Proof. This is [27]; we explain the local hypothesis and the collision assertion needed here. In a coordinate-and-input-frame cover a bounded step is a product with a split Schubert model. For an arbitrary preceding space \(Y\) and a complex \(K\) on it, the comparison for the next twisted product is therefore the exterior-product map \[\Psi_YK\boxtimes\Psi_{\mathop{\mathrm{Gr}}}\operatorname{Sat}(V) \xrightarrow{\sim} \Psi_{Y\times_S\mathop{\mathrm{Gr}}}(K\boxtimes\operatorname{Sat}(V)).\] The first factor need not be smooth. The specialized Grassmannian kernel is the Satake kernel with the same label and normalization; [27] fixes a single tensor identification for these kernels. Frame descent and proper push give the one-step comparison.

For successive steps, take \(Y\) to retain the preceding modifications and all leg parameters. The same calculation is valid after setting any collection of legs equal. Proper convolution then gives the fused comparison, and naturality of the exchange maps identifies it with the comparison obtained by successive pushes. Thus the diagonal exchange, which need not be invertible on arbitrary sheaves, is invertible on these Hecke constructions and respects nested diagonals. Permutations are the fusion symmetry of the kernels, and the identity-modification kernel gives the unit. Transported levels at all marked points, and node types away from the disc, change none of these frame calculations. This verifies the application of the cited proposition to every stack and family used here. Finally (9) identifies the specialization of the eigenvalue side, naturally in all its tensor maps, and hence specializes the coherence diagrams themselves. ◻

Proof. We adapt the rational-reduction proof of [27] to all the marked flags. This is the parabolic form of Ginzburg’s isotropy argument [17]; compare the separable-lifting criterion and its bundle-stack application in [1]. For \(\mathrm{SL}_n\) the required flag is already rational. First take ordinary flags, and let \(Z\) be a reduced irreducible component of \(\Lambda_S\), with function field \(F\). Its generic point supplies \((P,(\beta_i),\phi)\) on \(X_F\), together with its lift to \(S\). Over \(F(X)\), choose a rational full flag in the underlying rank-\(n\) vector space that makes the nilpotent field strictly triangular. Properness of the associated flag bundle extends this Borel reduction to the whole smooth curve \(X_F\). Denote it by \(Q\). Its nilradical bundle is a subbundle of \(\mathop{\mathrm{ad}}(P)\), so generic containment extends to \[\phi\in H^0\bigl(X_F, (Q\times^B\mathop{\mathrm{Lie}}N_B)\otimes\omega_{X_F}(D)\bigr).\] No separable or inseparable extension of \(F\) has been used.

For each \(i\), fix the relative position of \(Q_{x_i}\) and \(\beta_i\). Let \(\mathcal Y\) classify a global Borel reduction and flags with these prescribed relative positions. This is a smooth stack: \(\mathop{\mathrm{Bun}}_B\) is smooth by the vanishing of curve \(H^2\), and its added fibers are products of Borel orbits in \(G/B\). These orbits are smooth, since intersections of two Borels have their smooth torus-and-root-group description. Its deformation bundle consists of sections of \(\mathop{\mathrm{ad}}(Q)\) whose values at \(x_i\) lie in \(\mathop{\mathrm{Lie}}Q_{x_i}\cap\mathfrak b_{\beta_i}\).

The pullback of \(\phi\) as a covector on \(\mathcal Y\) is zero. Its pairing with this deformation bundle has no pole at any \(x_i\) by the original residue conditions. Generically the pairing is zero because the nilradical annihilates the Borel Lie algebra under the trace form. It is therefore zero everywhere, and Serre duality gives the asserted vanishing on deformations.

Spread the reduction and the fixed positions to a dense open of \(Z\), then restrict to its smooth locus. The canonical one-form on \(T^*S\) vanishes on this open: it evaluates \(\phi\) on the induced bundle deformation, which factors through \(\mathcal Y\). Directions relative to \(\mathcal B\) are also annihilated, since the chart covector comes from \(T^*\mathcal B\). Its exterior differential, the symplectic form, vanishes there as well. An isotropic subspace in a \(2d\)-dimensional symplectic space has dimension at most \(d\), proving the bound. The unlevelled case is the same proof with the conditions at the \(x_i\) omitted.

Products preserve the bound. A smooth cotangent pullback adds the relative dimension of the smooth map to the dimension of the cone; its canonical one-form is the pullback of the original one. The cone descriptions above therefore give the enhanced and quotient cases. In characteristic zero the rational-flag argument may start with any reduced irreducible subvariety of \(\Lambda_S\), giving the stated isotropy on its smooth locus. ◻

Proof. Write \(a(0)=s+v\) for its commuting semisimple and nilpotent parts. Both have trace zero, and \(s\ne0\). Diagonalize \(s\) and let \(M\) be its block centralizer. On the off-block matrices \(\mathop{\mathrm{ad}}(s)\) is invertible, whereas \(\mathop{\mathrm{ad}}(v)\) is nilpotent and commutes with it. Their sum is therefore invertible there.

For block sizes \(d_i\), the differentials of central cocharacters span \(\mathfrak z(\mathfrak m)=\{(c_i):\sum d_ic_i=0\}\). Explicitly, the integer vectors having entries \(d_j,-d_i\) in positions \(i,j\) and zero elsewhere lie in the cocharacter lattice, and span this hyperplane over the ground field because the \(d_i\) are invertible. The trace form on the center is nondegenerate, as checked above, and \(v\) is orthogonal to all block scalars. Some integral central cocharacter thus has \(\kappa(a(0),d\lambda)=\kappa(s,d\lambda)\ne0\).

Finally remove the off-block coefficient of \(t^r\) inductively for \(r\ge1\). Conjugation by an element congruent to \(1+t^rY\) modulo \(t^{r+1}\) changes that coefficient by \([Y,a(0)]\). The invertibility on off-block matrices supplies \(Y\), and smoothness of \(\mathrm{SL}_n\) supplies a group element with this first-order value. The successive changes converge in \(\mathrm{SL}_n[[t]]\), giving the first assertion of (11) without changing the constant term. ◻

Proof. We give the local calculation of [27] and explain its compatibility with all the levels. Choose \(y\) away from every marking where the field representing \(A\) is nonnilpotent. Write it as \(a(t)\,dt\) in a formal frame. Apply Lemma 7 and modify by \(\lambda(t)\), with adjoint lattices \[L=\mathfrak g[[t]],\qquad L'=\operatorname{Ad}(\lambda(t))L, \qquad J=L\cap L'.\] As \(a(t)\in\mathfrak m[[t]]\) and \(\lambda\) centralizes \(M\), this field is regular in both lattices. It extends to an output field \(A'\) with the same residues at all marked points.

The two fixed-side tangent spaces and their residue pairing are \[ T_hH_{\mathrm{in}}=L/J,\qquad T_hH_{\mathrm{out}}=L'/J, \qquad (D,C)\longmapsto\mathop{\mathrm{Res}}\kappa(D,C)\,dt. \tag{13}\] Indeed input disc gauge transformations move \(L'\), output disc gauge transformations move \(L\), and their common stabilizer has Lie algebra \(J\). For the integer weight decomposition \(\mathfrak g=\bigoplus_j\mathfrak g_j\) of \(\lambda\), these quotients are \[ L/J=\bigoplus_{j>0}\mathfrak g_j\otimes \Bbbk[[t]]/(t^j),\qquad L'/J=\bigoplus_{j<0}\mathfrak g_j\otimes t^j\Bbbk[[t]]/\Bbbk[[t]]. \tag{14}\] Opposite weights pair perfectly, and \(t^i\) pairs with \(t^{-1-i}\). Thus (13) is a perfect pairing.

Holding output and leg fixed, a vector \(C\in L'\) changes the input by its principal part. Its pairing with \(p^*A\) is \(\mathop{\mathrm{Res}}\kappa(a(t),C)\,dt=0\), because \(a(t),C\in L'\) and \(L'\) is its own annihilator under residue. The smooth cotangent sequence for \(q_H\) now says that \(p^*A\) descends to output bundle times leg. Its bundle component is \(A'\): principal-part deformations supported away from \(y\) and the markings test this assertion where the bundles are identified. They span the deformation space, since adding a sufficiently large pole divisor there kills \(H^1\) of the deformation bundle. For the leg component replace \(t\) by \(t-z\) while holding output fixed. The logarithmic derivative of the transition \(\lambda(t-z)\) is \(-d\lambda/t\). Serre duality therefore gives \[\xi=\pm\kappa(a(0),d\lambda)\,dz\ne0,\] where the common sign depends only on the gluing convention. This proves (12). It also shows that the second section vanishes at \(h\), since on \(\widetilde p\)-vertical vectors both \(p^*A\) and the leg differential vanish.

To compute the derivative of the first section, move the output lattice in the input fiber in direction \(D\in L/J\) and transport a \(q_H\)-vertical test vector \(C\in L'/J\) with it. Differentiation gives, up to sign, \[ \mathop{\mathrm{Res}}\kappa(a(t),[D,C])\,dt =\mathop{\mathrm{Res}}\kappa([a(t),D],C)\,dt. \tag{15}\] This is independent of the lifts: \(\mathop{\mathrm{ad}}(a(t))\) preserves \(L,L'\) and \(J\). It preserves every \(\lambda\)-weight summand, and on each nonzero weight its constant term is invertible by (11). It is consequently invertible on the truncated modules in (14). The perfect pairing (13) shows that (15) is perfect. It is exactly the derivative of the section at its zero. For the second section, hold the output fixed and move \(L\) by \(C\in L'/J\). The calculation exchanges \(C\) and \(D\), giving the transpose of the same perfect pairing, up to sign. Both zeros are therefore nondegenerate and isolated. Only integer truncation lengths occur; no cocharacter weight is inverted in the ground field.

Every calculation was made at \(y\). Away from \(y\) all the flags and their residue conditions are unchanged, so the proof applies with any number of ordinary or enhanced levels. On a product it acts in one factor. On the quotient it descends from that calculation: unchanged residues preserve annihilation of the diagonal torus, and the Hecke correspondence is obtained by the smooth quotient base change of Section 2.1. ◻

Proof. The Hecke eigenrelation first gives a vanishing after proper pushforward. The two nondegenerate zeros of Proposition 8 then have different roles: one isolates a single stalk in that pushforward, and the other gives local coordinates in which the equation is the original hypersurface equation plus a split quadratic form. A projective compactification will recover the cycles of the original hypersurface from those of the stabilized one.

1. The modified hypersurface and the eigenrelation. If \(df_d\) is nonzero on the tangent space relative to \(b\), the map \((b,f):D\to\mathcal B\times\mathbb A^1\) is smooth near \(d\) in the field case, and \(\{f=0\}\to\mathcal B\) is smooth near \(d\) in the trait case. Smooth base change proves the assertion. We may therefore assume \(df_d=b^*A\) for a covector \(A\notin\Lambda\) at \(b(d)\).

Choose the modification \(h\) supplied by Proposition 8, at a leg \(y\). Write \(H\) for its exact-position Hecke stratum, with leg map \(l:H\to L\), and write \[p:H\longrightarrow\mathcal B,\qquad o:H\longrightarrow\mathcal B,\qquad q_H=(o,l):H\longrightarrow\mathcal B\times L, \qquad \widetilde p=(p,l).\] Both \(q_H\) and \(\widetilde p\) are smooth of relative dimension \(m\). At \(h\) the proposition gives \[ p^*A=q_H^*(A',\xi),\qquad \xi\ne0. \tag{16}\] Use the two fibers named in that proposition: \[H_{\mathrm{in}}=\widetilde p^{-1}(b(d),y),\qquad H_{\mathrm{out}}=q_H^{-1}(o(h),y).\] On \(H_{\mathrm{in}}\), the restriction of \(p^*A\) to the \(q_H\)-relative tangent has a nondegenerate zero at \(h\); on \(H_{\mathrm{out}}\), the restriction of \(o^*A'\) to the \(\widetilde p\)-relative tangent does too. Each fiber retains the identification with its fixed-side bundle.

In the trait case these are special-fiber statements; take the corresponding relative Hecke stratum over \(S_0\). In the field case introduce the completed local trait \(S_0\) at \(0\) in the function line, with parameter \(z\), extend all spaces constantly, and impose \(f=z\). We will apply vanishing cycles over this trait. Thus the two cases have a common notation: \(e=0\) in the trait case and \(e=z\) in the field case, and the operation to be computed is, respectively, \(\Psi\) and \(\Phi\).

Take two copies \(H_1,H_2\) of \(H\) and form \[ W_0=H_2\times_{p,\mathcal B}D, \qquad P=H_1\times_{q_H,\mathcal B\times L,q_H}W_0. \tag{17}\] Thus the two modifications have the same output bundle and leg, while the input of the second lies in the chart \(D\); Figure 1 records these two different roles.

Let \(c:P\to D\) be the projection through \(W_0\). Set \(W=\{f=e\}\subset W_0\) and \(P_c=\{f(c)=e\}\subset P\). The distinguished points are \(w=(h,d)\) in \(W_s\) and \(a=(h,h,d)\) in \((P_c)_s\). Products here are over the trait when one is present.

The output-bundle map \(W\to\mathcal B\) is smooth near \(w\). Indeed, \(W_0\to\mathcal B\times L\) is smooth, and (16) says that the cut differential, on directions relative to the output bundle, has the nonzero leg component \(\xi\). The eigenisomorphism therefore implies that the cycles of the pulled-back Hecke transform vanish at \(w\). In the trait case this is smooth base change from \(\Psi F=0\), followed by the lisse tensor compatibility. In the field case \(W\to\mathcal B_{S_0}\) is smooth over the function trait, where \(\mathcal B_{S_0}\) is the constant stack. Pullback of the constant family with coefficient \(F\) therefore has zero vanishing cycles, and tensoring by the lisse eigenvalue preserves this assertion. Only the smooth map to the bundle stack enters this argument.

2. Isolating one stalk in the proper pushforward. Replace \(H_1\) by its Schubert bound \(\overline H_1\) and let \(\mathop{\mathrm{IC}}_\lambda\) be the simple Satake kernel of this bound. Set \[\overline P_c=\overline H_1\times_{\mathcal B\times L}W, \qquad \pi_c:\overline P_c\longrightarrow W.\] This map is proper. Let \(\mathcal J\) be the pullback to \(\overline P_c\) of the Hecke integrand \(F\,\widetilde\boxtimes\,\mathop{\mathrm{IC}}_\lambda\) on \(\overline H_1\), with the twisted-product convention of (8). In the field case extend the integrand constantly in \(z\) before pulling back to the cut; thus \(\mathcal J\) is defined on the whole \(z\)-family. In the trait case it is defined on the geometric generic fiber. Proper base change identifies \(R\pi_{c,*}\mathcal J\) with the pullback to \(W\) of the Hecke transform used in Step 1.

Let \(T=\pi_c^{-1}(w)\) and let \(C\) be the restriction to \(T\) of \(\Phi\mathcal J\) in the field case, or of \(\Psi\mathcal J\) in the trait case. Proper compatibility of cycles and the vanishing in Step 1 give \[ R\Gamma(T,C)=0. \tag{18}\] On the open stratum \(P_c\subset\overline P_c\) the kernel is \(E[m](m/2)\). We will isolate the contribution of \(a\) there.

The open part of \(T\) is exactly \(H_{\mathrm{out}}\) for \(H_1\), since \(w\) fixes \(H_2=h\) and \(c=d\). Thus (16) at the fixed second modification identifies \(d(f\circ c)\) along this whole fiber with the pullback of \((A',\xi)\) along \(q_{H,1}\). Before cutting, \(P\to\mathcal B\) by the first input bundle is smooth. At a point of \(T\) in its open-orbit neighborhood, failure of smoothness after cutting would mean that \(d(f\circ c)\) comes from the cotangent space of that first input bundle. Since \(P\to H_1\) is smooth, its cotangent map is injective; thus failure would already give on \(H_1\) an equality \[q_{H,1}^*(A',\xi)=p_1^*A_1\] for some \(A_1\). On \(\widetilde p_1\)-relative tangents this forces \(o_1^*A'\) to vanish. The nondegenerate zero on \(H_{\mathrm{out}}\) shows that \(a\) is isolated among such points of \(T\). Consequently \(C\) vanishes on a punctured neighborhood of \(a\) in \(T\): there the cut map to the first input stack is smooth and the open-orbit Satake kernel is an invertible constant shift and twist. This smoothness is over the trait. In the field case the integrand therefore pulls back the constant \(F\)-family on \(\mathcal B_{S_0}\), so it has zero \(\Phi\), as required; the trait case uses \(\Psi F=0\).

Here is the precise consequence of this isolation. If a constructible complex on a proper scheme vanishes on a punctured neighborhood of a closed point, its contribution at that point is a direct summand of its global cohomology. To see this, let \(j:T\setminus\{a\}\to T\) and \(i:\{a\}\to T\). The restriction \(j^*C\) is zero near \(a\), so \((Rj_*j^*C)_a=0\). The localization triangle consequently splits as the direct sum of \(i_*C_a\) and \(Rj_*j^*C\); both connecting morphisms vanish by these disjoint supports. The same localization argument applies to algebraic spaces. Equation (18) now gives \[ C_a=0. \tag{19}\]

3. An exact split quadratic equation at the isolated point. We identify the local equation at this isolated point exactly. The diagonal \(W_0\to P\), given by making the two modifications equal, is a section of the smooth map \(P\to W_0\). Étale locally at \(a\), there is a map \(r:P\to D\) lifting the first-input map to \(\mathcal B\), whose restriction to this diagonal is the original chart point, including its identification with the first input. Indeed, the smooth morphism \(P\times_{\mathcal B}D\to P\) has that section on the diagonal. To extend it, choose relative smooth coordinates, giving an étale map from a neighborhood of its value to \(\mathbb A^\delta_P\), where \(\delta\) is the relative dimension of \(D\to\mathcal B\) at \(d\). The coordinates of the prescribed section are functions on the diagonal; lift these functions to a neighborhood in \(P\) and take their graph in \(\mathbb A^\delta_P\). Pullback of the étale map along this graph gives an étale neighborhood of \(P\) with a lift to \(P\times_{\mathcal B}D\). Retain the branch carrying the prescribed section along the diagonal. The resulting projection to \(D\) is \(r\), and the fiber product retains the specified stack isomorphism between \(b\circ r\) and the first input. Thus \(c\) records the second input everywhere, whereas \(r\) records the first input in this étale neighborhood; the two agree on the diagonal. The diagonal, being a section of the smooth morphism \(P\to W_0\) of relative dimension \(m\), is a regular immersion of codimension \(m\). Choose relative coordinates \(v_1,\ldots,v_m\) generating its ideal. Since \(r=c\) on it, there are functions \(u_i\) with \[ f(c)-f(r)=\sum_{i=1}^m u_i v_i. \tag{20}\] In particular, the subsequent change of coordinates will give the exact cut equation.

On the diagonal fiber with \(c=d\) and leg \(y\), which is the fixed-input, fixed-leg fiber of \(H\) through \(h\), the conormal coefficients \(u_i\) are the negatives of the restriction of \(p^*A\) to the \(q_H\)-relative tangent, in the frame dual to the \(v_i\). In fact variation of the first modification with the second fixed has \(dc=0\), whereas \(b\circ r\) is its first input, so differentiating (20) gives precisely this restriction. This calculation is independent of the chart-vertical part of \(dr\): \(df_d=b^*A\) annihilates \(\ker(db_d)\). The description holds along that entire diagonal fiber. The nondegenerate zero on \(H_{\mathrm{in}}\) shows that \(u_i(a)=0\) and that the derivatives of the \(u_i\) on that fiber form a basis. The map \(W_0\to D\times L\) is the smooth base change of \(\widetilde p\), of relative dimension \(m\). On the diagonal, \(r=c\) and the leg provide the other coordinates; the \(v_i\) provide its normal coordinates. Hence \[ (r,l,u_1,\ldots,u_m,v_1,\ldots,v_m): P\longrightarrow D\times L\times\mathbb A^{2m} \tag{21}\] is étale near \(a\), with the evident relative interpretation over \(S_0\). On this neighborhood of the cut, \(\mathcal J|_{P_c}\simeq r^*K[m](m/2)\). Thus (19) says that the cycles of the pulled-back \(K\) vanish at the vertex of \[ f+\sum_i u_i v_i=e. \tag{22}\] For \(m=0\) this is already the required assertion, after removing the smooth leg factor.

4. Recovering the original hypersurface. We have proved vanishing after adjoining the split quadratic variables. To remove them, we use a proper family whose extra cohomology is supported precisely on the original cut \(\{f=e\}\). For \(m>0\), put \(B=D\times L\) and compactify (22) to the proper quadric family \[ \pi:Q=\left\{\sum_{i=1}^m u_i v_i+(f-e)w^2=0\right\} \subset\mathbb P^{2m}_B\longrightarrow B. \tag{23}\] In the field case \(B\) also carries the constant extension in \(z\). Every point above \((d,y)\) other than the vertex \([0:\cdots:0:1]\) is smooth for \(\pi\), since some \(u_i\) or \(v_i\) is nonzero. The cycles of \(\pi^*K\) there vanish by smooth base change; they vanish at the vertex by (21). Proper compatibility therefore gives zero cycles for \(R\pi_*\pi^*K\) at \((d,y)\).

The split-quadric calculation in [27] gives the precise extra term we need. Write \(i:Z=\{f=e\}\hookrightarrow B\). Hyperplane powers give a triangle \[ \bigoplus_{a=0}^{2m-1}E_B(-a)[-2a]\longrightarrow R\pi_*E_Q \longrightarrow i_*E_Z(-m)[-2m]\xrightarrow{+1}. \tag{24}\] Here the first arrow is an isomorphism outside \(Z\): a smooth split quadric of dimension \(2m-1\) has the same \(E\)-cohomology as \(\mathbb P^{2m-1}\). Over \(Z\) the family is the constant projective cone on the split quadric of dimension \(2m-2\). Its cohomology has one additional copy of \(E(-m)\) in degree \(2m\), beyond the hyperplane powers. Proper base change identifies the restriction of the cone of the first arrow with this constant complex on \(Z\); localization therefore identifies the cone as the last term displayed. This explains why the triangle is a statement about sheaves, not merely fiber cohomology dimensions.

Tensor (24) with \(K\) and use the projection formula. In the trait case take this triangle on the geometric generic fiber. The cycles of its first term vanish at \((d,y)\) because \(\Psi K=0\); those of its middle term vanish by the proper calculation above. Its last term therefore has zero nearby cycles there, and proper compatibility for \(i\) proves the claimed vanishing on \(\{f=0\}\). In the field case use the triangle on the whole \(z\)-family. Its first term has zero vanishing cycles because \(K\) is pulled from the constant family \(D\). The identical argument with \(\Phi\) proves \((\Phi_f K)_d=0\). Smooth base change removes the leg factor in both cases. The sheaf operations used throughout are those in Proposition 3. ◻

Proof of Theorem 9. Apply Lemma 11 on every smooth chart and every open subchart. It says that every function whose differential at a point is outside \(\Lambda_D\) is locally acyclic there relative to \(b^*F\). The cone includes the zero section, so these are precisely the nonzero directions that must be excluded. Weak singular support is the closure of the differentials of functions with nonzero vanishing cycles; over an infinite field closed-point tests suffice by constructibility. Barrett proves this description for the present \(E\)-coefficients and proves that weak singular support equals full singular support [3], extending Beilinson’s torsion-coefficient theorem [4]. The hypersurface calculation therefore gives \(\relax(b^*F)\subset\Lambda_D\). The same calculation on charts with additional smooth parameters is available, so the conclusion is compatible with the smooth-chart definition of singular support on the stack. ◻

Proof. The case \(r=1\) is Lemma 11. For \(r>1\) consider the proper incidence projection \[\pi:J=\{(t,[a_1:\cdots:a_r])\in D\times\mathbb P^{r-1}: \textstyle\sum_i a_i f_i(t)=0\}\longrightarrow D.\] At \((d,[a])\) the differential of the defining function on a standard projective chart is \(\sum_i a_i df_i\); derivatives in the projective directions are zero since all \(f_i(d)\) vanish. It is nonzero and lies outside the pulled-back nilpotent cone. The map \(D\times\mathbb A^{r-1}\to\mathcal B\) on each such chart is smooth, so Lemma 11 proves that the nearby cycles of \(\pi^*K\) vanish on the whole fiber \(\mathbb P^{r-1}\) over \(d\). Proper compatibility gives \((\Psi R\pi_*\pi^*K)_d=0\).

Let \(i:Z=\{f_1=\cdots=f_r=0\}\hookrightarrow D\). On the geometric generic fiber hyperplane powers give the triangle \[\bigoplus_{a=0}^{r-2}E_D(-a)[-2a]\longrightarrow R\pi_*E_J \longrightarrow i_*E_Z(-(r-1))[-2(r-1)]\xrightarrow{+1}.\] Indeed, over \(D\setminus Z\) the incidence is a projective bundle of relative dimension \(r-2\), so the first arrow is an isomorphism there. Over \(Z\) it is the product \(Z\times\mathbb P^{r-1}\), and the cone has only its constant top cohomology sheaf. Proper base change and localization identify the cone as written, not merely its stalk dimensions. Tensor by \(K\). The first term has zero nearby cycles by \(\Psi K=0\), and the middle term by the preceding proper calculation. Proper compatibility for \(i\) proves the lemma. ◻

Proof of Theorem 10. Suppose that the closure of the nonzero-stalk locus meets \(D_s\). Shrink to an affine chart smooth of constant relative dimension \(N\) over \(S_0\). The closure of that locus in \(D_{\bar\eta}\) has finitely many irreducible components \(Z_{\bar\eta,\alpha}\). Constructibility gives, on each component, a dense open subset where the stalk of \(K\) is nonzero. After a finite extension of the trait, descend the components and the proper closed complements of these opens. We only descend these finite algebraic data, and continue to use \(K\) over \(\bar\eta\). Let \(Z_\alpha\subset D\) and \(E_\alpha\subset Z_\alpha\) be their horizontal closures. Components not meeting \(D_s\) may be removed.

Write \(j\) for the largest dimension of a generic component whose closure meets \(D_s\), and choose an irreducible component \(Z_0\) of the reduced special fiber of such a closure. A horizontal integral finite-type scheme over this excellent trait, with generic dimension \(j\), has every nonempty special-fiber component of dimension \(j\); its local dimension at a special-fiber closed point is \(j+1\). For these two assertions see, respectively, [30]. The same dimension calculation shows that the special fibers of the horizontal exceptional closures \(E_\alpha\) have dimension at most \(j-1\) whenever the corresponding generic component has dimension at most \(j\).

Choose a general closed point \(d\) of \(Z_0\). We may arrange that \(Z_0\) is smooth at \(d\), that \(d\) is in none of the exceptional closures, and that every reduced special-fiber component of the support closure through \(d\) equals \(Z_0\) near \(d\). Distinct horizontal components may have this same special component; this causes no difficulty. By Proposition 6, \[\dim\Lambda_D\leq N.\] After shrinking a dense open in \(Z_0\), the fiber dimension theorem therefore gives \[ \dim(\Lambda_D)_d\leq N-j. \tag{25}\]

Choose a \(j\)-plane \(V\subset T_d^*D_s\) which avoids \((\Lambda_D)_d\setminus\{0\}\) and whose restriction to \(T_dZ_0\) is an isomorphism onto \(T_d^*Z_0\). Both are nonempty open conditions on the Grassmannian. When \(j>0\), for the first condition projectivize the cone in (25): it has dimension at most \(N-j-1\), and a general \(\mathbb P^{j-1}\) in \(\mathbb P^{N-1}\) is disjoint from it by the elementary incidence dimension count. The second is the usual complement-to-a-fixed-subspace condition. The base field is infinite, so the two opens meet. If \(j=0\), take \(V=0\) and use no cutting functions. Lift a basis of \(V\) to differentials of regular functions \(f_1,\ldots,f_j\) vanishing at \(d\). Their restrictions are parameters on the smooth scheme \(Z_0\). Put \(Y=\{f_1=\cdots=f_j=0\}\subset D\). Lemma 12 says \[ (\Psi(K|_{Y_{\bar\eta}}))_d=0. \tag{26}\]

We finish by showing directly that this stalk is nonzero. Near \(d\), the special fiber of the closed support closure intersected with \(Y\) is zero-dimensional. On any of its selected horizontal components \(Z_\alpha\) of generic dimension \(j\) through \(d\), the local ring has dimension \(j+1\), so its quotient by the \(j\) functions has dimension at least one [30]. Its further quotient by the uniformizer is zero-dimensional. Thus this intersection has a generization of \(d\) in the generic fiber. Such a generization lies outside \(E_\alpha\), since \(d\) was chosen outside its closure, and \(K\) has a nonzero geometric stalk there.

For completeness, the passage from this generization to nearby cycles is local and finite. Let \(Z\subset Y\) be the reduced intersection \(Y\cap\bigcup_\alpha Z_\alpha\). Its generic fiber contains the nonzero-stalk locus of \(K|_{Y_{\bar\eta}}\), the preceding generization lies in \(Z\), and \(d\) is isolated in \(Z_s\). The morphism \(Z\to S_0\) is quasi-finite at \(d\). Take an affine neighborhood in \(Y\) and use the henselian decomposition of its closed support: the local ring of \(Z\) at \(d\) is a finite local \(R\)-algebra and defines an open-and-closed piece \(T\subset Z\) [30]. Its unique special-fiber point is \(d\). Remove the closed complement \(Z\setminus T\) from the ambient neighborhood. On the resulting neighborhood, which has the same nearby-cycle stalk at \(d\), the support closure is the finite \(S_0\)-scheme \(T\).

The generic restriction of \(K\) is the extension by zero from this finite closed support. Proper base change now identifies its nearby-cycle stalk at \(d\) with \[\bigoplus_{z\in T_{\bar\eta}} (K|_{Y_{\bar\eta}})_z.\] This is a finite direct sum of complexes of \(E\)-vector spaces, at least one of which is nonzero. Hence the sum is nonzero, contradicting (26). The argument includes \(j=0\), when there are no cutting functions and the original support already has the required finite local piece. This proves the theorem. ◻

Proof. We use the identification with excursion operators in [24]; the scalar calculation is [27]. Here is the calculation for the free group of rank \(2g+s-1\). Matrix coefficients span \(E[Y]\). Exactness of invariants for the reductive group \(\check G\) implies that every invariant is a finite sum of functions \[ (g_1,\ldots,g_a)\longmapsto \operatorname{Tr}_{W}\left(A\circ\bigotimes_{i=1}^{a}V_i(g_i)\right), \qquad W=\bigotimes_{i=1}^{a}V_i,\quad A\in\operatorname{End}_{\check G}(W). \tag{28}\] Indeed products of matrix coefficients are images of \(\operatorname{End}_E(W)\) under this trace map. A finite sum of these finite-dimensional equivariant images contains a prescribed invariant; exactness of invariants lifts it to invariant endomorphisms in the corresponding finite sum of source spaces.

The excursion for (28) inserts the coevaluation tensor in \(W\otimes W^*\), transports each \(V_i\) around \(\gamma_i\), applies \(A\), and evaluates. The dual factor can remain at \(u\) as an identity leg. Naturality, the tensor constraints, and the unit constraint of the eigenstructure identify this operation with the same tensor calculation on the parameter. Its value is (28) at \(y\). This proves the assertion for every invariant function. In particular the scalar follows from the coherent eigenmaps, not merely from fixed-point eigenisomorphisms. ◻

Proof. This is the closed-orbit frame argument of [27]. The stabilizer of \(y\) is the centralizer of a dense subgroup, hence \(Z(\check G)=1\). Its orbit \(O\) is closed: a one-parameter subgroup giving a limit of the conjugated tuple forces every entry into its associated parabolic. Density forces that parabolic to be all of \(\check G\), so the one-parameter subgroup is central. The affine Hilbert–Mumford closed-orbit criterion proves the claim. Thus \(O\simeq\check G\).

A basis \(v_1,\ldots,v_b\) at \(y\) defines equivariant sections on \(O\) by \(s_j(hy)=hv_j\). Restriction \(E[Y]\otimes V\twoheadrightarrow E[O]\otimes V\) is surjective because \(O\) is closed and \(Y\) is affine. Reductivity in characteristic zero makes invariants exact, so these sections extend equivariantly to \(Y\). Use the extensions as the columns of \(\alpha_V\). Every character of the semisimple group \(\check G\) is trivial, so \(\det V\) is trivial. After choosing a volume form, \(f_V=\det\alpha_V\) is invariant and nonzero at \(y\). The adjugate defines a regular equivariant map \(\beta_V\) on all of \(Y\): on the dense open set \(f_V\ne0\) it equals \(f_V\alpha_V^{-1}\), and this identity verifies equivariance everywhere. The two adjugate identities give (29). ◻

Proof. For a degree-zero natural endomorphism \(f\) of the identity, naturality for \({}^p\tau_{\le r}K\to K\) and the universal property of truncation give \[f_{{}^p\tau_{\le r}K}={}^p\tau_{\le r}(f_K).\] Indeed both sides are the unique endomorphism of the truncation whose composite to \(K\) equals \(f_K\) after the truncation map. The other truncation has the analogous property. Invertibility therefore passes to both truncations, proving that \(\mathscr D_S\) inherits the perverse \(t\)-structure. No exactness of an arbitrary spectral operator has been assumed.

Applying the spectral action to (29) gives maps \[K^{\oplus b}\xrightarrow{\alpha_{V,K}}\mathop{\mathrm{Hecke}}_{V,u}(K) \xrightarrow{\beta_{V,K}}K^{\oplus b}.\] Symmetric monoidality identifies \(f_{\mathop{\mathrm{Hecke}}_{V,u}(K)}\) with \(\mathop{\mathrm{Hecke}}_{V,u}(f_K)\), so the Hecke functor preserves \(\mathscr D_S\). On this subcategory the adjugate identities make \(f_V^{-1}\beta_{V,K}\) an inverse to \(\alpha_{V,K}\). Hence \[ \mathop{\mathrm{Hecke}}_{V,u}|_{\mathscr D_S} \simeq\operatorname{id}_{\mathscr D_S}^{\oplus\dim V}, \tag{30}\] which proves exactness. Hecke transport along a path from \(u\) proves the assertion at any allowed point. The zero representation acts by zero. Compositions, including convolution at a shared point, are therefore exact and preserve \(\mathscr D_S\) as well. ◻

Proof of Proposition 13. Let \(K\) be the Betti realization of \(F\), using Section 2.4. Theorem 9 and singular-support comparison put \(K\) in \(\mathscr D^{\mathrm{lcc}}_\Lambda(\mathcal B)\). Its topological monodromy remains Zariski dense. The topological fundamental group has dense image in the profinite fundamental group; a polynomial equation on the image, after adjoining its coefficients to a finite extension of \(L\), is closed for the adic topology. An equation on topological monodromy therefore holds on the entire adic image. The tuple \(y\) associated with \(K\) satisfies Lemma 15; use this tuple to choose \(S\) and \(\mathscr D_S\). By Lemma 14, \(f_{V,K}=f_V(y)\operatorname{id}_K\), so \(K\in\mathscr D_S\). Lemma 16 puts all of its perverse truncations and cohomologies in this subcategory.

We first prove the bounds needed to truncate moving transforms. For \(A\in\mathscr D_S\) and \(m=|I|\), the functor \[A\longmapsto\mathop{\mathrm{Hecke}}_{I,\boldsymbol V}(A)[m]\] is perverse \(t\)-exact. On a contractible leg patch, (27) identifies its unshifted value with a constant family of fixed-point transforms. The fiber is a composition of fixed-point Hecke functors, using convolution where the legs coincide, and hence has the bounds of \(A\) by Lemma 16. The constant sheaf shifted by \(m\) on the smooth parameter space is perverse. The product description therefore proves the asserted bounds on smooth charts. Applying this exact functor to a truncation triangle and using its universal property yields natural isomorphisms \[ {}^pH^j\bigl(\mathop{\mathrm{Hecke}}_{I,\boldsymbol V}(A)[m]\bigr) \simeq\mathop{\mathrm{Hecke}}_{I,\boldsymbol V}({}^pH^jA)[m]. \tag{31}\]

The same bounds hold for successive operations with all previous leg variables retained. Locally in those variables the input is a constant family whose fixed-leg fibers belong to \(\mathscr D_S\). Applying the next Hecke operator uses (30) on these fibers and (27) for its parameter dependence. Thus the truncation comparison holds throughout each composition diagram. The shift is the dimension of the total retained parameter space, counted once; successive operations at a shared leg add no parameter.

For clarity, let \(\pi:I\twoheadrightarrow J\) be a surjection, let \(\Delta_\pi:U^J\to U^I\) be its collision diagonal, and put \(W_j=\bigotimes_{i\in\pi^{-1}(j)}V_i\). The fusion convention gives \[(1\times\Delta_\pi)^*\mathop{\mathrm{Hecke}}_{I,\boldsymbol V}(A) \simeq\mathop{\mathrm{Hecke}}_{J,\boldsymbol W}(A).\] On the local product families just considered, \[ (1\times\Delta_\pi)^*[\,|J|-|I|\,] \tag{32}\] is perverse exact: it replaces a perverse constant factor on a smooth \(|I|\)-dimensional base by the perverse constant factor on the \(|J|\)-dimensional base. This proves compatibility of (31) with collision restriction, including iterated collisions. Local constancy only away from diagonals would not suffice for this step.

These bounds construct the required comparisons in the adic category itself. Start with the adic truncation triangles of \(F\) and apply the adic Hecke functors. Betti realization preserves perversity and is conservative on the finite-type charts, so the bounds just proved show that their images are truncation triangles. Their universal property therefore supplies (31) adically for \(F\). The same argument applies to the retained-leg compositions and the shifted diagonal restriction (32). In particular no algebraization of maps constructed only in the Betti category is needed.

Write \(\mathcal V_{I,\sigma}=\boxtimes_{i\in I}(V_i)_\sigma\). Shift the original adic eigenmap by \(m\), apply \({}^pH^j\), and use (31). Exterior product with the lisse factor \(\mathcal V_{I,\sigma}[m]\) is perverse exact. We obtain \[\mathop{\mathrm{Hecke}}_{I,\boldsymbol V}({}^pH^jF)[m] \simeq({}^pH^jF\boxtimes\mathcal V_{I,\sigma})[m].\] Removing the common shift gives precisely the required eigenmap \[ \mathop{\mathrm{Hecke}}_{I,\boldsymbol V}({}^pH^jF) \simeq{}^pH^jF\boxtimes\mathcal V_{I,\sigma}, \tag{33}\] with no moving-leg shift.

Finally apply the functorial truncation comparisons to the diagrams of the original coherent eigenstructure. Naturality in representations, the unit, and permutations commute with truncation. The retained-leg comparison proves compatibility with convolution, and (32) proves compatibility with fusion. Every vertex over \(U^I\), after the single shift \([|I|]\), lies in the perverse heart: locally it is a fixed-point exact transform of \({}^pH^jF\) times a lisse parameter factor. After collision to \(U^J\), the adjustment \([|J|-|I|]\) places all vertices in the new heart. For objects \(P,Q\) of such a heart, \(\pi_r\operatorname{Map}(P,Q)=\operatorname{Hom}(P,Q[-r])=0\) for \(r>0\). Thus their mapping spaces are discrete, and the commutative diagrams give all the coherent constraints. The auxiliary choices of frames played no role in constructing (33).

Theorem 9 applied to these eigenmaps gives singular support in \(\Lambda\); one may also use perverse microlocal dévissage. All bounds and comparisons were local on finite-type smooth charts. If \(F\) is nonzero, its bounded restriction to some such chart has nonzero perverse cohomology. Smooth perverse pullback identifies this with a restriction of some \({}^pH^jF\), proving nonvanishing. ◻

Proof. By the cone description following Definition 5, the nilpotent cone on \(\mathcal A^+\) is the smooth cotangent pullback of the ordinary-level cone: a nilpotent Borel-valued residue has zero toral component at every marked point. On a smooth torsor chart \(S\times H\) the singular support is therefore contained in \(\Lambda_S\times0_H\). The cone \(\Lambda_S\) is isotropic by Proposition 6. Choose a microlocal stratification of \(S\) whose conormals contain it. The product stratification criterion used in [24] and [27] makes \(Q\) constructible for \(\{S_\alpha\times H\}\). Restriction to a slice over a contractible patch of \(H\) is an equivalence for this constructible category; hence it identifies the entire complex with the pullback from that slice. Continuation gives the assertion on overlaps. These descriptions persist under base change and, in particular, under passage to the base of a single enhancement torus. ◻

Proof. This is [27]. Quotient the enhancement torus by its positive real radial subgroup \(A\simeq\mathbb R^d\), to factor \(P\xrightarrow{\rho}\overline P\xrightarrow{\bar\pi}C\) over the whole disc. The second map is a compact-torus bundle and therefore proper. Relative monodromicity gives \(F=\rho^*\overline F\): restriction across a contractible radial factor is an equivalence, so the local descriptions glue. In trivializations extending across zero, \[R\rho_*\rho^*\overline F=\overline F, \qquad \Psi F=\rho_0^*\Psi\overline F.\] These identities persist after base change and tensoring with a factor from the target. Proper base change for \(b\bar\pi\) proves all assertions. The maps are the natural exchanges for restriction and direct image from the universal cover of \(\Delta^*\), so they respect composition. The same proof applies to products of tori. ◻

Proof. We give the several-level version of [27], including the comparisons needed to apply (35).

Moving correspondence and its single-step comparison. Let \(\mathfrak H_{i,\Delta}\) parametrize a leg \(z\in\Delta\), two bundles \(P_0,P_1\) with enhancements \(\eta_{0j},\eta_{1j}\) at every \(x_j\), and a modification \(P_0\simeq P_1\) off \(z\). At \(x_i\) the two enhancements are independent. At each \(x_j\) with \(j\ne i\) require \(\eta_{1j}\) to be the transport of \(\eta_{0j}\). Write \(p\) for input and \(o\) for output-and-leg. We bound the underlying modifications by closed spherical Schubert bounds, retaining the full independent flags at \(x_i\). Choose these bounds to contain the Satake supports; their closed central fibers then contain the specialized supports.

Forgetting the levels at \(j\ne i\) makes this the pullback of the one-point moving correspondence by the stack of those levels on its input. Transport gives the same description using its output. This cartesian assertion also holds for successive modifications: there is one choice of each other enhancement, and the intermediate and final choices are determined. It does not assert that \(Q\) is pulled back from the one-point stack.

In an input frame compatible with \(\eta_{0i}\) the local family is the one used to define \(Z\): its punctured fiber is \(\mathop{\mathrm{Gr}}_G\times G/N_B\) and its central fiber is \(LG/\mathrm{Iw}^0\). The kernel \(K_V\) on the punctured fiber is the relative Satake kernel on \(\mathop{\mathrm{Gr}}_G\), tensored with \(\mathbf1_T\) on \(T=B/N_B\subset G/N_B\), extended by zero. Thus it requires equality of the two ordinary flags at \(x_i\) and records their enhancement difference. Its specialization is \(Z(V)\). With the relative normalization of Section 2, the moving-leg perverse shift is absent on both sides of this statement.

The kernels and their maps descend from these frames. On a bound one can take finite jets on a sufficiently thick neighborhood of the sum of the graphs of \(x_i\) and \(z\), so the same smooth frame space works at collision. The equivariance of Satake and the \(N_B\)-invariance of the flag factor give descent off zero; at zero it is the \(\mathrm{Iw}^0\)-equivariance of \(Z(V)\). The finite jet quotients of this last group are unipotent, hence Betti contractible, so descent retains the maps and their cocycles. The extra levels are outside this neighborhood and enter only in the bundle factor.

For an input family \(D\) over \(\Delta^*\), write \[A_V(D)=o_*(D\,\widetilde\boxtimes K_V), \qquad B_V(R)=Z(V)\star_{x_i}R,\] where the twisted product includes the fixed relative normalization. For \(Q\) viewed as a constant family, the unit calculation gives \[ A_V(Q)\simeq\mathop{\mathrm{Hecke}}_V(Q)|_{\Delta^*} \simeq Q\boxtimes V_\sigma|_{\Delta^*}. \tag{39}\] Indeed the only integration beyond the usual Hecke correspondence is over the input enhancement at \(x_i\), with kernel \(\mathbf1_T\).

The output map \(o\) factors over the full disc by forgetting just \(\eta_{0i}\), keeping its ordinary flag, and then by a proper map. The first map is a \(T\)-torsor. For the second, bounded input modifications with fixed output lie in a proper Grassmannian bound, and their ordinary input flags form a proper \(G/B\)-bundle. All other enhancements are determined by transport from the output; they add no nonproper directions. At \(z=0\) the output enhancement is still fixed, and choosing the ordinary input flag is still the full proper flag bundle on the modified input fiber. Thus the quotient is proper at collision as well; we have not replaced it by an exact-position affine-flag stratum. The integrand is relatively monodromic along the forgotten torus: \(Q\) has this property by hypothesis, and \(K_V\) has a local system in the enhancement difference with support conditions independent of that difference. Lemma 18 therefore applies.

In the input frames, the nearby-cycle tensor map for this integrand is an exterior-product comparison with an arbitrary coefficient from the bundle factor. It is an isomorphism both for constant \(D=Q\) and for \(D=Q\boxtimes L\), where \(L\) is a finite-rank local system on \(\Delta^*\): trivialize \(L\) on the puncture cover. This is a Betti calculation on finite-dimensional bounds, as in [27]. On sufficiently small chart neighborhoods, the local Milnor data adapted to the constructible factors have finite triangulations. Finite cellular complexes then compute the comparison, permitting the infinite-dimensional stalks of \(\mathscr L_T\); no global finite triangulation of the correspondence is needed. There is no adic inverse limit in this calculation. Smooth frame descent preserves the comparison and its naturality.

Combining this tensor map with inverse push exchange defines an actual natural isomorphism \[ \begin{split} c_V(D):B_V(\Psi D) &=o_{0,*}(\Psi D\,\widetilde\boxtimes Z(V))\\ &\longrightarrow o_{0,*}\Psi(D\,\widetilde\boxtimes K_V) \xrightarrow{\sim}\Psi A_V(D). \end{split} \tag{40}\] For constant \(D=Q\), its composite with the specialization of (39) gives (37) and identifies \(m_V\) with \(M_{V,i}\). We still need to check that this single-step identification remains valid on each factor of a convolution.

The two-step comparison. First, for \(D=Q\boxtimes L\) the construction gives the commutative square \[ \begin{CD} B_V(\Psi(Q\boxtimes L)) @>{c_V(Q\boxtimes L)}>> \Psi A_V(Q\boxtimes L)\\ @V{\sim}VV @VV{\sim}V\\ B_V(Q)\otimes L_\partial @>{c_V(Q)\otimes\mathrm{id}}>> \Psi A_V(Q)\otimes L_\partial. \end{CD} \tag{41}\] On the puncture cover both routes are the same tensor comparison for \(Q,K_V\), tensored with the fiber of \(L\), followed by the same push exchange. Finite rank allows that fiber through the direct images. The square is natural in every local-system endomorphism of \(L\), including its monodromy: the latter commutes with the cyclic fundamental group of \(\Delta^*\) and is an endomorphism of \(L\) itself.

Now let \(\mathfrak H^{(2)}_{i;W,V}\) parametrize successive bounded modifications \(P_0\to P_1\to P_2\) at the same \(z\in\Delta\). The three enhancements at \(x_i\) are independent; those at every other mark are transported throughout. Its integrand is \(Q\,\widetilde\boxtimes K_W\,\widetilde\boxtimes K_V\). There are two useful pushes. The map \(f\) forgetting \(P_0\) and the first modification is the base change of the first output map. It factors by forgetting \(\eta_{0i}\), then by a proper map. The map \(g\) composing the modifications and forgetting \(P_1\) factors by forgetting \(\eta_{1i}\), then by a proper map: the bounded intermediate bundles form the closed convolution fiber in a proper Grassmannian bound, and the intermediate ordinary flag contributes a proper \(G/B\)-bundle. Choose a target bound containing all composites. Both factorizations are over the whole disc. Transport again determines the other marked-point enhancements.

The integrand is relatively monodromic for both forgotten tori. For \(f\) this is the single-step check, with the second kernel pulled back from the target. For \(g\), \(Q\) is independent of \(\eta_{1i}\), and varying \(\eta_{1i}\) varies the two enhancement differences inversely. Their universal local systems remain a local system in this variable; their ordinary-flag and Schubert conditions are independent of it. Lemma 18 gives the push, base-change, and projection-formula comparisons for \(f\) and \(g\).

The unit convolution on \(\Delta^*\) and the specified monoidal map of \(Z\) at zero induce \[b:A_VA_W(Q)\xrightarrow{\sim}A_{V\otimes W}(Q),\qquad b_0:B_VB_W(Q)\xrightarrow{\sim}B_{V\otimes W}(Q).\] Tensor factors here are in action order. The essential compatibility is \[ \begin{CD} B_VB_W(Q) @>{b_0}>> B_{V\otimes W}(Q)\\ @V{c_V(A_W(Q))\,B_V(c_W(Q))}VV @VV{c_{V\otimes W}(Q)}V\\ \Psi(A_VA_W(Q)) @>{\Psi b}>>\Psi A_{V\otimes W}(Q). \end{CD} \tag{42}\] The left side is defined because (39) identifies \(A_W(Q)\) with \(Q\boxtimes W_\sigma|_{\Delta^*}\), an allowed input for \(c_V\).

To prove the square, start on the two-step correspondence with the natural map from the twisted product of the nearby cycles of \(Q,K_W,K_V\) to the nearby cycles of their twisted product. The local kernel comparison of [6], together with exterior comparison for \(Q\), makes it an isomorphism. Pushing first through \(f\) identifies it with the left side of (42), by base change and projection formula. Pushing first through \(g\) identifies it with \(b_0\) followed by \(c_{V\otimes W}(Q)\): the specialized kernel map is precisely the monoidal map of \(Z\) in the input frames. After the final output push these are equal, since tensor exchange, push exchange, and the intervening projection-formula squares are natural and compose. Every nonproper exchange in this assertion has one of the torus factorizations just checked. On the punctured family the same equality uses the compatible successive unit integrations specified after (34). Thus the square compares these particular maps, not merely two isomorphic outputs.

Individual monodromy and trace. Apply (41) with \(L=W_\sigma|_{\Delta^*}\). The comparison \(c_W(Q)\) sends \(m_W\) to \(M_{W,i}\); naturality of the square in that local-system endomorphism carries it to the \(W_{\partial i}\) factor alone after applying \(B_V\). For the outer factor, the left vertical identification in (41) is a projection-formula map, natural in endomorphisms of the kernel \(Z(V)\). It therefore intertwines \(m_V\) acting on \(B_V(\Psi(Q\boxtimes L))\) with \(m_V\) on \(B_V(Q)\) tensored with \(\mathrm{id}_{L_\partial}\). The bottom arrow is \(c_V(Q)\otimes\mathrm{id}\), which identifies this operator with \(M_{V,i}\) on its factor alone. More explicitly, let \[\Phi_{V,W}:B_VB_W(Q)\xrightarrow{\sim} Q\otimes V_{\partial i}\otimes W_{\partial i}\] be the successive comparison and eigenidentification, with the fixed symmetry placing the factors in action order. What we have proved is the pair of identities \[\begin{align*} \Phi_{V,W}(m_V\star\mathrm{id}) &=(\mathrm{id}_Q\otimes M_{V,i}\otimes\mathrm{id})\Phi_{V,W},\\ \Phi_{V,W}(\mathrm{id}\star m_W) &=(\mathrm{id}_Q\otimes\mathrm{id}\otimes M_{W,i})\Phi_{V,W}. \end{align*}\] The square (42) and the original coherent eigenmaps identify this comparison with the one for \(V\otimes W\). Naturality in representation morphisms includes coevaluation and evaluation; the tensor unit is covered by (34). Iteration gives the same assertion for any number of factors.

Consequently, applying (35) to \(Q\) gives insertion and contraction of \(V_{\partial i}\) with \(M_{V,i}\) on its indicated factor, hence \(\operatorname{Tr}(M_{V,i})\mathrm{id}_Q\). The right side of that local formula acts by \(\chi_V(\mu_i)\) through the universal-unit identification. This proves (38) with all other enhancements retained. ◻

Proof. We extend the final step of [27] to the full \(T^s\)-torsor. Take a nonzero perverse cohomology \(Q\) of the given eigencomplex. Proposition 13 supplies its coherent eigenstructure, and Theorem 9 puts its singular support in the nilpotent cone. Lemma 17 and Betti comparison therefore allow us to apply Lemma 19 at every \(x_i\). Since \(M_{V,i}\) is unipotent, \[ \chi_V(\mu_i)=(\dim V)\mathrm{id}_Q \qquad\text{for every }i,V. \tag{43}\]

Restrict to one \(H=T^s\) fiber and to a finite-dimensional stalk cohomology space \(W\) there. Its commuting monodromies have joint generalized eigencharacters, each a tuple \((a_1,\ldots,a_s)\in\widehat T(E)^s\). Equation (43) says \(\chi_V(a_i)=\chi_V(1)\) for all representations \(V\). Characters span \(\mathcal O(\widehat T)^{W_G}\), where \(W_G\) is the Weyl group, and separate semisimple classes. Thus \(a_i\) has the same image as \(1\) in the finite Weyl quotient; the fiber through \(1\) is the singleton \(\{1\}\). Every \(a_i=1\), so all commuting monodromy operators along the whole \(H\) are unipotent. This conclusion uses no regularity or fixed Jordan type.

Return to adic sheaves and put \(F=q_!Q\). The representable morphism \(q\) is of finite type with fixed relative dimension, so \(F\) is lcc. To prove \(F\ne0\), choose a fiber \(H=q^{-1}(a)\) on which \(Q\) has a nonzero stalk. Compute on its Betti realization, whose cohomology sheaves are finite-rank local systems. Let \(j\) be the largest index with \(\mathcal H^j(Q|_H)\ne0\), and let \(W\) be its fiber. Write \(r=\dim H=s\dim T\) and \(\Gamma_H=\pi_1(H^{\mathrm{an}})\). Then \[H_c^{2r}(H,\mathcal H^j(Q|_H)) \simeq W_{\Gamma_H}(-r)\ne0.\] Indeed the finitely many commuting operators \(M-1\) are nilpotent, so the augmentation ideal \(\mathfrak a\subset E[\Gamma_H]\) acts nilpotently on this particular \(W\). If its coinvariants vanished, \(W=\mathfrak aW=\mathfrak a^2W=\cdots=0\), a contradiction. The displayed identification is Poincaré duality. In the compact-support hypercohomology spectral sequence \[E_2^{a,b}=H_c^a(H,\mathcal H^b(Q|_H)) \Longrightarrow H_c^{a+b}(H,Q|_H),\] the nonzero term \((a,b)=(2r,j)\) has no incoming differential, because higher cohomology sheaves vanish, and no outgoing differential, because \(H_c^a\) vanishes for \(a>2r\). It survives, proving nonzero compactly supported hypercohomology. Betti comparison and base change for \(q_!\) identify this with a nonzero stalk of \(F\).

Finally the unramified Hecke correspondences transport all levels. The enhanced correspondence is cartesian from the ordinary one by \(q\) using either projection, and has the same relative Satake kernel. Base change and projection formula give, for every leg set \(I\), \[\begin{align*} \mathop{\mathrm{Hecke}}_{I,(V_i)}(q_!Q) &\simeq(q\times\mathrm{id}_{U^I})_!\mathop{\mathrm{Hecke}}_{I,(V_i)}(Q)\\ &\simeq q_!Q\boxtimes\mathop{\boxtimes}_{i\in I}(V_i)_\sigma. \end{align*}\] On bounds the Hecke pushforwards are proper. The same cartesian description holds on successive-modification diagrams and after restriction to every collision diagonal; their natural exchanges compose. Thus these isomorphisms transport all the coherence diagrams, in the original relative normalization without a moving-leg shift. The nonzero lcc eigencomplex \(F\) has a nonzero perverse cohomology. Apply Proposition 13 to obtain the asserted coherent eigensheaf and Theorem 9 for its nilpotent singular support. ◻

Proof. We first calculate on a normalized root disc of the first component. Put \(t=u^m\). Locally on a test scheme \(S\), choose an equivariant frame in which inertia acts by \(\lambda(\zeta)\). Frames lift from the origin through infinitesimal neighborhoods by smoothness and exactness of \(\mu_m\)-invariants. Their changes satisfy \[h(\zeta u)=\lambda(\zeta)h(u)\lambda(\zeta)^{-1}, \qquad h(0)\in T.\] In the invariant punctured-disc frame the same change is \(g(t)=\lambda(u)^{-1}h(u)\lambda(u)\). For a positive root \(\alpha\) and \(j=\langle\alpha,\lambda\rangle\), its positive and negative root series transform as \[ u^j a_\alpha(u^m)\longmapsto a_\alpha(t), \qquad u^{m-j}b_\alpha(u^m)\longmapsto t b_\alpha(t). \tag{47}\] Toral series are series in \(t\). These are exactly the series in the Iwahori group \[I_B(S)=\{g\in G(\mathcal O_S[[t]]):g(0)\in B\}.\] For completeness the root calculation identifies the whole group: \(h(0)\in T\) puts \(h\) in the formal big cell \(R_u(B^-)TR_u(B)\); the formulas in (47) transform its root-ordered factorization into that of \(I_B\). Conversely reduction of an Iwahori element belongs to \(B\), hence to that big cell, and the inverse formulas give a regular equivariant frame change.

Gluing to the component off its marked points therefore turns a bundle of this root type into an ordinary \(B\)-level bundle. Evaluation \(h\mapsto h(0)\) becomes the torus projection \(I_B\to T\). Its kernel is the inverse image of \(N_B\) under \(I_B\to B\), so identifying the gerbe restriction with the fixed type torsor is precisely the extra datum of an enhanced flag.

On the second branch a positive coordinate generator is \(\eta=\zeta^{-1}\). Its fiber action is \(\lambda(\eta^{-1})\), so the same calculation uses \(-\lambda\) and gives \(B^-\). In the common node generator both fixed type torsors have automorphism group \(T\). A bundle on the nodal curve consists of its bundles on the normalization and an identification at every pair of residual gerbes. Choosing frames of both gerbe restrictions makes that identification automatic; changing the two frames simultaneously leaves it unchanged. Applying this construction at the \(s\) disjoint pairs gives (46). Multiple nodes between the same components impose no additional relation: nodal descent specifies one independent gluing isomorphism at each node. With reversed right-action conventions the same quotient would have inversion on its second torus factor; here we use simultaneous gerbe frames.

A modification away from the markings transports the gerbe frames and their identifications. The corresponding Hecke diagram therefore pulls back to the diagram on the indicated factor. ◻

Proof. Let \(H=\rho_1(\pi_1^{\mathrm{et}}(U_1))\subset\check G(L')\) be the compact image, for a finite coefficient field \(L'\). Choose a cofinal decreasing sequence of open normal subgroups \(H_r\) and put \(Q_r=H/H_r\). One obtains such a sequence from congruence kernels of an \(H\)-stable lattice in a faithful representation. Both \(U_j\) carry connected \(Q_r\)-torsors. Choose indices \(m_r\) divisible by the fixed \(m\) and by the orders of all their boundary images, with \(m_r\mid m_{r+1}\).

By the root-stack description of tame covers [22], the torsors extend to finite étale torsors on the normalization of \(C(m_r)_0\). At each pair of gerbes their actions agree: the node identifies a positive generator on the first branch with a negative generator on the second, and (48) cancels that inversion. Reduce the chosen boundary-fiber identifications modulo \(H_r\) to identify these restrictions. They are compatible at every level of the tower. The torsors glue to a finite étale \(Q_r\)-torsor \(Y_{r,0}\) on \(C(m_r)_0\). Locally this is nodal gluing of finite étale algebras with identified fibers: on the completed branch discs the covers are constant, with the same \(\mu_{m_r}\)-action at the origin, so their gluing is constant on the nodal cover and descends equivariantly. This also proves compatibility under the power maps of the twisted models.

Proper henselian invariance of finite étale covers lifts \(Y_{r,0}\) to a torsor \(Y_r\) over \(C(m_r)\) [28]. Its hypotheses hold because the models are proper with finite diagonal over complete henselian traits. Full faithfulness lifts the group actions, transition maps after pullback to divisible models, and all relations between them. All generic fibers become \(C_{K_1}\). Choose a base section specializing in \(U_1\) and compatible points of the torsor fibers above it; finite étale covers of this strictly henselian section are constant. The tower therefore gives a continuous homomorphism \[ \rho_{K_1}:\pi_1^{\mathrm{et}}(C_{K_1})\longrightarrow \varprojlim_r Q_r=H. \tag{49}\]

We check its image. Each \(Y_{r,0}\) is connected, since its two normalization torsors are connected and are glued along nonempty fibers. The total torsor \(Y_r\) is proper and flat with reduced geometric fibers. Its tame coarse space is flat as well: locally its coordinate ring is the \(\mu_{m_r}\)-invariant direct summand of an \(R_{m_r}\)-flat ring, and invariant formation commutes with base change. The connected reduced proper special fiber has \(H^0(\mathcal O)=\mathbb C\). Semicontinuity, applied to the coarse space, gives \(h^0(Y_{r,K_1},\mathcal O)=1\), so the geometric generic torsor is connected. Thus (49) surjects onto every \(Q_r\). Its image is compact and therefore closed in \(H\); surjectivity to this cofinal system makes it all of \(H\). In particular the parameter remains Zariski dense.

For an algebraic representation of \(\check G\), enlarge its finite coefficient field if necessary and choose an \(H\)-stable lattice. Each finite reduction factors through some \(Q_r\). Its lifted torsor supplies a lisse extension over the smooth leg locus after the corresponding finite trait extension, with the specified special reductions on both components. All twisted models agree away from the nodes. The tower thus gives the required tensor compatibilities; maps between lattices are compared after clearing denominators. As allowed in Proposition 4, different reductions may require different finite trait extensions. ◻

Proof. The construction gives \(F_1^+\) on the enhanced stack for \(X\). Proposition 20 applies because the parameter is dense and each boundary monodromy is unipotent, and gives a nonzero perverse eigenobject on the ordinary-flag stack. Its singular support lies in the required cone by Theorem 9. No regularity of any local nilpotent is used. ◻