomology is therefore annihilated by the maximal ideal, and extension of scalars from \(Z_x\) to \(R_x\) changes none of its cohomology groups. The natural map from the uncompleted spherical Koszul test to its completion is thus a quasi-isomorphism. On this specialized cohomology the polynomial coefficients of the chain projector reduce to their residue-field values. Its image is exactly the maximal-modulus group for the holonomy slide \(S_x\), by its Hensel factorization. Raising an invertible slide to the positive power \(\kappa\) preserves the ordering of eigenvalue moduli. Hence the surviving degree \(N\) belongs to the extreme summand to which Theorem 18 applies, giving \[ N\le C+dm\max_{e\in E_{t_x}} \langle\lambda_0,h_e\rangle. \tag{57}\]

The restriction of \(t_x\) to \(L^0\) is that of \(t_0\). Hence \(t_x\) differs from \(t_0\) only by a connected-central factor. The adjoint eigenspace \(E_{t_x}\), its compatible neutral cocharacters, and the semisimple exponent are consequently independent of that factor. Because \(t_0\) is a violation and its central exponent is zero, the strict part of Theorem 18 gives \[\delta=2\|\lambda_0\|^2- \max_{e\in E_{t_x}}\langle\lambda_0,h_e\rangle>0.\] The same theorem makes \(C\) uniform under the possible connected-central changes of \(t_x\). Its hypotheses also allow the fixed auxiliary label and all the finitely many hyperspecial variants; replace their constants by their maximum. No level, auxiliary kernel, or semisimple holonomy datum varies with \(m\).

Subtracting Equation (57) from Equation (56) now yields \(dm\delta\le C-b\), impossible for arbitrarily large \(m\). There is no violating weight. This completes the induction and the proof of Theorem 25. ◻

Proof. Finite dimensionality of \(C_{D,\mathbb Q}\) implies that of its Hecke image algebra. On the complex cusp space use the positive automorphic \(L^2\) inner product. For a spherical function \(a\), the adjoint of convolution by \(a\) is convolution by \(a^*(g)=\overline{a(g^{-1})}\). This function is again spherical and its operator preserves the cusp space. The spherical operators at one place commute, and operators at different places commute. Consequently all their restrictions are commuting normal operators. Simultaneous diagonalization follows in the finite-dimensional space, and implies that the Hecke image algebra is reduced. A finite reduced algebra over \(\mathbb Q\) is a product of number fields, so it splits over \(\overline{\mathbb Q}\) and gives Equation (58). The normalized Satake isomorphism is defined over \(\mathbb Q(\sqrt{q_x})\); hence an algebraic Hecke character determines an algebraic point of the dual adjoint quotient, and therefore an algebraic semisimple conjugacy class. ◻

Proof. Fix one embedding and conjugate its proposed decomposition so that its product is \(\iota(t)\) itself. Let \(A\) be the image of the diagonal torus of \(\operatorname{SL}_2\). Its centralizer \(L=Z_{\widehat G}(A)\) is a connected Levi subgroup (Milne 2017, Theorem 17.38 and Section 25.25). The semisimple element \(c_\iota\) lies in \(L\), so it lies in a maximal torus \(S\) of \(L\). The central torus \(A\) belongs to every maximal torus of \(L\). Therefore \(S\) contains both \(A\) and \(\iota(t)\) and is a maximal torus of \(\widehat G\). Both \(S\) and \(T_{\mathbb C}\) are maximal tori of the connected group \(Z_{\widehat G}(\iota(t))^\circ\); maximal-torus conjugacy (Milne 2017, Theorem 17.10) therefore puts the diagonal cocharacter in \(T\) without moving \(\iota(t)\). This argument never requires the full semisimple centralizer, or the centralizer of the triple, to be connected.

After this alignment, let \(\mu_\iota\in X_*(T)\) be the diagonal cocharacter. The compact factor belongs to \(T(\mathbb C)\), and the closure of its cyclic subgroup is compact. Every torus character therefore has modulus one on it. Equation (59) gives \[ 2\lambda_\iota(t)=\mu_\iota. \tag{60}\]

We prove that these integral cocharacters are independent of the embedding. Choose a positive definite Weyl-invariant inner product on \(X_*(T)\otimes\mathbb R\), and extend it to an invariant complex bilinear form on \(\widehat{\mathfrak g}_{\mathbb C}\). Such a form is obtained by choosing a positive multiple of the Killing form on each simple factor. For two embeddings write \(\lambda,\lambda'\) for their exponents and \(\delta=\lambda'-\lambda\). The aligned triple at the first embedding has neutral element \(H_\mu=d\mu(1)=2\lambda\) under the usual identification of the real cocharacter space with the real subspace of \(\operatorname{Lie}T\). If a root component \(e_\alpha\) of its raising element is nonzero, then \[\iota(\alpha(t))=r, \qquad \alpha(\lambda)=1.\] Since \(\alpha(t)\) is algebraic and \(r\) is rational, this is the algebraic identity \(\alpha(t)=r\). At the second embedding it implies \(\alpha(\lambda')=1\). The corresponding argument for the lowering element uses the identity \(\alpha(t)=r^{-1}\). It follows that \(\delta\) centralizes both the raising and lowering elements, say \(e\) and \(f\). Invariance gives \[(\delta,2\lambda) =(\delta,[e,f])=([\delta,e],f)=0.\] Applying the same argument to the aligned triple at the second embedding gives \((\delta,2\lambda')=0\). Subtracting proves \((\delta,\delta)=0\), whence \(\lambda'=\lambda\). Equation (60) proves (i).

The common cocharacter \(\mu\) is defined over \(\mathbb Q\), since \(T\) is split. Put \(H_\mu=d\mu(1)\). The equations in \(e,f\in\widehat{\mathfrak g}\) \[ \begin{gathered} [H_\mu,e]=2e,\qquad [H_\mu,f]=-2f,\qquad [e,f]=H_\mu,\\ \operatorname{Ad}(t)e=re,\qquad \operatorname{Ad}(t)f=r^{-1}f \end{gathered} \tag{61}\] are polynomial equations over \(\overline{\mathbb Q}\) and have a complex solution by the alignment above. They consequently have a solution over \(\overline{\mathbb Q}\). In characteristic zero, the resulting \(\mathfrak{sl}_2\) homomorphism integrates to an algebraic homomorphism \(\phi:\operatorname{SL}_2\to\widehat G\). Its diagonal cocharacter is \(\mu\), because cocharacters in characteristic zero are determined by their differentials. This includes the zero triple and the trivial homomorphism.

Choose an algebraic square root of \(r\) and set \(c=t\mu(\sqrt r)^{-1}\in T(\overline{\mathbb Q})\). Equation (61) says that \(c\) centralizes the triple, hence its connected algebraic image \(\phi(\operatorname{SL}_2)\). At every embedding and for every \(a\in X^*(T)\), \[|\iota(a(c))| =r^{\langle a,\lambda\rangle} r^{-\langle a,\mu\rangle/2}=1.\] Thus the cyclic subgroup generated by \(\iota(c)\) has compact closure in \(T(\mathbb C)\). This closure is contained in the closed subgroup \(Z_{\widehat G}(\iota\phi(\operatorname{SL}_2))(\mathbb C)\), and is conjugate into a maximal compact subgroup of that reductive group (Adams and Taibi 2016, Theorems 3.6 and 3.7). This argument also applies to a disconnected triple centralizer and proves (ii).

For uniqueness, first observe that any compact–Jacobson–Morozov decomposition of \(t\) has diagonal cocharacter \(\mu\) after the same torus alignment. Decompose the Lie algebra by \(H_\mu\) as \(\widehat{\mathfrak g}=\bigoplus_j\widehat{\mathfrak g}_j\). For any triple \((e,H_\mu,f)\), elementary \(\mathfrak{sl}_2\) representation theory gives \[[\widehat{\mathfrak g}_0,e]=\widehat{\mathfrak g}_2:\] in every irreducible \(\mathfrak{sl}_2\) summand having weight two, raising maps weight zero onto weight two. The orbit of \(e\) under \(Z_{\widehat G}(H_\mu)^\circ\) is therefore open in the irreducible vector space \(\widehat{\mathfrak g}_2\). There can be only one such open orbit. Hence the neutral cocharacter determines the nilpotent orbit, without an exception for very-even orbits in type \(D\).

Finally conjugate \(\phi\) over \(\overline{\mathbb Q}\) to the selected representative \(\phi_{\mathcal O}\) of its orbit. The conjugated remainder satisfies precisely the all-embeddings condition defining \(H_{\mathcal O}^c(\overline{\mathbb Q})\) (Gaitsgory et al. 2026, sec. 3.1.1 and 3.4.1–3.4.2). This proves (iii). ◻

Proof. Take the irreducible constituents \(\mathcal W_j\) of a Jordan–Hölder filtration of \(\mathcal V\). After a finite extension of coefficients they are defined over finite extensions of \(\mathbb Q_\ell\). At each closed point the characteristic polynomial of \(\mathcal V\) is the product of those of its constituents, so their eigenvalues are still algebraic. It suffices to treat one constituent \(\mathcal W\) of rank \(n\).

The rank-one sheaf \(\det\mathcal W\) has finite geometric monodromy (Deligne 1980, Proposition 1.3.4). Fix a closed point \(y\in|U|\) of degree \(d_y\) and choose an algebraic number \(b\) such that \[b^{n d_y} =\det(\operatorname{Frob}_y\mid\mathcal W).\] Normalize the degree map on the Weil group by \(\deg(\operatorname{Frob}_x)=d_x\). The number \(b\) is an \(\ell\)-adic unit: the original representation has compact image, so the eigenvalues of every group element, and in particular its determinant, are \(\ell\)-adic units. The constant-field character \(b^{\deg}\) consequently extends continuously from the Weil group to the arithmetic fundamental group. Set \(\mathcal W^0=\mathcal W\otimes b^{-\deg}\).

Its determinant still has finite geometric monodromy, and has value one on \(\operatorname{Frob}_y\). A power killing its geometric monodromy factors through \(\operatorname{Gal}(\overline{\mathbb F}_q/ \mathbb F_q)\); its value on an element of degree \(d_y\) is one, so this constant-field character has finite order. Thus \(\det\mathcal W^0\) has finite order. This is also the determinant normalization described in (Deligne 1980, sec. 1.3.6).

Laurent Lafforgue’s purity theorem applies to the irreducible lisse sheaf \(\mathcal W^0\) on the smooth curve \(U\): its finite-order determinant implies that all its Frobenius eigenvalues are algebraic and have modulus one under every complex embedding (Lafforgue 2002, Theorem VII.6(i)–(ii)). Therefore every Frobenius eigenvalue \(\alpha\) of \(\mathcal W\) at \(x\) satisfies \[\log_{q_x}|\alpha|=\log_q|b|.\] This value is independent of \(x\). Taking the union over the fixed list of constituents, with their ranks and multiplicities, proves the assertion. Neither the rank-one theorem nor the purity theorem requires \(U\) to be proper. ◻

Proof. Choose an embedding \(\overline{\mathbb Q}\hookrightarrow\overline{\mathbb Q}_\ell\) and view \(\chi\) as an \(\ell\)-adic character. V. Lafforgue’s global parameter theorem gives a continuous semisimple parameter \[\sigma:\pi_1(U)\longrightarrow\widehat G(\overline{\mathbb Q}_\ell),\] defined over a finite extension of \(\mathbb Q_\ell\), for which the semisimple class of \(\sigma(\operatorname{Frob}_x)\) is the Satake class of \(\chi\) at every \(x\in|U|\) (Lafforgue 2017, Theorem 1.1). Here is why one parameter can be chosen for this joint character. Its nonzero joint eigenspace is preserved by the commuting excursion operators. Their finite-dimensional image has a character on that eigenspace. The parameter attached to such an excursion character has the required good-place Satake evaluations. We need neither uniqueness of this lift of \(\chi\) nor reducedness of the excursion algebra.

For an algebraic representation \(V\) of \(\widehat G\), the composition \(V\circ\sigma\) is a lisse sheaf on \(U\). Its Frobenius eigenvalues are algebraic, since the corresponding semisimple Satake classes are algebraic. Lemma 40 shows that their multiset of logarithmic absolute values, with base \(q_x\), is independent of \(x\).

For completeness, this determines the exponent in the dual torus. Let \(V_a\) have dominant highest weight \(a\). If \(\lambda_{\chi,x}\) is dominant, every weight \(a'\) of \(V_a\) satisfies \[\langle a',\lambda_{\chi,x}\rangle \le\langle a,\lambda_{\chi,x}\rangle.\] The largest Frobenius exponent in \(V_a\circ\sigma\) is therefore \(\langle a,\lambda_{\chi,x}\rangle\), and it is independent of \(x\). The dominant weights in the character lattice of the dual maximal torus span its real weight space: a sufficiently divisible positive multiple of each fundamental weight belongs to this lattice. Their pairings determine \(\lambda_{\chi,x}\). This argument applies to every isogeny type of the semisimple dual group. ◻

Proof. If the cusp space is zero, the assertion is immediate. Otherwise fix \(\chi\in\Sigma_D\) and a good place \(x\). At any complex realization, a nonzero vector of \(C_\chi\) occurs in the discrete cuspidal automorphic spectrum. Choose a nonzero projection of such a vector to an irreducible cuspidal constituent. This projection retains the joint Hecke character and is spherical at \(x\). Lemma 26 shows that its Iwahori invariants contain the dominant ordering of its spherical Bernstein character. Theorem 25 therefore puts the spherical parameter in compact–Jacobson–Morozov form.

The same statement applies to every Galois-conjugate character: \(C_{D,\mathbb Q}\) and its Hecke action are defined over \(\mathbb Q\), so every such character occurs again in Equation (58). By Equation (62) and Remark 39, every embedding of an algebraic torus representative \(t_{\chi,x}\) itself has compact–Jacobson–Morozov form. Lemma 38 then gives a unique orbit \(\mathcal O_{\chi,x}\) with \[[t_{\chi,x}]\in R_{q_x,\mathcal O_{\chi,x}}.\] Proposition 41 makes its dominant exponent independent of \(x\). The orbit uniqueness in Lemma 38 thus makes \(\mathcal O_{\chi,x}\) independent of \(x\) as well; denote it by \(\mathcal O_\chi\).

We have partitioned the finite set \(\Sigma_D\) into the subsets \(\Sigma_{D,\mathcal O}=\{\chi:\mathcal O_\chi=\mathcal O\}\). They are Galois-stable by Remark 39 and Equation (62). Let \(e_\chi\) be the primitive idempotent of \(\overline{\mathbb Q}\otimes\mathbb T_{D,\mathbb Q}\) associated with \(\chi\). The sum \[e_{\mathcal O} =\sum_{\chi\in\Sigma_{D,\mathcal O}}e_\chi\] is Galois-fixed, so belongs to \(\mathbb T_{D,\mathbb Q}\). These idempotents are orthogonal and sum to one. They yield a rational direct sum decomposition with summands \(e_{\mathcal O}C_{D,\mathbb Q}\).

It remains to identify them with the given cyclic-support spaces. For \(f\in C_{D,\mathbb Q}\) write \(f=\sum_\chi f_\chi\) after scalar extension using Equation (58). At a fixed place \(x\), the character support of \(\overline{\mathbb Q}\otimes A_x f\) is exactly \[\{\chi|_{A_x}:f_\chi\ne0\}.\] Indeed, the local eigenspace for a restricted character is the direct sum of the joint eigenspaces with that restriction, and the projection of \(f\) to it is nonzero precisely when one of the corresponding \(f_\chi\) is nonzero. It follows that this cyclic support is contained in \(R_{q_x,\mathcal O}\) for every good \(x\) if and only if every \(\chi\) with \(f_\chi\ne0\) has \(\mathcal O_\chi=\mathcal O\). Here we use the pairwise disjointness of the orbit sets. The condition is therefore equivalent to \(e_{\mathcal O}f=f\), proving \(e_{\mathcal O}C_{D,\mathbb Q}=C_{D,\mathcal O,\mathbb Q}\). This is the spectral-support definition of (Gaitsgory et al. 2026, sec. 3.4.3), with the closed points restricted to \(U\). Equation (63) follows. ◻

Proof of Theorem 1. Proposition 42 proves the rational assertion. Tensor Equation (63) with \(\overline{\mathbb Q}_\ell\) over \(\mathbb Q\). The resulting summands are exactly \(C_{D,\mathcal O,\ell}\) by their definition. Scalar extension is exact, so their natural addition map to \(C_{D,\ell}\) is an isomorphism. ◻

Proof of Corollary 2. Choose the split model of \(G\) over the constant field with the same root datum. Its base change to \(F\) is isomorphic to the given group. Let \(S\) be the finite set of places at which \(\pi\) is ramified. For a split adjoint group all hyperspecial maximal compact subgroups at a place are conjugate, so unramifiedness is equivalent to having \(G(\mathcal O_x)\)-fixed vectors for this model. Smoothness of the local representations and their restricted tensor product allow an effective divisor \(D\) supported on \(S\) with \(\pi^{K_D}\ne0\): at each point of \(S\), take a principal congruence subgroup fixing a nonzero local vector. The spherical Hecke algebras outside \(S\) act on \(\pi^{K_D}\) through the joint character determined by the local components \(\pi_x\). This character occurs in the complex cusp space \(C_{D,\mathbb C}=\mathbb C\otimes_{\mathbb Q}C_{D,\mathbb Q}\). By Lemma 37, it and its Satake classes are algebraic. Choose \(\iota:\overline{\mathbb Q}\hookrightarrow\mathbb C\) realizing this character, and write \([t_x]\) for its algebraic normalized Satake classes. The complex scalar extension of Theorem 1 and the pairwise disjointness in Lemma 38 give a single orbit \(\mathcal O\) such that \[[t_x]\in R_{q_x,\mathcal O} \qquad (x\notin S).\] We use positive real square roots in the complex normalized Satake parameters of \(\pi_x\). Their comparison with \(\iota(t_x)\) has at most the finite central correction in Remark 39, which changes neither the orbit nor the polar exponent.

At the place \(v\), where \(\pi_v\) is generic, choose an algebraic representative \(t_v\in\widehat T(\overline{\mathbb Q})\) in a maximal torus of \(\widehat G\). Let \(\nu_v\in X_*(\widehat T)\otimes\mathbb R\) be its polar exponent, so \(|\iota(a(t_v))|=q_v^{\langle a,\nu_v\rangle}\) for every \(a\in X^*(\widehat T)\). The torus alignment in Lemma 38 identifies \[\nu_v=\tfrac12\mu_{\mathcal O},\] where \(\mu_{\mathcal O}\in X_*(\widehat T)\) is the diagonal cocharacter of a Jacobson–Morozov homomorphism for \(\mathcal O\). For every weight \(a\) of an algebraic representation of \(\widehat G\), \[\langle a,\nu_v\rangle =\tfrac12\langle a,\mu_{\mathcal O}\rangle \in\tfrac12\mathbb Z.\] Thus the complex Satake parameter of \(\pi_v\) is geometric in the sense of (Ciubotaru and Harris 2023, Definition 6.3). Since \(G\) is adjoint, \(\widehat G\) is simply connected; in particular, the adjoint and fundamental representations required in their Corollary 6.32 are algebraic representations of \(\widehat G\) and satisfy the displayed condition.

The cuspidal representation \(\pi\) occurs in the unitary cuspidal \(L^2\) spectrum, so each local component is unitarizable. The component \(\pi_v\) is therefore spherical, generic, and unitary. By (Ciubotaru and Harris 2023, Corollary 6.32(1)), its polar exponent is zero. Hence \(\mu_{\mathcal O}=0\). The \(\mathfrak{sl}_2\) relations then make the raising and lowering elements zero, so \(\mathcal O=\{0\}\). For this orbit the Jacobson–Morozov homomorphism is trivial, so every class in \(R_{q_x,\{0\}}\) has compact image under \(\iota\). A spherical representation with compact normalized Satake parameter is tempered. This proves the assertion at every \(x\notin S\).

Finally, a nonzero global Whittaker functional on \(\pi\) induces a nonzero local Whittaker functional at each place. Indeed, choose a pure tensor on which the global functional is nonzero and fix all its factors except the chosen local one. The resulting functional on that factor has the required local equivariance and is nonzero. A globally generic \(\pi\) is therefore generic at every place and unramified outside the finite set \(S\), so it satisfies the hypothesis of the corollary. ◻