Proof. Let \(e_C\in\mathop{\mathrm{QCoh}}(Y)\) be the structure sheaf of \(C\) extended by zero. Its action is the projection onto \(\mathcal D_C\). On \(\mathcal C_s\) this object acts by zero, since \(C\cap Y'=\varnothing\) and the primed equivalence is spectral-linear. Hence, for \(D\in\mathcal D_C\), \[\Psi_{\mathcal B}(D) \simeq\Psi_{\mathcal B}(e_C\cdot D) \simeq e_C\cdot\Psi_{\mathcal B}(D)=0.\] By (5), the category \(\mathcal D_C\) is identified with \(\mathop{\mathrm{IndCoh}}_{\mathop{\mathrm{Nilp}}}(C)\). It is nonzero: the fully faithful functor from \(\mathop{\mathrm{QCoh}}(C)\) into \(\mathop{\mathrm{IndCoh}}_{\mathop{\mathrm{Nilp}}}(C)\) sends the nonzero object \(\mathcal O_C\) to a nonzero object; see (Gaitsgory and Raskin 2025a, secs. 1.3.1–1.3.4).

The category \(\mathcal D_Y\) is compactly generated (Gaitsgory and Raskin 2025a, sec. 3.1.7). The binary decomposition \[\mathcal D_Y\simeq\mathcal D_C\times\mathcal D_{Y\setminus C}\] shows that projection onto \(\mathcal D_C\) preserves compactness: its right adjoint, the inclusion of that factor, is continuous. Project compact generators of \(\mathcal D_Y\). At least one projection is nonzero, because otherwise their colimit closure would give \(\mathcal D_C=0\). Choose such an object \(F\).

The inclusion of either direct factor also preserves compactness, since its right adjoint is the continuous projection. Applying this first to \(\mathcal D_C\subset\mathcal D_Y\) and then to \(\mathcal D_Y\subset\mathcal C_K\) shows that \(F\) is compact in \(\mathcal C_K\). The nilpotent inclusion \(\mathcal C_K\hookrightarrow\mathop{\mathrm{Shv}}(\mathcal B_K)\) preserves compactness by (Gaitsgory and Raskin 2025a, Theorem 1.1.7). Finally, a compact sheaf on an algebraic stack is constructible on smooth finite-type charts (Arinkin et al. 2022b, sec. F.2.2). This proves the last assertion. ◻

Proof. The first equality follows from smooth base change and \(\Psi_{\mathcal B}(F)=0\). For an exterior product in (8), ordinary pullback gives \[(a,\ell_T)_K^*(D\boxtimes\mathcal L_K) =a_K^*D\otimes\ell_{T,K}^*\mathcal L_K.\] The sheaf \(\ell_T^*\mathcal L_S\) is lisse on \(T\), regardless of the smoothness of \(\ell_T\). Formula (3) and Proposition 6 therefore give \[\Psi_T\bigl(a_K^*D\otimes\ell_{T,K}^*\mathcal L_K\bigr) \simeq a_s^*\Psi_{\mathcal B}(D)\otimes\ell_{T,s}^*\mathcal L_s=0.\] Both pullback and specialization are continuous, so the same vanishing holds for the stable colimit closure of these exterior products, which contains \(\mathrm H_V(F)\). The calculation is local on the smooth locus, and the finite-extension assertion follows from the invariance of geometric specialization already proved. ◻

Proof. The \(T\)-weight decomposition is \(\mathfrak g=\mathfrak t\oplus\bigoplus_\alpha\mathfrak g_\alpha\). Invariance under \(T\) makes \(\mathfrak t\) orthogonal to the root spaces and makes \(\mathfrak g_\alpha\) pair only with \(\mathfrak g_{-\alpha}\). Nondegeneracy of \(\kappa\) therefore gives nondegeneracy on \(\mathfrak t\) and on each opposite-root pairing. Choose compatible root vectors \(e_\alpha,e_{-\alpha}\) with \([e_\alpha,e_{-\alpha}]=d\alpha^\vee\). This identity follows by differentiating the rank-one root homomorphism from \(\mathrm{SL}_2\); it is valid for every isogeny type; see the root-subgroup construction in (Springer 1998, secs. 8.1.1–8.1.4) and the bracket formula in (Conrad 2014, Corollary 5.1.12, Equation (5.1.4)). Invariance of the form gives (10) with \(c_\alpha=\kappa(e_\alpha,e_{-\alpha})\ne0\). Thus one of \(d\alpha,d\alpha^\vee\) vanishes if and only if the other does. Their simultaneous vanishing would put \(\alpha\in p\Lambda^\vee\) and \(\alpha^\vee\in p\Lambda\), contradicting the integral identity \(\langle\alpha,\alpha^\vee\rangle=2\), since \(p^2\) does not divide \(2\). These arguments use torus characters, rather than just their differentials, and also apply in characteristic two.

Let \(B\supset T\) be a Borel subgroup with Lie algebra \(\mathfrak b\). The incidence morphism \[G\times^B\mathfrak b\longrightarrow\mathfrak g, \qquad [g,x]\longmapsto\mathop{\mathrm{Ad}}(g)x,\] is proper: its source is closed in \((G/B)\times\mathfrak g\). Choose \(c\in\mathfrak t\) outside the root differential hyperplanes. At \([1,c]\), the negative-root tangent directions map isomorphically onto the negative root spaces, and \(\mathfrak b\) supplies the remaining directions. The morphism is consequently dominant, hence surjective.

For a semisimple element in \(\mathfrak b\), write it as \(c+u\), with \(c\in\mathfrak t\) and \(u\) in the positive-root Lie algebra. Conjugate by positive root groups in increasing root height. If \(d\alpha(c)\ne0\), conjugation by the \(\alpha\)-root group changes the \(\alpha\) coefficient by \(-z\,d\alpha(c)\) and changes the other positive-root terms only in greater height. It therefore eliminates that coefficient. After all such steps the result is \(c+u_0\) with \([c,u_0]=0\). In a faithful representation, \(c\) is semisimple and \(u_0\) is nilpotent. Since their sum is semisimple, the matrix Jordan decomposition forces \(u_0=0\). ◻

Proof. Lemma 9 conjugates \(s\) into \(\mathfrak t\). If \(s=0\), apply the Borel assertion of that lemma to \(n\). The projection \(\mathfrak b\to\mathfrak t\) preserves restricted \(p\)-operations. Since \(n\) is killed by an iterated \(p\)-operation and that operation is injective on \(\mathfrak t\), its toral projection is zero. A strictly dominant cocharacter contracts the positive-root Lie algebra to zero. Hence every invariant polynomial has the same value on \(n\) and on zero, a contradiction. Thus \(s\ne0\).

By Hypothesis 1, \(M=Z_G(s)\) is a Levi subgroup. Its Lie algebra is the kernel of \(\operatorname{ad}_s\), and it contains \(a_0\). On \(\mathfrak g/\mathfrak m\), the operator \(\operatorname{ad}_s\) is diagonalizable with nonzero eigenvalues. The commuting operator \(\operatorname{ad}_n\) is nilpotent, since \(\operatorname{ad}_n^{p^e}=\operatorname{ad}_{n^{[p^e]}}=0\) for large \(e\). On each eigenspace of \(\operatorname{ad}_s\), the sum \(\operatorname{ad}_{a_0}=\operatorname{ad}_s+\operatorname{ad}_n\) is therefore invertible.

To prove (11), conjugate \(n\) inside \(M\) into the Lie algebra of a Borel subgroup of \(M\) containing \(T\). The incidence argument of Lemma 9 applies to \(M\): its roots have nonzero differentials, and \(\kappa\) is nondegenerate on \(\mathfrak m\) by the same weight decomposition. Restricted nilpotence again makes the toral projection of \(n\) zero. Thus the conjugated \(n\) lies in the positive-root Lie algebra of \(M\), which is orthogonal to \(\mathfrak t\). Since \(d\lambda\in\mathfrak t\) is fixed by \(M\), invariance of \(\kappa\) gives \(\kappa(n,d\lambda)=0\) before conjugation as well. ◻

Proof. After choosing a positive system, let \(I=\{\alpha_1,\ldots,\alpha_r\}\) be the simple roots of \(M\). Set \(J_I=\operatorname{span}_k\{d\alpha_i^\vee\}\subset\mathfrak t\). Because every root differential is nonzero, a central element of \(\mathfrak m\) has no root-space component. Equation (10) therefore identifies \[\mathfrak z(\mathfrak m) =\bigcap_i\ker(d\alpha_i)=J_I^\perp.\] The form is nondegenerate on this center by hypothesis, and on \(\mathfrak t\) by Lemma 9; it follows that it is nondegenerate on \(J_I\).

Define integral maps \[U:\Lambda\longrightarrow\mathbb Z^r,\quad y\longmapsto(\langle\alpha_i,y\rangle)_i, \qquad V:\mathbb Z^r\longrightarrow\Lambda,\quad e_j\longmapsto\alpha_j^\vee.\] Their composite is the Cartan matrix \(C_I=UV=(\langle\alpha_i,\alpha_j^\vee\rangle)_{i,j}\). A subscript \(k\) will mean reduction to \(k\). Equation (10) gives \(V_k^*\kappa=D U_k\), where \(D=\operatorname{diag}(c_{\alpha_1},\ldots,c_{\alpha_r})\) is invertible. The radical of \(V_k^*\kappa V_k\) is exactly \(\ker V_k\), by nondegeneracy on \(J_I=\mathop{\mathrm{im}}V_k\). Consequently \[ \operatorname{rank}(C_{I,k}) =\operatorname{rank}(V_k)=\operatorname{rank}(U_k). \tag{13}\]

If some Levi Cartan determinant were divisible by \(p\), the ordinary determinant list would give a further Levi of type \(A_{p-1}\). Indeed, the Cartan matrices of the classified Dynkin diagrams (Milne 2022, Theorem C.56 and §24(c)) have determinants \[\begin{array}{c|ccccccccc} \text{type}&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2\\ \hline \det&r+1&2&2&4&3&2&1&1&1. \end{array}\] In type \(A_r\), take a consecutive string of \(p-1\) vertices. In the determinant-two and determinant-four cases take one vertex, since \(p=2\). In type \(E_6\) take two adjacent vertices, since \(p=3\). Such a simple-root subdiagram defines a Levi of \(G\), so the preceding rank calculation applies to it.

For this \(A_{p-1}\) Levi, \(r=p-1\) and \(\det C_I=p\). Choose dual bases of the full character and cocharacter lattices, of rank \(N_T\). The matrices of \(U,V\) have sizes \(r\times N_T\) and \(N_T\times r\). By (13), both reductions have rank less than \(r\), so every maximal minor of each matrix is divisible by \(p\). The integral Cauchy–Binet identity gives \[\det C_I= \sum_{\substack{J\subset\{1,\ldots,N_T\}\\ |J|=r}} \det(U_{:,J})\det(V_{J,:}),\] which is divisible by \(p^2\), contrary to \(\det C_I=p\). Using the full lattices here retains all central torus directions and makes no assumption on the isogeny type.

For an arbitrary Levi, put \(d_I=\det C_I\) and define an integral endomorphism of \(\Lambda\) by \[\Pi_I=d_I\mathop{\mathrm{id}}_\Lambda-V\operatorname{adj}(C_I)U.\] It satisfies \(U\Pi_I=0\). The lattice \(\ker U\) consists precisely of cocharacters into \(Z(M)\): their images commute with \(T\) and with every root group of \(M\). For \(z\in\ker U_k\), \(\Pi_{I,k}(z)=d_Iz\). Since \(d_I\) is invertible in \(k\), the differentials of elements of \(\ker U\) span \(\ker U_k=\mathfrak z(\mathfrak m)\), proving (12). For \(r=0\), take \(d_I=1\) and \(\Pi_I=\mathop{\mathrm{id}}_\Lambda\); the same proof covers tori. ◻

Proof. Suppose that the current arc belongs to \(\mathfrak m[[t]]\) modulo \(t^r\), with \(r\ge1\), and let \(c_r\in\mathfrak g/\mathfrak m\) be the class of its coefficient of \(t^r\). Choose \(D\in\mathfrak g\) such that \(c_r+[D,a_0]=0\) in the quotient. Smoothness of \(G\) gives an element \(g_r(t)\in G(k[[t]])\) with \[g_r(t)\equiv1+t^rD\pmod {t^{r+1}}.\] Here the displayed expression specifies a point in the tangent kernel for the square-zero ideal \((t^r)/(t^{r+1})\); formal smoothness lifts it successively to every higher order. Conjugation by \(g_r(t)\) changes the coefficient of \(t^r\) by \([D,a_0]\) and leaves all lower coefficients unchanged. It therefore places the arc in \(\mathfrak m[[t]]\) modulo \(t^{r+1}\).

Iterate this construction. Since \(g_r(t)\equiv1\pmod {t^r}\), the successive products converge in the \(t\)-adic topology. The affine group \(G\) takes this compatible system of points modulo \(t^j\) to a point \(g(t)\in G(k[[t]])\) with \(g(0)=1\). Its conjugation sends the arc into \(\mathfrak m[[t]]\). No exponential series is used. ◻

Proof of Proposition 8. Under Hypothesis 2, use the restricted-power construction and Lemma 11. Under Hypothesis 3, use the additive-polynomial construction and integral Weyl-group averaging. In each case Lemma 10 supplies the Levi and quotient invertibility, with \(\kappa(a_0,d\lambda)\ne0\). Apply Lemma 12 after the initial constant conjugation. The formal change of frame fixes \(a_0\), so all three assertions in (9) hold. ◻

Proof. Use the split integral affine Grassmannian, whose loops are \(G(A((t)))\) and whose positive loops are \(G(A[[t]])\). Its integral Schubert bound is projective, and the positive-loop orbit is smooth, open, and fiberwise dense, of relative dimension \(m\); see (Richarz and Scholbach 2021, sec. 1.2 and 2.1). We use these models by base change from \(\mathbb Z\). The variable \(t\) is the curve parameter, including when the base trait has mixed characteristic.

A coordinate along the graph of \(x\) and a frame on the formal disc identify modifications of a fixed input bundle with the corresponding Grassmannian spaces. These descriptions glue by Beauville–Laszlo descent (Beauville and Laszlo 1995). More explicitly, the graph is an effective Cartier divisor. The gluing theorem for modules also glues the coordinate algebra, multiplication, and coaction of an affine \(G\)-torsor; the torsor identity can be checked on the two gluing pieces. A disc frame exists locally: lift a frame on the graph successively through its infinitesimal neighborhoods using smoothness of \(G\). Changes of frame and formal coordinate preserve the orbit and its bound. Consequently the properness, smoothness, and dimensions descend to the input-and-leg projection.

Swapping the bundles and inverting \(\alpha\) is an involution of the Hecke correspondence. On double cosets it replaces \(\mu\) by \(-\mu\), whose dominant representative is \(\mu^\vee\). It carries the bound to the oppositely labelled bound. This proves the assertions for \(q\), since \(\langle 2\rho,\mu^\vee\rangle=m\). Separatedness of the open correspondence follows from that of the proper bound. This involution is used on the correspondence, rather than as a purported inversion map on the right-quotient affine Grassmannian.

The fibers here include an identification of the fixed bundle with the specified bundle. They are relative-position spaces: an automorphism of the moving bundle compatible with its identification off the graph is the identity. In particular, no further quotient by automorphisms of the fixed bundle is taken.

For the last assertion, geometric Satake gives the intersection complex of the bound. Its restriction to the smooth open orbit is the shifted constant sheaf, with the indicated optional Tate normalization; see (Richarz 2014, Proposition 4.1). The Hecke construction tensors this kernel with the ordinary pullback \(p_K^*F\) and pushes forward along the proper output-and-leg projection; see (Gaitsgory and Raskin 2025a, secs. 4.2.1–4.2.4). Relabelling by \(\mu^\vee\) if necessary matches these projections with the chosen representation. A fixed overall shift or Tate line in the Hecke normalization changes neither this description nor any vanishing assertion below. Only the open-orbit restriction is being identified; the kernel on the boundary remains its intersection complex. ◻

Proof. Since \(A\) is non-nilpotent, choose \(x\in X(k)\) where its value is non-nilpotent. Choose a formal coordinate \(t\) and a bundle frame at \(x\), and write \(A=a(t)\,dt\). Chevalley restriction identifies the nilpotent fiber with \(\chi^{-1}(\chi(0))\), so \(\chi(a(0))\ne\chi(0)\). Proposition 8 then supplies a formal change of frame, a Levi subgroup \(M\), and a cocharacter \(\lambda\) of \(Z(M)\) such that \[ a(t)\in\mathfrak m[[t]],\qquad \operatorname{ad}(a(0)) \text{ is invertible on }\mathfrak g/\mathfrak m, \qquad \kappa(a(0),d\lambda)\ne0. \tag{16}\] Here \(d\lambda\) is the value of the cocharacter differential on the canonical generator of \(\mathop{\mathrm{Lie}}(\mathbb G_m)\). After constant conjugation, we regard \(\lambda\) as an element of the fixed cocharacter lattice; it therefore also labels the integral model of Lemma 14. Modify by \(\lambda(t)\). Since \(\lambda\) centralizes \(M\), the transported field is regular in the output frame and still has expression \(a(t)\,dt\). Together with \(A\) away from \(x\), it defines the global Higgs field \(A'\) on \(P'\). This central-in-a-Levi construction and the resulting leg covector are the modification used in (Arinkin et al. 2022b, secs. 20.4–20.6). We prove the derivative assertions explicitly.

The two tangent spaces. In the input frame put \[L=\mathfrak g[[t]],\qquad L'=\mathop{\mathrm{Ad}}(\lambda(t))L,\qquad I=L\cap L'.\] Let \(\mathfrak g_j\) be the weight-\(j\) space for the algebraic \(\mathbb G_m\)-action defined by \(\lambda\). The integers \(j\) denote characters of \(\mathbb G_m\), without reduction modulo \(p\). Then \[ \begin{split} L'&=\bigoplus_j t^j\mathfrak g_j[[t]],\\ P_1:=L/I&=\bigoplus_{j>0} \mathfrak g_j\otimes_k k[[t]]/(t^j),\\ P_2:=L'/I&=\bigoplus_{j<0} \mathfrak g_j\otimes_k(t^jk[[t]]/k[[t]]). \end{split} \tag{17}\] An infinitesimal disc automorphism \(1+\epsilon D\), \(D\in L\), acts on the output lattice while fixing \((P,x)\). Its stabilizer tangent is \(I\). The orbit is smooth, and the root formula gives \(\dim_k(L/I)=m\), so this computes its full tangent. The same argument with the output fixed gives \[ T_hH^{\mathrm{in}}_{P,x}=P_1, \qquad T_hH^{\mathrm{out}}_{P',x}=P_2. \tag{18}\] The notation \(1+\epsilon D\) uses the canonical identification of the kernel over dual numbers with the Lie algebra; no exponential map is required.

There is a perfect pairing \[ \beta:P_1\times P_2\longrightarrow k, \qquad \beta(D,C)=\mathop{\mathrm{Res}}_{t=0}\kappa(D,C)\,dt. \tag{19}\] Indeed, \(L\) is its own annihilator for the residue pairing on \(\mathfrak g((t))\), and invariance of \(\kappa\) gives the same assertion for \(L'\). Thus the pairing descends to the displayed quotients. More explicitly, \(\kappa\) pairs \(\mathfrak g_j\) perfectly with \(\mathfrak g_{-j}\); the coefficient of \(t^a\) in the first space, for \(0\le a<j\), pairs with the coefficient of \(t^{-a-1}\) in the second. This proves perfection, including when some \(j\) is divisible by \(p\).

Transport of the covector and motion of the leg. Choose the gluing sign convention in which an infinitesimal transition \(1+\epsilon C\) gives the boundary class of the principal part \(C\). Serre duality pairs this class with \(A\) by \(\mathop{\mathrm{Res}}\kappa(a,C)\,dt\). For an output-fixed tangent \(C\in L'\), this residue is zero, since \(a\in L'\) and \(L'\) is self-annihilating. Consequently \(dp_h^*A\) vanishes in the \(q\)-relative cotangent, so it is \(dq_h^*\theta\) for a unique covector \(\theta\).

Its bundle component is \(A'\). To check this, choose a point \(x'\ne x\) and vary both bundle transitions there while leaving the modification at \(x\) unchanged. Principal parts at \(x'\) surject onto \(H^1(X,\mathfrak g_{P'})\): for sufficiently large \(N\), Serre vanishing gives \(H^1(X,\mathfrak g_{P'}(Nx'))=0\). The pairings on these directions agree because \(A=A'\) near \(x'\). Write therefore \(\theta=(A',\xi)\).

Now keep the input bundle fixed and move the leg from \(t=0\) to \(t=\epsilon\). The output transition becomes \(g_\epsilon=\lambda(t-\epsilon)\), and its first-order displacement is \[(\delta g)g^{-1}=-\frac{d\lambda}{t}.\] Evaluating the cotangent identity on this motion yields \[0=-\kappa(a(0),d\lambda)+\xi(\partial_t), \qquad \xi=\kappa(a(0),d\lambda)\,dt\big|_x\ne0.\] This is also the residue formula of (Arinkin et al. 2022b, Equation (20.20)). Reversing the gluing boundary convention would reverse the common sign and would leave all subsequent assertions unchanged.

The first relative derivative. The section \(S_A\) is defined along the entire fixed-input fiber, using its specified identification with \(P\). Its value at \(h\) is zero by the preceding calculation. For \(D\in L\), deform the output lattice to \[L'_\epsilon=\mathop{\mathrm{Ad}}(1+\epsilon D)L'.\] A test vector \(C\in L'\) extends as \(C_\epsilon=\mathop{\mathrm{Ad}}(1+\epsilon D)C\) in the deformed lattice. The input field remains the fixed \(a(t)\,dt\). Differentiating the residue pairing gives \[ \begin{split} \big\langle(dS_A)_h(D\bmod I),C\bmod I\big\rangle &=\mathop{\mathrm{Res}}\kappa(a,[D,C])\,dt\\ &=\mathop{\mathrm{Res}}\kappa([a,D],C)\,dt =\beta(\operatorname{ad}(a)D,C). \end{split} \tag{20}\] Because \(S_A(h)=0\), this derivative is independent of how the test vector is extended. Also \(\operatorname{ad}(a)\) preserves \(L\), \(L'\), and \(I\), so the formula is independent of representatives.

It remains to prove invertibility. The operator \(\operatorname{ad}(a(t))\) preserves every weight summand in (17). Since \(\lambda\) is central in \(M\), each nonzero weight space injects as an invariant summand into \(\mathfrak g/\mathfrak m\). Its endomorphism induced by \(\operatorname{ad}(a(0))\) is therefore invertible by (16). The determinant of the corresponding \(k[[t]]\)-linear block is a unit. It follows that \(\operatorname{ad}(a(t))\) is invertible on every truncated summand of \(P_1\) and \(P_2\). Equation (20), together with the perfect pairing (19), now proves that \((dS_A)_h:P_1\to P_2^*\) is an isomorphism. No regularity of \(\lambda\) outside \(M\) is required; any additional zero-weight directions do not occur in these quotients.

The reverse derivative. The value of \(S_{A'}\) at \(h\) is zero because \(a\in L\) and \(L\) is self-annihilating. Keeping the output fixed, vary the input lattice by \(C\in L'\) and extend a test vector \(D\in L\) to \(\mathop{\mathrm{Ad}}(1+\epsilon C)D\). The same calculation, with the same gluing convention, gives \[\big\langle(dS_{A'})_h(C\bmod I),D\bmod I\big\rangle =\mathop{\mathrm{Res}}\kappa(a,[C,D])\,dt =\mathop{\mathrm{Res}}\kappa([a,C],D)\,dt.\] Invertibility on \(P_2\) and residue duality prove the second assertion. Both sections are sections of rank-\(m\) vector bundles on smooth \(m\)-dimensional fibers. Their invertible derivatives make their zeros at \(h\) reduced and isolated.

When \(m=0\), both tangent spaces are zero and both derivatives are isomorphisms between zero spaces. The smooth zero-dimensional fibers already isolate \(h\), and the moving-leg computation still applies. Thus the conclusion includes central modifications and torus factors. ◻

Proof. Write \(h=c_1(\mathcal O_{Q_f}(1))\in H^2(Q_f,E(1))\). Adjunction applied to each \(h^j\) gives the displayed morphism from \(E_B(-j)[-2j]\) to \(Rg_*E_{Q_f}\). Denote the cone of their direct sum by \(\mathcal K\), and write \(j_B:B\setminus D\hookrightarrow B\).

We recall the elementary cohomology calculation that controls this map. We use the projective-space and projective-bundle calculations of (Milne 2013, Example 16.3 and Theorem 23.2), together with localization. A split smooth projective quadric of dimension \(n\) has no odd cohomology, and has one copy of \(E(-r)\) in degree \(2r\) for \(0\leq r\leq n\), with an additional copy when \(n\) is even and \(r=n/2\). One obtains the calculation inductively by writing its equation as \(x_0x_1+q'(x_2,\ldots)=0\). The locus \(x_0\ne0\) is \(\mathbb A^n\); its complement is the projective cone over the split quadric of dimension \(n-2\). Removing the vertex of that cone gives a line bundle over the smaller quadric. This yields a filtration by affine cells, with one cell of every dimension \(0,\ldots,n\) and a second middle-dimensional cell when \(n\) is even. The initial cases are a conic, isomorphic to \(\mathbb P^1\), and the split zero-dimensional quadric, consisting of two points. Localization and \(R\Gamma_c(\mathbb A^r,E)=E(-r)[-2r]\) give the asserted groups. On a smooth quadric of positive dimension, the top hyperplane power has degree \(2\), so is nonzero in \(E\). All lower hyperplane powers are therefore nonzero as well.

Over \(B\setminus D\), the geometric fibers of \(g\) are smooth quadrics of odd dimension \(2m-1\). The preceding calculation shows that the hyperplane-power map is an isomorphism on their cohomology. Proper base change therefore gives \(j_B^*\mathcal K=0\).

Over \(D\) the family, together with its hyperplane line bundle, is the product with \[V_m=\left\{\sum_{a=1}^{m}u_av_a=0\right\} \subset\mathbb P_{K_0}^{2m}.\] Let \(e=[0:\cdots:0:1]\) be its vertex. Projection away from \(e\) identifies \(V_m\setminus\{e\}\) with the total space of a line bundle over the split smooth quadric \[C_m=\left\{\sum_{a=1}^{m}u_av_a=0\right\} \subset\mathbb P_{K_0}^{2m-1}.\] Compactly supported cohomology of this line bundle and localization at the vertex give \[ H^0(V_m,E)=E,\qquad H^1(V_m,E)=0,\qquad H^r(V_m,E)=H^{r-2}(C_m,E)(-1)\quad(r\geq2). \tag{30}\] Thus \(V_m\) has the cohomology of \(\mathbb P^{2m-1}\) with one extra copy of \(E(-m)\) in degree \(2m\).

The hyperplane powers remain nonzero on \(V_m\). To check the top one without a smoothness assertion about \(V_m\), resolve the vertex by the projective line bundle \[\pi:\widetilde V_m =\mathbb P_{\mathrm{lines},C_m} (\mathcal O_{C_m}(-1)\oplus\mathcal O_{C_m}) \longrightarrow C_m.\] Here the subscript specifies that the projective bundle parameterizes lines. The inclusion of \(\mathcal O_{C_m}(-1)\oplus\mathcal O_{C_m}\) into the trivial vector bundle with fiber \(K_0^{2m}\oplus K_0\) defines \(\widetilde V_m\to V_m\); its fiber over a point of \(C_m\) parameterizes the projective line joining that point to \(e\). If \(\zeta\) is the pullback of the hyperplane class and \(h_C=c_1(\mathcal O_{C_m}(1))\), the projective bundle relation is \(\zeta^2=\pi^*h_C\,\zeta\) (Milne 2013, sec. 23). Hence \[\int_{\widetilde V_m}\zeta^{2m-1} =\int_{C_m}h_C^{2m-2}=2.\] It follows that the top hyperplane power on \(V_m\) is nonzero. Multiplication gives the same conclusion for every lower power. This includes \(m=1\): \(C_1\) consists of two points, \(V_1\) consists of two projective lines meeting at \(e\), and the resolution is the disjoint union of those lines. The hyperplane class has degree \(1\) on each component.

Consequently the cone of the vector-space morphism \[\bigoplus_{j=0}^{2m-1}E(-j)[-2j] \longrightarrow R\Gamma(V_m,E)\] has exactly one nonzero cohomology group, a copy of \(E(-m)\) in degree \(2m\). Injectivity in every degree, established by the nonvanishing of the hyperplane powers, rules out any additional cohomology group in the cone. The cone is therefore isomorphic to \(E(-m)[-2m]\).

Proper base change identifies \(i^*\mathcal K\) with the constant pullback of this vector-space cone: both the family over \(D\) and its hyperplane-power morphism are pulled back from \(V_m\). Finally, open–closed localization gives \[(j_B)_!j_B^*\mathcal K\longrightarrow\mathcal K \longrightarrow i_*i^*\mathcal K\longrightarrow .\] Since its first term vanishes, this identifies \(\mathcal K\simeq i_*E_D(-m)[-2m]\), proving (29). In particular, the third term is constant along \(D\); no middle-cohomology local system is left unidentified. Localization and proper base change hold with finite coefficients (The Stacks Project Authors 2026, Tags 095L and 0DDE) and in their adic forms (The Stacks Project Authors 2026, Tags 09AH and 09C9); we use these compatibilities after inverting \(\ell\) and extending coefficients to \(E\). ◻

Proof. If \(d(f_s)_u\) is not in the image of \(T^*_{b(u)}\mathcal B_s\), its image in the relative cotangent space for \(b\) is nonzero. The smooth hypersurface criterion shows that \(D\to\mathcal B\) is smooth near \(u\). Lemma 7 then gives the assertion.

Otherwise \(d(f_s)_u=b^*A\) for a non-nilpotent covector \(A\in T^*_{b(u)}\mathcal B_s\). Proposition 16 provides a point \(x\in\mathscr X_s(k)\) and an integer \(m\geq0\) with the following property. Set \[T=U\times_S\mathscr X,\qquad t_0=(u,x),\qquad D_T=D\times_S\mathscr X,\] and let \(F_T\) be the pullback of \(F_U\) to \(T_K\). For the affine quadric over \(T\) \[Q^a=\left\{\sum_{a=1}^m u_av_a+f=0\right\} \subset T\times_S\mathbb A_S^{2m},\] the nearby cycles of the pullback of \(F_T\) vanish at \((t_0,0,0)\). The symbols \(u_a\) here are affine coordinates, distinct from the fixed point \(u\in U_s\). If \(m=0\), this affine model is \(D_T\) itself. Smooth compatibility for \(D_T\to D\) immediately gives the desired vanishing.

Suppose \(m\geq1\) and compactify to \[g:Q_f=\left\{\sum_{a=1}^m u_av_a+fw^2=0\right\} \subset\mathbb P_T^{2m}\longrightarrow T.\] The fiber over \(t_0\) is the split cone of Lemma 17. At every point of this fiber other than its vertex, some \(u_a\) or \(v_a\) is nonzero. In the chart \(u_a=1\), for example, its equation solves uniquely for \(v_a\), so \(g\) is smooth there. This argument works also in characteristic \(2\). Lemma 7 gives \(\Psi_TF_T=0\), and smooth compatibility therefore gives vanishing of the nearby cycles of \(g_K^*F_T\) at all these nonvertex points. The vertex lies in the chart \(w=1\), where its vanishing was supplied by Proposition 16.

The nearby-cycle complex restricts to zero on the entire proper fiber. Proper compatibility and the projection formula give \[ \left(\Psi_T\bigl(F_T\otimes Rg_{K*}E\bigr)\right)_{t_0}=0. \tag{31}\] Apply Lemma 17 on \(T_K\) and tensor its triangle with \(F_T\). The projection formula identifies the third term with \[i_{K*}(F_T|_{(D_T)_K})(-m)[-2m], \qquad i:D_T\hookrightarrow T.\] After applying \(\Psi_T\) and taking the stalk at \(t_0\), the first term vanishes because \(\Psi_TF_T=0\), and the second vanishes by (31). Proper compatibility for the closed immersion \(i\) identifies the third term with \[\bigl(\Psi_{D_T}(F_T|_{(D_T)_K})\bigr)_{t_0} (-m)[-2m].\] It too vanishes. Smooth compatibility for \(D_T\to D\) completes the proof. ◻

Proof. If \(d=1\), the equation in \(B\times\mathbb P^0\) is \(f_1=0\), so the stated triangle is immediate. Suppose \(d\geq2\). Off \(J\), the map \(\mathcal O_B^d\to\mathcal O_B\) with coefficients \((f_1,\ldots,f_d)\) is surjective. Its kernel is a vector bundle of rank \(d-1\), and \(I\) is its projective bundle of lines. The powers of the restricted hyperplane class identify its direct image with the first term of (32). This can also be checked on geometric fibers by proper base change and the cohomology of \(\mathbb P^{d-2}\).

On \(J\), the family is exactly \(J\times\mathbb P^{d-1}\) and its hyperplane class comes from the second factor. The restricted cone of the displayed morphism is therefore the constant top class \(E_J(-(d-1))[-2(d-1)]\). The global cone vanishes off \(J\), so open–closed localization identifies it with the pushforward of this restricted cone, proving the triangle. ◻

Proof. For \(d=0\) the assertion is Lemma 7. For \(d\geq1\) form the proper incidence morphism \[r:I=\left\{\sum_{a=1}^d c_af_a=0\right\} \subset U\times_S\mathbb P_S^{d-1}\longrightarrow U.\] Each affine projective-space chart in \(U\times_S\mathbb P_S^{d-1}\) is still smooth over \(\mathcal B\). On the chart \(c_a=1\), the hypersurface equation is \(f_a+\sum_{b\ne a}c_bf_b=0\). At a point \((u,[c])\) its special-fiber differential is \[\sum_{b=1}^d c_b\,d(f_{b,s})_u.\] Indeed its projective-parameter component is zero since every \(f_b(u)\) is zero. The displayed combination is nonzero by linear independence, and lies outside \((\mathcal N_U)_u\) by the hypothesis. The nilpotent cone on the product chart is the pullback of \(\mathcal N_U\) with zero parameter component. Proposition 19 therefore gives vanishing of the nearby cycles of \(r_K^*F_U\) at every closed point over \(u\).

The restriction of this nearby-cycle complex to the projective fiber is constructible. A nonzero constructible cohomology sheaf on a finite-type scheme over \(k\) has a nonzero stalk at a closed \(k\)-point. Thus the complex vanishes on the entire fiber. Proper compatibility gives \[(\Psi_U Rr_{K*}r_K^*F_U)_u=0.\] Tensor the triangle of Lemma 20 on \(U_K\) with \(F_U\) and apply the projection formula. The first term is a finite sum of shifts and twists of \(F_U\), the second is \(Rr_{K*}r_K^*F_U\), and the third is \[i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)], \qquad i:J\hookrightarrow U.\] After specialization and taking the stalk at \(u\), the first two terms vanish. Proper compatibility for \(i\) shows that the third is the corresponding shift and twist of \((\Psi_J(F_U|_{J_K}))_u\), proving the result. For \(d=1\) the first sum is empty, so the same argument remains valid. ◻

Proof. Let \(\mathcal P_K\) be the bundle. By (Drinfeld and Simpson 1995, Theorem 2 and Remark 2(b)), it is trivial on a nonempty open subset of \(\mathscr X_K\). Here Theorem 2 applies to connected reductive groups, and the base field \(K\) is algebraically closed. Choose a trivialization away from distinct points \(x_1,\ldots,x_r\in\mathscr X(K)\). Gluing at these points gives a chain of one-point modifications \[\mathcal P_0\dashrightarrow\mathcal P_1\dashrightarrow\cdots \dashrightarrow\mathcal P_r=\mathcal P_K, \qquad \mathcal P_0\text{ trivial},\] whose \(i\)th modification has leg \(x_i\). Choose a Schubert bound for each of the finitely many relative positions in this chain.

Starting with the trivial bundle over \(S=\mathop{\mathrm{Spec}}R\), form the space of chains satisfying these bounds, allowing the legs to vary on \(\mathscr X\). At each stage the bounded Hecke space is representable and proper over the preceding bundle and its new leg. Since \(\mathscr X\to S\) is proper, induction shows that the space of chains is a proper algebraic space of finite type over \(S\). It carries the universal final bundle and contains the chosen chain as a \(K\)-point. This point descends to a finite extension of \(\operatorname{Frac}(R)\): the space is of finite type and \(K\) is algebraic over that field. The valuative criterion extends the descended point over \(R'\), and the universal final bundle gives the required extension. The construction allows the legs to coincide on the closed fiber, since it uses successive one-point modifications. ◻

Proof. Descend the finite support data. Let \(A_1,\ldots,A_t\) be the irreducible components of the closure of the nonzero-stalk locus in \(U_K\). Bounded constructibility gives a proper closed subset \(B_i\subsetneq A_i\) such that every stalk on \(A_i\setminus B_i\) is nonzero. Give these closed sets their reduced structures. Their finitely generated ideals descend to a common finite extension of \(\operatorname{Frac}(R)\); enlarge that extension to descend all containments as well. Replace the trait accordingly. The new support closure still meets the special fiber: the finite map from the new model to the old one maps its closure onto the old closure, since it is closed and agrees on the geometric generic support. This step does not assert that \(\mathcal F\) descends.

Write \(\overline A_i\) and \(\overline B_i\) for their horizontal closures in \(U\), and put \(e_i=\dim A_i\). Each \(\overline A_i\) is integral and dominates \(S\). Every nonempty special fiber \((\overline A_i)_s\) is pure of dimension \(e_i\) (The Stacks Project Authors 2026, Tag 0B2J). Applying the same statement to the irreducible components of \(B_i\) shows that \[\dim(\overline B_i)_s<e_i \quad\text{whenever }(\overline B_i)_s\ne\varnothing.\] The trait is excellent, and hence universally catenary: a complete Noetherian local ring is excellent (The Stacks Project Authors 2026, Tag 07QW). Consequently, at every closed point \(v\in(\overline A_i)_s\), the dimension formula gives \[ \dim\mathcal O_{\overline A_i,v}=e_i+1 \qquad\text{\cite[Tag~02JT]{Stacks}}. \tag{34}\]

Choose the special point. Let \(d\) be maximal among the \(e_i\) for which \((\overline A_i)_s\) is nonempty. Such an index exists by the support assumption. Choose a reduced irreducible component \(T\) of dimension \(d\) in one of these special fibers. At a general closed point of \(T\), its reduced structure is smooth, no other distinct reduced component of \(\bigcup_i(\overline A_i)_s\) passes through the point, and none of the \(\overline B_i\) passes through it. To see the last assertion, a component with empty special fiber contributes nothing, while every remaining bad special locus has dimension strictly below \(d\). Different horizontal components \(\overline A_i\) may have the same reduced special component \(T\); we retain all of them.

We can also arrange that \[ \dim\Lambda_u\leq N-d. \tag{35}\] Indeed, restrict \(\Lambda\) to \(T\). Its components that do not dominate \(T\) can be excluded over a proper closed subset of \(T\). The generic fiber of each remaining component has dimension at most \(N-d\), since \(\dim\Lambda\leq N\). The same bound holds over a nonempty open subset of \(T\). Choose \(u\in T(k)\) satisfying all these conditions.

Choose transverse equations. Let \(V=T_u^*U_s\), so \(\dim V=N\). If \(d>0\), a general \(d\)-plane \(L\subset V\) satisfies \(L\cap\Lambda_u\subseteq\{0\}\). Indeed, (35) gives \(\dim\mathbb P(\Lambda_u)\leq N-d-1\), and the incidence of \(d\)-planes meeting this projective set is a proper closed subset of \(\mathop{\mathrm{Gr}}(d,V)\). The condition that \[L\longrightarrow T_u^*T\] be an isomorphism is another nonempty open condition. These two open subsets meet because the Grassmannian is irreducible. Choose such an \(L\) and a basis of it. For \(d=0\), take \(L=0\).

Lift the basis to functions \(f_1,\ldots,f_d\) vanishing at \(u\), on an affine neighborhood in \(U\). Their restrictions to the smooth \(d\)-dimensional variety \(T\) form a regular system of parameters at \(u\). Hence, for \(J=V(f_1,\ldots,f_d)\) and \[Z=J\cap\bigcup_i\overline A_i,\] the point \(u\) is isolated in \(Z_s\) after shrinking. This assertion concerns the union of all the support closures: its reduced special fiber near \(u\) is precisely \(T\), even if several horizontal components specialize to \(T\).

We have obtained the required differentials. It remains to show that the slice retains a nonzero generic stalk and that this stalk survives specialization.

Find a generic generization in the slice. Choose an index \(i\) for which \(e_i=d\) and \(T\) is a component of \((\overline A_i)_s\). By (34), the local ring of \(\overline A_i\) at \(u\) has dimension \(d+1\). Quotienting by \(f_1,\ldots,f_d\) leaves dimension at least one. Its quotient by a uniformizer \(\pi\) has dimension zero, by the isolation just proved. Thus some prime of the cut local ring does not contain \(\pi\): otherwise \(\pi\) would be nilpotent and the ring would have dimension zero. This gives a generic-fiber point \(z\) specializing to \(u\).

The morphism \(Z\to S\) is quasi-finite at \(u\), since \(u\) is an isolated closed point of its fiber (The Stacks Project Authors 2026, Tag 01TH). Its quasi-finite locus is open, so it contains \(z\). Thus \(z\) has residue field finite over \(\operatorname{Frac}(R)\). Its geometric lifts to \(K\) avoid \(B_i\): a point in the closed set \(\overline B_i\) could not specialize to \(u\notin\overline B_i\). Each such lift therefore has a nonzero stalk of \(\mathcal F|_{J_K}\).

Compute nearby cycles on a finite neighborhood. Take an affine neighborhood of \(u\) inside the quasi-finite locus of \(Z\to S\). The henselian decomposition of a finite-type algebra (The Stacks Project Authors 2026, Tag 04GJ) provides a finite local factor containing \(u\). It defines an open neighborhood \(Q\) of \(u\) in \(Z\), finite over \(S\), whose special fiber has only the point \(u\). Equivalently, the separated form of Zariski’s Main Theorem embeds the quasi-finite neighborhood into a finite \(S\)-scheme; its henselian local factor at \(u\) lies in the original open because an open subset of a local spectrum containing the closed point is the entire spectrum. Every generization of \(u\), including \(z\), lies in \(Q\).

The restriction \(\mathcal F|_{J_K}\) is supported on the closed subset \(Z_K\). To use the finite neighborhood, remove \(Z\setminus Q\) from \(J\). We realize this removal in the ambient chart: the complement is closed in \(J\), hence in \(U\), so remove it from \(U\) and then choose an affine neighborhood of \(u\). This neighborhood contains all of \(Q\), since every point of the finite local scheme \(Q\) specializes to \(u\). The new \(J\) is still cut out by the same functions in a smooth affine chart; the change leaves its nearby-cycle stalk at \(u\) unchanged and makes \(Q\) closed in \(J\). If \(\iota:Q\hookrightarrow J\) denotes this closed immersion and \(\mathcal H=\mathcal F|_{Q_K}\), localization gives \[\mathcal F|_{J_K}\simeq\iota_{K,*}\mathcal H.\] Proper compatibility for \(\iota\) therefore reduces (33) to the stalk of \(\Psi_Q\mathcal H\) at \(u\). Let \(a:Q\to S\) be the finite structural map. Its special fiber has only the point \(u\), so proper compatibility gives \[(\Psi_Q\mathcal H)_u \simeq \Psi_S\bigl(\mathop{\mathrm{R\Gamma}}(Q_K,\mathcal H)\bigr) \simeq \mathop{\mathrm{R\Gamma}}(Q_K,\mathcal H).\] The second identification uses geometric specialization on the trait, which is the identity on coefficient complexes. The finite \(K\)-scheme \(Q_K\) has the étale topos of a finite discrete set. Consequently \[\mathop{\mathrm{R\Gamma}}(Q_K,\mathcal H) \simeq\bigoplus_{x\in|Q_K|}\mathcal H_x.\] At least one summand is nonzero, as proved above, so this complex is nonzero. This proves (33).

These are geometric nearby cycles with the monodromy action forgotten; no inertia invariants are taken. The displayed finite direct-sum calculation is compatible with finite coefficients, their \(\ell\)-adic limit, inversion of \(\ell\), and extension to \(E\). It therefore applies to the specialization functor of Section 2 without requiring a field of descent for \(\mathcal F\). ◻

Proof of Theorem 4. If \(Y'\ne Y\), Proposition 6 supplies the nonzero compact object \(F\) used above. Lemma 22 and the subsequent chart construction give a smooth affine \(U\to\mathcal B\) of constant relative dimension \(N\) for which the closure of the nonzero-stalk locus of \(F_U\) meets \(U_s\).

Let \(\mathcal N_U\subset T^*U_s\) be the pullback of the global nilpotent cone. Under either displayed hypothesis regime, the nilpotent-parabolic assumption and invariant form give the half-dimensionality theorem of (Arinkin et al. 2022b, Appendix D.1.1–D.1.8). In its smooth-chart formulation, (Arinkin et al. 2022b, Appendix D.1.2), this says exactly that \[\dim\mathcal N_U\leq\dim U_s=N.\] Apply Lemma 23 with \(\mathcal F=F_U\) and \(\Lambda=\mathcal N_U\). It gives a slice \(J=V(f_1,\ldots,f_d)\) and \(u\in J_s(k)\) for which \[\bigl(\Psi_J(F_U|_{J_K})\bigr)_u\ne0,\] while the independent differentials of its equations span a plane meeting \((\mathcal N_U)_u\) only at zero. The higher-codimension vanishing of Proposition 21 applies to this same chart and slice and says that this stalk is zero. This contradiction also covers \(d=0\), when the test is the original smooth-chart vanishing.

Thus the complementary union is empty. The primed equivalence recalled in Section 2 is consequently the full restricted equivalence asserted in Theorem 4. ◻

Proof. The decomposition \(Y=Z\sqcup(Y\setminus Z)\) decomposes quasi-coherent and nilpotent ind-coherent sheaves into their two corresponding summands. Consequently \[\mathop{\mathrm{QCoh}}(Z)\underset{\mathcal A}\otimes\mathcal D \simeq\mathop{\mathrm{IndCoh}}_{\mathop{\mathrm{Nilp}}}(Z).\] Tensor the \(\mathcal A\)-linear equivalence of Theorem 4 with \(\mathop{\mathrm{QCoh}}(Z)\) over \(\mathcal A\). This proves the asserted identification by the specified functor.

Every component contains a semisimple \(E\)-point (Arinkin et al. 2022b, Proposition 3.7.2 and Corollary 3.7.4). Pullback to this point shows that \(\mathcal O_Z\ne0\). The fully faithful embedding \(\mathop{\mathrm{QCoh}}(Z)\hookrightarrow\mathop{\mathrm{IndCoh}}_{\mathop{\mathrm{Nilp}}}(Z)\) therefore proves nonvanishing. ◻

Proof. The primed union \(Y'\) is Frobenius-stable. The spectral trace calculation of Beraldo–Lin–Reeves (Beraldo et al. 2024, secs. 6.4.12–6.4.13) enters the known primed formula (Gaitsgory and Raskin 2025a, secs. 1.5.3–1.5.5 and Corollary 1.5.6): \[\mathop{\mathrm{Funct}}_c\bigl(\mathop{\mathrm{Bun}}_{G_0}(\mathbb F_q),E\bigr) \simeq \Gamma^{\mathop{\mathrm{IndCoh}}}\bigl((Y')^{\mathop{\mathrm{Frob}}}, \omega_{(Y')^{\mathop{\mathrm{Frob}}}}\bigr).\] Its automorphic trace input is (Arinkin et al. 2022a, Theorem 0.2.6), with the endofunctor given by pushforward along geometric Frobenius on \(\mathop{\mathrm{Bun}}_G\). By Theorem 4, the actual inclusion \(Y'\hookrightarrow Y\) is equality. Taking derived fixed points identifies \((Y')^{\mathop{\mathrm{Frob}}}\) with \(Y^{\mathrm{arithm}}\), and transports the same dualizing object and global-sections functor. Substitution gives the result. ◻

Proof. Presentability gives the right adjoint. Every compact object of \(\mathcal A_0\) is dualizable, so its image under \(a\) is a perfect complex of \(E\)-vector spaces. Thus \(a\) preserves compact objects. Testing against compact generators and using adjunction shows that \(R_a\) preserves filtered colimits. The right adjoint is exact, since both categories are stable. Exactness and preservation of filtered colimits imply preservation of all colimits.

The adjunction has a canonical projection morphism \[b\otimes R_a(V)\longrightarrow R_a\bigl(a(b)\otimes V\bigr).\] For dualizable \(b\), its invertibility follows by testing maps out of an arbitrary \(c\in\mathcal A_0\): \[\begin{align*} \mathop{\mathrm{RHom}}_{\mathcal A_0}(c,b\otimes R_a(V)) &\simeq\mathop{\mathrm{RHom}}_{\mathcal A_0}(b^\vee\otimes c,R_a(V))\\ &\simeq\mathop{\mathrm{RHom}}_E(a(b)^\vee\otimes a(c),V)\\ &\simeq\mathop{\mathrm{RHom}}_E(a(c),a(b)\otimes V). \end{align*}\] Both sides of the projection morphism preserve colimits in \(b\). Compact generation therefore proves invertibility for every \(b\). These morphisms are the adjunction mates of the module structure on \(a\); hence their unit and associativity compatibilities hold before invertibility is imposed. They now give a coherent module-functor structure on \(R_a\). Finally, \[\mathop{\mathrm{RHom}}_{\mathcal A_0}(\mathbf1,R_a(E)) \simeq\mathop{\mathrm{RHom}}_E(E,E)\simeq E,\] so \(R_a(E)\ne0\). This last argument does not require a compact unit. ◻

Proof. Apply Lemma 26 to \(a=\sigma^*:\mathcal A\to\mathop{\mathrm{Vect}}_E\), and write \(\sigma_*\) for its continuous right adjoint. The composite \[\Phi_\sigma: \mathop{\mathrm{Vect}}_\sigma\xrightarrow{\sigma_*}\mathcal A \xrightarrow{\jmath_Y}\mathcal D \xrightarrow{(\mathbb L_G^{\mathrm{restr}})^{-1}}\mathcal C_s\] is a continuous \(\mathcal A\)-linear functor. Here \(\mathop{\mathrm{Vect}}_\sigma\) denotes \(\mathop{\mathrm{Vect}}_E\) with its action through \(\sigma^*\). Set \(M_\sigma=\Phi_\sigma(E)\). It is nonzero by the lemma, full faithfulness of \(\jmath_Y\), and the Langlands equivalence. If \(\sigma\) lies in \(Z\), the idempotent \(e_Z\) pulls back to \(E\), while its complementary idempotent pulls back to zero. Linearity therefore places \(M_\sigma\) in \(\mathcal C_{s,Z}\).

To verify the eigenstructure, fix a finite set \(I\). The action of \(\mathcal E_V^I\) on the automorphic category is the multi-leg Hecke functor \(H_I(V,-)\) (Arinkin et al. 2022b, secs. 14.2–14.3). Extend \(\Phi_\sigma\) by the identity on \(\mathcal Q_I\). Its module structure gives \[H_I(V,M_\sigma) \simeq(\Phi_\sigma\otimes\mathop{\mathrm{id}})\bigl(\sigma^I(V)\bigr) \simeq M_\sigma\boxtimes\sigma^I(V).\] The second equivalence uses that a continuous \(E\)-linear functor from \(\mathop{\mathrm{Vect}}_E\) is tensoring with its value on \(E\).

These equivalences are natural in \(V\). They preserve tensor products and the unit by the coherent module structure of \(\Phi_\sigma\), and are compatible with collisions of legs and maps between finite sets by the universal symmetric monoidal evaluation system. Thus they supply the homotopy-coherent Hecke eigenstructure of (Arinkin et al. 2022b, sec. 15.2). Equivalently, restricting \(\Phi_\sigma\) along the symmetric monoidal Ran-to-spectral action gives the eigenobject in (Arinkin et al. 2022b, sec. 15.1). The evaluation uses \(\sigma\) itself, so its eigenvalue is the actual parameter, whether or not it is semisimple. ◻

Proof. Constructible coefficient extension. We first prove the ambient statement. For a finite extension \(L/E_0\), coefficient extension and restriction preserve bounded constructibles. On objects extended from \(E_0\), the mapping complexes are obtained by tensoring their original mapping complexes with \(L\). Moreover, every \(L\)-constructible \(M\) is a retract of \(L\otimes_{E_0}\operatorname{Res}_{L/E_0}M\): the multiplication map admits an \(L\)-linear section given by the separability idempotent of \(L/E_0\). Thus the induced compact objects have the required mapping complexes and generate after taking retracts. Constructible \(E\)-complexes and their morphisms are obtained by passage through the finite coefficient fields. These coefficient conventions and the compact-object description are supplied by (Hemo et al. 2023, Proposition 5.2, Theorem 7.7, Lemma 7.9, Proposition 8.2 and Corollary 8.3); see also (Hansen and Scholze 2023, sec. 2). The cohomological finiteness condition (8.1) of (Hemo et al. 2023) holds on every finite-type chart over the separably closed field \(k\), with either algebraic \(\mathbb Q_\ell\)-coefficient field, by Lemma 8.6(1) there. Thus its compactness statements apply to the charts used here. Passing to ind-completions proves (36).

Detecting the cohomological support condition. View the left side of (36) as internal \(E\)-module objects in \(\mathop{\mathrm{Shv}}_{E_0}(T)\). We will show that such a module satisfies the \(E\)-coefficient support condition exactly when its underlying \(E_0\)-object satisfies the \(E_0\)-condition. This will identify the required subcategories degree by degree, including their unbounded objects.

Write \(F\) for the free-module functor and \(U\) for its forgetful functor. Both are exact for ordinary and perverse cohomology. For \(U\), this assertion includes infinite coefficient extension: if a constructible \(E\)-object has a form \(M_L\) over a finite extension \(L/E_0\), its underlying \(E_0\)-object is \[ U(M_L\otimes_L E)= \mathop{\operatorname{colim}}_{L\subset L'\subset E, \,[L':E_0]<\infty} \operatorname{Res}_{L'/E_0}(M_L\otimes_L L'). \tag{39}\] Finite extension and restriction are t-exact, and the ind-t-structures commute with filtered colimits. This proves the assertion first on constructibles, then on the ambient ind-categories.

For bounded constructibles, coefficient extension preserves and reflects the singular-support bound. The defining test pairs express this bound by universal local acyclicity. We use its dualizability criterion in the algebraic \(\mathbb Q_\ell\)-coefficient setting of (Hansen and Scholze 2023, Theorem 1.6). The correspondence category in §3 of that paper has compositions formed from pullback, tensor product and \(f_!\). These operations commute with coefficient extension by the coefficient formalism and projection formula of §2 there. Extending the duality data therefore preserves universal local acyclicity.

For reflection, suppose a bounded constructible complex over a finite extension \(L/E_0\) becomes universally locally acyclic over \(E\). The relative dual of its \(E\)-extension is constructible by (Hansen and Scholze 2023, Proposition 3.4(ii)). The dual, evaluation and coevaluation maps, and both triangle identities are finite data on the fixed finite-type test schemes and their products. By (Hemo et al. 2023, Lemma 7.9), they descend to a common finite coefficient extension. Thus the complex is already universally locally acyclic over that finite extension. Finite extension reflects local acyclicity: after forgetting the extension, the local-acyclicity test is a nonzero finite direct sum of the original test. This proves reflection for each test pair and hence for the singular-support bound. Finite restriction preserves the same bound, since after a splitting coefficient extension it is a finite direct sum of coefficient conjugates. The same arguments, or the finite-rank local-system description, apply to lissity.

It follows that \(F\) and \(U\) preserve the relevant ind-perverse hearts. For \(U\), use (39) and filtered colimits; a finite rank over \(E\) is not being treated as a finite rank over \(E_0\). To check the converse explicitly, let \(P\) be a perverse internal \(E\)-module and suppose that \(U(P)\) lies in the required ind-heart over \(E_0\). The counit \[F(U(P))\longrightarrow P\] is an epimorphism in the module heart: its underlying map splits by \(u\mapsto1\otimes u\). Its source belongs to the required ind-heart over \(E\). That heart is closed under quotients, by the Serre property of the singular-support heart. Hence \(P\) belongs to it as well. Thus, in every degree, the cohomological support condition for an \(E\)-module is precisely the condition on its underlying \(E_0\)-object. For \(\mathop{\mathrm{QLisse}}\), apply this argument to the zero-cone ind-perverse heart and use its equivalence with the ordinary-cohomology definition (Arinkin et al. 2022b, sec. 1.2.9). Ordinary lisse sheaves need not form a Serre subcategory of the full ordinary heart; no such assertion is used.

Scalar extension and smooth descent. We have identified which ambient \(E\)-modules satisfy the support condition. To realize them as the categorical scalar extension of the corresponding \(E_0\)-subcategory, we use their free-module resolutions.

The perverse Serre condition gives closure under finite cones. Since perverse cohomology commutes with filtered colimits, these subcategories are closed under filtered colimits and arbitrary direct sums, hence under all colimits, including bar realizations. They are also closed under scalar tensors. Internal modules in each such subcategory are therefore exactly the ambient modules whose underlying objects belong to it. The free-forgetful bar resolution realizes every module from free modules within that subcategory. This identifies these module categories with the corresponding categorical scalar extension and proves (37) and the corresponding singular-support comparison on smooth charts. Ordinary tensor products and their structural maps commute with coefficient extension, giving the monoidal assertion. Connective free modules generate the connective module category under connective colimits, by the same bar resolution, so the induced tensor t-structures agree.

Finally, take the smooth-chart limits defining sheaves on a stack. Internal modules commute with these limits. Equivalently, \(\mathop{\mathrm{Vect}}_E\) is the module category of the continuous monad \(E\otimes_{E_0}(-)\) on \(\mathop{\mathrm{Vect}}_{E_0}\) and is dualizable as a presentable \(E_0\)-linear category (Gaitsgory et al. 2022, Lemma 1.6.3). Tensoring with a dualizable category has tensoring with its dual as a left adjoint, so it preserves limits. The smooth pullbacks defining these limits are continuous. Applying the chartwise support comparison proves (38). ◻

Proof. Let \(A\) be a connective commutative \(E\)-algebra and write \(\operatorname{Mod}_A=\mathop{\mathrm{QCoh}}(\mathop{\mathrm{Spec}}A)\). Lemma 29 and associativity give a symmetric monoidal equivalence \[\operatorname{Mod}_A\otimes_{E_0}\mathop{\mathrm{QLisse}}_{E_0}(X) \simeq \operatorname{Mod}_A\otimes_E\mathop{\mathrm{QLisse}}_E(X).\] The tensor t-structures are generated by connective objects of the factors; equivalently they are the module t-structures with t-exact forgetful functor (Arinkin et al. 2022b, sec. 1.4.1). They agree under this identification by the connective assertion in Lemma 29.

For the split dual group, scalar extension identifies \(\mathop{\mathrm{Vect}}_E\otimes_{E_0}\mathop{\mathrm{Rep}}_{E_0}(\check G_0)\) with \(\mathop{\mathrm{Rep}}_E(\check G)\) as a symmetric monoidal category with t-structure. In characteristic zero, finite-dimensional representations in the heart generate its connective objects; over a split reductive group the irreducible highest-weight representations are defined over \(E_0\) and remain irreducible after extension (Milne 2022, Theorem 22.2, §§22.3–22.5 and Theorem 22.42). The universal property of scalar extension therefore identifies tensor functors into the displayed target, and right t-exactness is equivalent before and after this extension. This proves the first assertion on every connective affine \(E\)-algebra \(A\), naturally in \(A\), using precisely the moduli definition (Arinkin et al. 2022b, sec. 1.4.2).

Present \(Y_0\) as the colimit of its affine test schemes. Base change presents the left side of (40) by the base-changed affines. Quasi-coherent categories form the corresponding limit. On an affine, the coefficient comparison is the usual equivalence \(\mathop{\mathrm{Vect}}_E\otimes_{E_0}\operatorname{Mod}_B \simeq\operatorname{Mod}_{B\otimes_{E_0}E}\). The affine pullbacks in this diagram are continuous. The categorical dualizability of \(\mathop{\mathrm{Vect}}_E\), justified in Lemma 29, therefore lets tensoring commute with this limit and proves the second assertion, including its monoidal structure. No finite-dimensionality of \(E\) over \(E_0\) is required.

Evaluation of a representation at the tensor functor parametrized by \(\mathop{\mathrm{Spec}}A\) is unchanged by these identifications. The assertion is natural in representations, affine tests and maps of finite sets. It therefore identifies the entire universal evaluation system, including tensor products and collisions of legs. ◻

Proof. First, \(\mathop{\mathrm{QLisse}}_E(X)=\mathop{\mathrm{IndLisse}}_E(X)\) for this curve (Arinkin et al. 2022b, sec. 1.3.3 and §E.2.8). The same equality holds over \(E_0\). To see this, extend an object of \(\mathop{\mathrm{QLisse}}_{E_0}(X)\) and express it using the constructible lisse generators over \(E\). Forgetting coefficients takes those generators into the colimit closure of the \(E_0\)-constructible lisse objects, by (39). The original object is a retract of its extension followed by forgetting: choose an \(E_0\)-linear retraction \(E\to E_0\) of the unit. Constructible lisse objects are compact in the ambient ind-constructible category. Thus these \(\mathop{\mathrm{QLisse}}\) categories are compactly generated and dualizable. Their exterior product and affine-limit evaluation constructions consequently commute with the coefficient comparisons above.

Fix the geometric Satake identification over \(E_0\), available in (Mirković and Vilonen 2018, Theorem 14.1), and use its scalar extension over \(E\). Take the same dual-group identification, cohomological shifts and geometric Tate-line conventions. We compare the whole finite-set monoidal diagram giving the Hecke functors, not only its objects indexed by highest weights. On each bounded Schubert support, coefficient extension commutes with exterior products, pullbacks, descent and proper convolution pushforwards. It is perverse t-exact, so it also commutes with intermediate extension, expressed as the image of the map from perverse extension by zero to perverse direct image. Hence it takes the Satake IC objects, their convolution maps, and the fusion diagrams with their unit and symmetry constraints to the corresponding diagrams over \(E\). The fiber and weight functors in geometric Satake identify the same split dual group under this operation (Mirković and Vilonen 2018, secs. 4–6 and Theorem 14.1).

For completeness, the derived functors used here are the derived extension of naive geometric Satake. In (Gaitsgory et al. 2022, secs. B.3.5–B.3.8) the compatible monoidal diagram in the shifted perverse hearts is extended using the bounded-derived universal property. The induced right-lax monoidal structure is strict; one then restricts to compact representations and ind-extends. Apply this construction to the entire coefficient-comparison diagram. Naturality gives the comparison of the derived functors and all finite-set structural maps; the monoidal equivalences persist under the exact coefficient functor. The correspondence action of (Gaitsgory et al. 2022, secs. B.2.5–B.2.10) then compares the continuous Hecke systems, including the units, composition maps and base-change compatibilities used in their action.

The factorization into nilpotent sheaves with \(\mathop{\mathrm{QLisse}}\)-valued legs is through the fully faithful exterior-product functors (Arinkin et al. 2022b, sec. 14.2.5). Lemma 29 compares these subcategories, so the comparison also holds after this factorization. Lemma 30 identifies universal evaluation. By the assumed rational datum, restricting the extended spectral action along evaluation therefore gives exactly the established coherent Hecke system over \(E\). The equivalence of spaces of actions in (Arinkin et al. 2022b, Theorem 8.1.4), applied over the algebraically closed field \(E\), identifies it with the established spectral action of (Arinkin et al. 2022b, Theorem 14.3.2). ◻

Proof. Let \(e_U\) denote \(\mathcal O_U\) extended by zero in \(\mathop{\mathrm{QCoh}}(Y_0)\). Its extension is the analogous idempotent \(e_{U_E}\), where \(U_E=U\times_{E_0}E\). Compatibility of the actions and generation by scalar-extended objects show that \(e_{U_E}\) acts by zero on all of \(\mathcal C_E\).

The curve and split group descend to a finite field, so Theorem 4 in regime A identifies this action with the action on \(\mathop{\mathrm{IndCoh}}_{\mathop{\mathrm{Nilp}}}(Y)\). Apply \(e_{U_E}\) to the object \(\jmath_Y(\mathcal O_Y)\), with \(\jmath_Y\) as defined in Section 8. Its value is \(\jmath_Y(e_{U_E})\). Full faithfulness of \(\jmath_Y\) implies \(e_{U_E}=0\), hence \(U_E=\varnothing\).

To descend this equality, take any affine test \(T=\mathop{\mathrm{Spec}}A\to Y_0\). The inverse image of \(U\) is an open-and-closed affine summand of \(T\), cut out by an idempotent of \(\pi_0(A)\). Its base change to \(A\otimes_{E_0}E\) is empty. Faithful flatness of \(E/E_0\) forces that idempotent to vanish. This holds on every affine test, so \(U=\varnothing\). No rational-point description of components is used. ◻

Proof of Theorem 5. For \(X\ne\mathbb P^1\), Lemma 31 supplies the hypothesis of Lemma 32. An open-and-closed localization is the image of its idempotent projection, so the vanishing of the localized category is exactly zero action of that idempotent. The lemma proves the assertion.

If \(X=\mathbb P^1\), the geometric spectral prestack \(Y\) has one connected component: its components are indexed by semisimple geometric local systems (Arinkin et al. 2022b, Proposition 3.7.2), and the geometric étale fundamental group of \(\mathbb P^1\) is trivial. Indeed, a connected finite étale cover \(C\to\mathbb P^1\) of degree \(n\) satisfies \(2g(C)-2=-2n\) by Riemann–Hurwitz (The Stacks Project Authors 2026, Tag 0C1B), so \(n=1\). By Lemma 30 and the faithful affine test in Lemma 32, \(Y_0\) has no nonempty proper open-and-closed substack. The category \(\mathcal C_0\) is nonzero; for instance its dualizing object has shifted constant restrictions on smooth charts and has zero singular support. Thus its only nonempty open-and-closed localization, the whole category, is nonzero. This also proves the theorem in genus zero.

Finally, the stipulated complement of \(Y'_0\) acts by zero, so it is empty. In particular a given spectral-linear primed equivalence has this property by linearity, and its specified inclusion is an equality of prestacks. ◻

Proof. Theorem 5 identifies the actual inclusion \(Y'_0\subset Y_0\) with equality. Its derived fixed-point stacks, dualizing objects and ind-coherent global-sections functors are therefore the same. Substitute this equality in the given map. ◻