Frobenius Structures on Tame Hecke Eigensheaves

1 Introduction

A Hecke eigensheaf realizes a local system on a curve by the geometry of modifications of bundles. Over a finite field, an arithmetic local system contains more information than its geometric restriction: it also specifies Frobenius. The corresponding automorphic object must therefore carry a Frobenius structure that preserves its eigenisomorphisms. We construct such objects with Borel level and tame regular-unipotent monodromy, after a finite extension of the field of constants.

1.1 The result

For a smooth projective pointed curve (X,x), let G=\SL_n and let B be its upper-triangular Borel. The stack \Bun_{G,B,x}(X) parametrizes G-bundles on X with a B-reduction at x. A sheaf on this stack is locally constructible perverse if its pullback to each smooth finite-type scheme chart, shifted by the relative dimension of that chart, is bounded constructible and perverse. Its parabolic global nilpotent cone \Lambda_{\mathrm{par}} consists of generically nilpotent Higgs fields with at most a simple pole at x and residue in the nilradical of the chosen Borel. Singular support is interpreted on the same smooth charts.

For representations \boldsymbol V=(V_i)_{i\in I} of the dual group \PGL_n, write \Hecke_{I,\boldsymbol V} for the Hecke functor with legs in (X\setminus\{x\})^I. We use the relative Satake normalization of Section 2, with no moving-leg shift. In particular, the tensor-unit functor is exterior product with the constant \E-sheaf on (X\setminus\{x\})^I.

A Weil structure over \F_{q^m} means an isomorphism between the q^m-Frobenius pullback of the geometric sheaf and the sheaf itself. We require the eigenisomorphisms to respect this structure and the Frobenius structure on the parameter.

Write k=\overline{\F}_q, X=X_0\times_{\F_q}k, x=(x_0)_k, and U=U_0\times_{\F_q}k for the geometric curve and its punctured complement. The Frobenius structure is required on the entire eigenstructure, rather than only on the underlying perverse sheaf. The finite extension in the theorem arises when a point on an auxiliary bundle stack is made invariant under a power of Frobenius. We use Weil structures, so a compatible action of the infinite cyclic group is sufficient.

1.2 Background and the geometric input

The geometric Langlands program replaces automorphic functions by sheaves on moduli stacks of bundles and asks that Hecke eigenvalues be realized as local systems on a curve. Drinfeld’s rank-two work and Laumon’s geometric formulation established central parts of this perspective [Dri, Lau]. Frenkel, Gaitsgory, and Vilonen constructed unramified \GL_n eigensheaves, including the finite-field Weil setting; Gaitsgory subsequently gave a geometric proof of the vanishing theorem underlying that construction [FGV, Gai04]. Thus the arithmetic compatibility of an eigenstructure is a classical requirement, not a new condition introduced here.

For general groups, the restricted spectral stack and its action on automorphic sheaves with nilpotent singular support were developed by Arinkin, Gaitsgory, Kazhdan, Raskin, Rozenblyum, and Varshavsky [AGKRRV]. Gaitsgory and Raskin prove the restricted geometric Langlands equivalence in characteristic zero and construct its positive-characteristic specialization; in positive characteristic their general-group result concerns a union of spectral components [GR]. We use the characteristic-zero equivalence and the spectral action, with their precise scopes, in Section 3.

Our geometric input is the companion manuscript Constructible tame Hecke eigensheaves in positive characteristic [CT]. It constructs geometric, locally constructible perverse eigensheaves with Borel level, parabolic nilpotent singular support, and the full fusion-compatible eigenstructure. Its method first creates level structure by a characteristic-zero degeneration to a separating node and then specializes to positive characteristic. We use its nearby-cycle, nonvanishing, perverse-truncation, and enhancement-descent theorems, stated at their uses below. Those results do not supply Frobenius descent. The present paper proves the additional arithmetic compatibility for \SL_n under the hypotheses of Theorem 1.1.

In characteristic-zero de Rham sheaf theory, Færgeman constructs coherent tame eigenobjects for irreducible regular-singular parameters, conditional on the spectral-action and localization-compatibility inputs specified in [Fær, Sections 1.3.6–1.4.3]. Regular holonomicity and generic perversity in the tame setting remain expectations in that preprint [Fær, Remark 1.4.1.2]. Our theorem establishes perversity in the geometric \ell-adic setting and equips the eigenstructure with a finite-field Weil action, for the dense parameters with regular-unipotent boundary monodromy specified above.

The nodal geometry is part of the automorphic gluing framework of Nadler and Yun [NY], with twisted curves as in Abramovich–Vistoli and Olsson [AV, Ols]. Root stacks and proper henselian invariance of finite covers allow us to retain finite-quotient monodromy through the two degenerations [LO, Ryd]. The sheaf-theoretic specialization used here is geometric nearby cycles, in the form developed by Hansen–Scholze and Gaitsgory–Raskin and adapted to level structure in [CT, HS, GR].

1.3 The arithmetic constructions and the proof

There are two extra difficulties in making the geometric construction equivariant. First, an invariant parameter need not make an arbitrarily chosen geometric eigenobject visibly invariant. Second, the two boundary parameters at a separating node must be glued by a conjugation that preserves both Frobenius and compact image.

Section 3 treats the first difficulty. For a dense unramified \PGL_n-parameter in characteristic zero, the relevant restricted spectral component is a smooth formal space with one point. Transport of the spectral action preserves that component. The inverse image of its skyscraper has scalar endomorphisms and no negative self-extensions; these properties recognize its transport up to shift. An invariant polarization on the curve rules out a nonzero shift on the automorphic side, giving a generator isomorphism on this particular eigenobject. Density and the trivial center of \PGL_n force compatibility with its full eigenstructure. This recognition argument avoids choosing an equivariant normalization of the Langlands equivalence itself.

Section 4 first lifts the original arithmetic parameter to a punctured curve C_1 in characteristic zero. At its puncture the Frobenius matrix A and the regular nilpotent logarithm N_1 satisfy \Ad(A)N_1=qN_1. A cocharacter commuting with A scales N_1. We choose a ramified cover C_2\to C_1 whose degree makes the required gluing conjugator lie in a fixed compact open subgroup. The resulting boundary identification commutes with A. This compact equivariant gluing is the second reusable ingredient: inverse boundary monodromies alone would not ensure a continuous compact-valued parameter on the smoothing.

The compatible finite torsor towers on the two punctured components glue on balanced twisted nodal models and lift to a dense unramified parameter on their smooth generic curve. Apply the equivariant unramified result there. In Section 5, nearby cycles on a prescribed node type, restriction to one component, and descent from enhanced to ordinary flags produce an equivariant perverse eigensheaf on C_1. The restriction uses a point defined over a finite extension of the characteristic-zero base; a power of the generator fixes that point. Finally, Section 6 specializes this eigensheaf to the original finite-field curve. Both specializations use a separate support-closure argument to prove nonvanishing. Naturality of the comparisons in Section 2 carries the entire eigenstructure through these steps.

2 Equivariant Hecke systems and specialization

We fix the sheaf conventions and explain which parts of the geometric specialization formalism carry a semilinear action. The issue is compatibility of the entire Hecke system, including collisions of moving points.

2.1 Sheaves, level structures, and moving legs

Write E=\E, G=\SL_n, and \Gd=\PGL_n, with the split forms understood over every base field. On a smooth stack \mathcal B locally of finite type, D_{\lcc}(\mathcal B,E) denotes the category of geometric adic complexes whose pullback to each smooth finite-type scheme chart is bounded and constructible. The bounds may depend on the chart. A complex F is perverse if b^*F[d] is perverse for every smooth chart b:Y\to\mathcal B of constant relative dimension d. These conventions agree with [CT, Section 2.1].

For a pointed smooth projective curve (X,x), put

An enhanced flag is a reduction to R_u(B); forgetting the enhancement is a representable T-torsor \mathcal A^+\to\mathcal A. At ordinary Borel level, the trace form identifies a cotangent vector with a Higgs field

where B_P is the chosen Borel in the fiber at x. The cone \Lambda_{\mathrm{par}} consists of those fields that are nilpotent over the function field of X. On a smooth chart, singular support means the geometric adic singular support of the pulled-back complex, contained in the smooth cotangent pullback of this cone; see [CT, Section 3.1] and [Bei, Bar].

Let U be the locus of allowed modifications, disjoint from the level point. For a finite set I, the Hecke stack \mathcal H_I has an input bundle map p_I and an output-and-leg map o_I:\mathcal H_I\to\mathcal B\times U^I. If \boldsymbol V=(V_i)_{i\in I} are representations of \Gd, define

The twisted product is ordinary exterior product in input frames, descended using equivariance of the Satake kernel. All pushforwards are taken on finite Schubert bounds, where the output maps are proper. On a fixed-point Schubert orbit of dimension d_\lambda=\langle2\rho_G,\lambda\rangle, the simple kernel is normalized by

Here \rho_G is the half-sum of positive roots. Since the cocharacter lattice of G is its coroot lattice, d_\lambda is even. In particular all twists in (2.2) are integral. There is no moving-leg shift in (2.1). We use geometric Satake with its fusion tensor structure [MV].

For a surjection a:I\twoheadrightarrow J, let \delta_a:U^J\to U^I be the collision diagonal and set W_j=\bigotimes_{i\in a^{-1}(j)}V_i. The relative fusion convention is

This is ordinary pullback, without a codimension shift.

Let \sigma be a semilinear automorphism of the ground field and the curve, preserving the split group and the level data. An equivariant sheaf has an isomorphism \sigma^*F\simeq F; an equivariant eigenstructure requires (2.4) to commute with this isomorphism and the specified isomorphism \sigma^*\vartheta\simeq\vartheta. We use the group generated by \sigma, so the one isomorphism determines all its iterates and inverses. Transport acts on the base, not on the coefficient field E. At a finite-field fiber the ground-field action a\mapsto a^q gives the arithmetic convention. Inverting the generator everywhere gives the usual geometric Frobenius convention for Weil sheaves.

2.2 Satake transport and its Weil normalization

2.3 The specialization input with an action

The geometric result we need is stronger than one-point Hecke commutation. It is the combination of Propositions 2.1 and 5.1 of [CT], with the following relative setup. Let S=\Spec R be an excellent complete strictly henselian discrete valuation trait with algebraically closed residue field and with \ell invertible. Fix a geometric generic point \bar\eta and write s for the closed point. Let \mathcal B/S be a smooth locally finite-type bundle stack and let U/S be a smooth scheme of relative dimension one of allowed legs. Bounded Hecke correspondences have proper representable output maps, and in coordinates and input frames they have the split Schubert local models with their spherical kernels. The setup includes ordinary and enhanced level disjoint from the legs, and the twisted nodal families used below.

2.4 The characteristic restriction for \SL_n

The remaining geometric inputs of [CT] have four Lie-theoretic hypotheses: a nondegenerate invariant symmetric form, also nondegenerate on every Levi’s Lie-algebra center; Chevalley restriction for every Levi; scheme-theoretic semisimple centralizers that are Levis; and containment of every nilpotent element, over every field extension, in the nilradical of a parabolic defined over that field. We verify them to make the characteristic restriction in the main theorem explicit.

3 An equivariant unramified eigencomplex

The first specialization will start with an unramified eigencomplex on a smooth projective curve in characteristic zero. The geometric existence theorem supplies such a complex, but does not by itself supply the semilinear action needed here. We obtain that action by using the particular eigencomplex attached to a spectral skyscraper. Its spectral component is preserved by transport, and the skyscraper can be recognized from its endomorphisms. A second rigidity argument then shows that every isomorphism with its transport respects the full eigenstructure.

Here f^* denotes transport on the curve and on its bundle stack, with the coefficient field \E fixed. We prove the proposition in three steps: identify the spectral summand and its behavior under transport, recognize the transported object without a shift, and prove uniqueness of the coherent eigenstructure on that object.

3.1 The spectral skyscraper and transport of the action

Write

The first stack parametrizes local systems with restricted variation; the second is the full automorphic category with global nilpotent singular support. We use the following precise characteristic-zero input from the proof of [CT, Lemma 8.1]. The Gaitsgory–Raskin equivalence

is linear for the \QCoh(\mathcal S)-action, and the inclusion of \mathcal D_C in the ambient automorphic category preserves compact objects [GR, Main Theorem 1.3.9(ii), Lemma 1.3.7, and Theorem 1.1.7]. These statements apply to geometric \E-adic sheaves over an arbitrary algebraically closed characteristic-zero field [GR, Section 2.3]. The spectral action agrees, through universal evaluation, with the coherent Hecke action [AGKRRV, Main Theorem 14.3.2 and Section 14.3.3]. As in [CT, Lemma 8.1], we identify the external Satake labels with our input–output convention: if a tensor relabeling \alpha is required, the external point is \tau\circ\alpha^{-1}. Evaluation at the point denoted by \tau below is therefore exactly V\mapsto V_\tau.

Let i_\tau:\Spec\E\to\mathcal S denote this point, and put

We recall the local description used in [CT, Lemma 8.1], since it will also recognize the transport of this object. The components of \mathcal S are indexed by semisimple parameters; the component of an irreducible parameter has only one isomorphism class of geometric points [AGKRRV, Proposition 3.7.2 and Corollary 3.7.5]. Density makes \tau irreducible and gives

The tangent complex of \mathcal S at \tau is \RGamma(C,\ad(\tau))[1] [AGKRRV, Section 21.2.1]. Its degree -1 cohomology is zero by (3.2). The invariant nondegenerate form on \operatorname{Lie}(\Gd) and Poincaré duality give

Thus the open-and-closed component \mathcal S_\tau containing \tau has no infinitesimal automorphisms or obstructions and is a smooth one-point formal polydisc, with tangent space H^1(C,\ad(\tau)).

For this formal component we use the completion description

as in [AGKRRV, Sections 21.1.7–21.1.10]; the right-hand category consists of objects set-theoretically supported at the origin. Its compact objects are bounded coherent complexes with this support. Indeed, coherent complexes with this support are compact and generate the supported category; on the smooth affine space this is also the usual generation by the Koszul complex at the origin. In particular their cohomology modules have finite length. The skyscraper \delta_\tau is nonzero and compact. Since the formal component is smooth, it has no spectral obstruction directions, so \delta_\tau has nilpotent spectral singular support. This is the ind-coherent singular-support condition, rather than the microlocal support of a constructible skyscraper.

Choose an equivalence (3.1), and let D_\tau be the inverse image of \delta_\tau. It is compact in \mathcal D_C and in the ambient automorphic category. Compact automorphic objects are bounded constructible on every finite-type smooth chart [AGKRRV, Appendix F, Section F.2.2], so D_\tau is locally bounded constructible. Its full eigenstructure is the one constructed in [CT, Lemma 8.1]: the module identities

applied to universal evaluation over C^I, give

The unit, tensor, permutation, and fusion maps come from the same universal evaluation system. Spectral linearity transports all of them together. This proves the required geometric assertions about D_\tau.

The remaining issue is the action of f. It is enough at this stage to preserve the summand corresponding to \mathcal S_\tau; no equivariance of the chosen equivalence (3.1) is required.

3.2 Recognizing the transported object

The preceding lemma places f^*D_\tau in the same spectral component as D_\tau. We now recognize its image there. The required elementary criterion is useful independently of the spectral setting.

For the skyscraper, and therefore for D_\tau, we have

These are the ambient automorphic Hom groups, since \mathcal D_C is a full subcategory. Transport preserves both these groups and compactness. Lemma 3.2 places f^*D_\tau in the \mathcal S_\tau summand. Its image under (3.1) is therefore a bounded coherent complex supported at the origin of the smooth formal component. Applying Lemma 3.3 to the local ring at that origin gives

for some r\in\Z.

We eliminate the shift on a bounded open, so no global cohomological bound on \Bun_G(C) is needed. Use the f-invariant ample line bundle \mathcal L in the statement. For each sufficiently large integer m, let \mathcal U_m\subset\Bun_{\SL_n}(C) be the open substack on which the associated vector bundle \mathcal V satisfies

These conditions are open and the opens cover the bundle stack by Serre vanishing. Each \mathcal U_m is of finite type: rank and trivialized determinant fix the Hilbert polynomial, the displayed conditions fix h^0, and a choice of basis of sections gives the usual finite-type Quot presentation. Adding the determinant trivialization is also a finite-type condition. Because f preserves \mathcal L, every \mathcal U_m is invariant under transport.

Choose m with D_\tau|_{\mathcal U_m}\ne0. On a finite-type smooth atlas of \mathcal U_m, local bounded constructibility gives a finite cohomological interval; descent gives the same boundedness on \mathcal U_m. Consequently

is finite and nonempty. Pullback along the automorphism of \mathcal U_m is exact and conservative, so it preserves S_m. Equation (3.6) gives S_m=S_m-r, which forces r=0. We have obtained an isomorphism b:f^*D_\tau\simeq D_\tau. It remains to verify its compatibility with the Hecke eigenmaps.

3.3 Uniqueness of the full eigenstructure

For I a finite set and L_1,L_2 finite-rank lisse \E-sheaves on C^I, there is a natural external-product formula

We explain its applicability to the non-quasicompact bundle stack. On every finite-type smooth scheme chart, the restrictions of D_\tau are bounded constructible, and the usual constructible Hom–Künneth formula applies. Equivalently, one can use adjunction for the projection to the chart, proper base change for C^I, and the projection formula. The second factor in (3.7) is a bounded finite-dimensional \E-complex, hence perfect. Tensoring with it commutes with the limits computing smooth descent and restriction over finite-type opens of the bundle stack. The chartwise formulas therefore give the displayed global formula. On constructible chart objects these are also the Hom complexes in the ambient ind-sheaf category.

Both factors on the right are concentrated in nonnegative degrees, by (3.5) and the fact that the L_i are ordinary lisse sheaves. Thus

The first identification sends a lisse-sheaf map a to \id_{D_\tau}\boxtimes a.

4 Arithmetic towers and equivariant gluing at a node

We construct the parameter to which the unramified input of Proposition 3.1 will be applied. There are two steps. First we lift the arithmetic parameter \rho to a punctured curve in characteristic zero, retaining its Frobenius action and its entire boundary homomorphism. We then place this lift on one component of a separating nodal curve and extend it to an unramified parameter on a smooth generic fiber. In the second step the gluing must preserve both Frobenius and compactness of the monodromy image. A ramified auxiliary component will allow us to satisfy these two conditions simultaneously.

4.1 Lifting the arithmetic parameter

With k=\overline{\F}_q, set

Here \Dbar is the algebraic closure of D inside K. The ring R is an excellent complete strictly henselian discrete valuation ring with residue field k. Extend the Witt-vector automorphism of R induced by a\mapsto a^q on k to an automorphism \varphi of K. It fixes D and preserves \Dbar. On roots of unity of \ell-power order it acts by \zeta\mapsto\zeta^q, as follows from their Teichmüller lifts in R. We use the same letter for these compatible actions, with the coordinate orientation fixed in Section 2.

Choose a smooth projective pointed lift

Pointed deformations are unobstructed because H^2(X_0,T_{X_0}(-x_0))=0; an ample line bundle also lifts, and formal algebraization gives the projective family. This is the lifting construction of [CT, Section 8.5], here performed over R_0 so as to retain the field of definition. Write

By smoothness, a function t on a neighborhood of \mathfrak x over R_0 may be chosen as a relative parameter. Its differential trivializes the conormal line of the section. We also regard t as a rational function on C_1.

For an integer d invertible in R_0, the coordinate t identifies the residual gerbe of \mathscr X(\sqrt[d]{\mathfrak x}) with B\mu_d over R_0: it trivializes the divisor line by 1/t and thereby specifies its trivial dth root. Pulling a finite torsor back to this neutral object gives its boundary fiber. A frame over the strictly henselian base R identifies that fiber with its finite structure group, acting on itself on the right. In this frame, both inertia and semilinear transport are recorded by left multiplication. The construction below chooses such frames compatibly through the tower.

We prove the proposition after a finite-cover lemma that will also be used in the nodal smoothing. It compares monodromy orbits on finite torsor fibers, allowing disconnected torsors as required when we enlarge the compact group that records monodromy.

4.2 A compact conjugator and an auxiliary component

The first parameter is now lifted with its arithmetic action. To extend it across a separating node, the two boundary actions must agree when expressed using the opposite orientations of the two branches. The following linear-algebra observation gives an identification that also commutes with A and belongs to a fixed compact group.

Fix c,J,e,P as in the lemma, enlarging the coefficient field once and again denoting it by L'. Retain the notation for all parameters. Define a cover

The valuation of t at x_1 is one. The polynomial w^e-t is therefore Eisenstein there, even after extension to \Dbar, so this is a degree-e geometrically connected cover. The curve C_2 is smooth and projective over D, and there is a unique point x_2 over x_1. It is D-rational and totally ramified, with local parameter w; these assertions are also visible in the completed local extension D[[t]]\subset D[[w]]. Riemann–Hurwitz gives g(C_2)\geq2.

Put U_2=C_2\setminus\{x_2\} and \sigma_2=h^*\sigma_1 on (U_2)_{\Dbar}. Any other branch points of h lie over U_1 and cause no ramification in this pullback local system. In the coordinate w, its boundary homomorphism is \exp(e\,t_\ell(\gamma)N_1).

For the rest of the construction choose open normal subgroups J_r\subset J, decreasing and cofinal at 1, and set

The map H_0\to Q_r factors through one of the earlier quotients \Gamma_j, because H_0\cap J_r is open in H_0. Extension of structure group therefore gives a right Q_r-torsor on the first component with its original equivariance; pullback by h gives the one on the second. These torsors need not be connected. For any common root index b divisible by their stage-r inertia orders they extend over C_i(\sqrt[b]{x_i})_{\Dbar}. The cover h induces a map of these root stacks, since h^*x_1=e x_2. In coordinates it sends a bth root of t to the eth power of a bth root of w; on the residual gerbes its map is

The neutral objects fixed by t and w correspond under this map: in their divisor-line trivializations, h^*(1/t)=(1/w)^e, with no scalar factor. The second torsor and its frame are pulled back from the first, and this fiber identification is defined over D. Thus the induced frame on the second gerbe has exactly the same equivariance value A as the first. This equality, in addition to the factor e in the inertia, is the arithmetic information supplied by the algebraic cover.

4.3 The balanced smoothing and its parameter

Identify x_1 with x_2 and write C_0=C_1\cup_{x_1=x_2}C_2. Choose a projective smoothing \mathscr C\to\Spec D[[s]] of this separating node, with a relatively ample line bundle. The curve has unobstructed deformations and an independent smoothing parameter at the split node; lifting an ample line bundle algebraizes the deformation. As in [CT, Section 8.2], the completed local equation can be chosen as

where a restricts to a parameter on the first branch and b' to one on the second. Rescaling their tangent coordinates over D, and rescaling s accordingly, ensures

These equalities will identify the coordinate neutralizations of the branch gerbes without a constant unit twist.

Choose compatible roots s_b of s and put

Extend \varphi|_{\Dbar} coefficientwise to \Dbar((s)), then to the union obtained by adjoining the compatible roots s_b while fixing those roots, and finally to \Omega. We again denote this automorphism by \varphi. The smooth projective geometric generic fiber of \mathscr C will be denoted C/\Omega. It is connected of genus g(C_1)+g(C_2), and its polarization is \varphi-invariant.

For each b, let \mathcal C(b)\to S_b be the balanced twisted model with node chart

Away from the closed node this is the ordinary family after base change. The construction is the balanced twisted-curve construction of [Ols, Theorems 1.9–1.10, Remark 1.11, and Proposition 2.2(ii)]; the chart and its power maps are described in [CT, Section 8.2]. In particular each model is proper, flat, and tame, with projective coarse space and geometrically reduced fibers. Its special-fiber normalization is the disjoint union of the two root stacks C_i(\sqrt[b]{x_i})_{\Dbar}. For b\mid m the power maps on u,v give the corresponding maps between models over the map S_m\to S_b.

The construction may be made over D[[s_b]] with the group scheme \mu_b, so the models and these maps carry the chosen semilinear actions. The ratios a/t and b'/w have residue one. Their formal roots with constant term one are unique and compatible under power maps in characteristic zero. They are also preserved by \varphi. Consequently the node chart identifies the branch neutral objects with those already used for t,w. For general b, only the \ell-primary quotient of the gerbe acts on our torsors. On that quotient \varphi acts by the qth power; the other prime factors act trivially on the torsors.

We fix at this point the index that will produce Borel level on the special-fiber bundle stack. Choose a strictly dominant cocharacter \lambda\in X_*(T) for the split torus T\subset B\subset G and an integer b_* satisfying

Choose the parameter indices b_r so that b_*\mid b_r\mid b_{r+1} and b_r kills the two stage-r inertia actions. The sheaf construction will use the fixed model \mathcal C(b_*) and inertia type \lambda|_{\mu_{b_*}}; the varying indices b_r supply only finite-stage extensions of the parameter. Denote by \mathcal V\subset\mathcal C(b_*) its smooth scheme locus. Its geometric generic fiber is C and its special fiber is (U_1\sqcup U_2)_{\Dbar}.

5 From the separating node to an equivariant flag eigensheaf

The parameter \tau of Proposition 4.4 lives on the smooth projective generic curve C/\Omega of the twisted degeneration. Proposition 3.1 therefore supplies an equivariant unramified eigencomplex. We now specialize this complex to the node, retain the first normalization component, and remove the enhancement of its flag. The two issues are nonvanishing on the prescribed node type and preservation of the semilinear action when passing to a perverse eigensheaf.

5.1 The prescribed node type and its two flags

Retain the fixed index b_* and strictly dominant cocharacter \lambda from Section 4; thus

Put S=S_{b_*}=\Spec\Dbar[[s_{b_*}]]. In the relative bundle stack of \mathcal C(b_*)/S, remove from the closed fiber every inertia type except the conjugacy class of \lambda|_{\mu_{b_*}}. Denote the resulting open stack by \mathcal B. The condition is open because the finitely many conjugacy classes of homomorphisms \mu_{b_*}\to G are locally constant on the residual gerbe. The centralizer of the chosen type is T: the inequalities ensure that no root is trivial on \lambda(\mu_{b_*}). The Hom-stack theorem gives algebraicity and local finite presentation [HR, Theorem 1.2]. Smoothness follows from the vanishing of H^2 of the adjoint bundle: the curve is tame, coarse pushforward is exact, and its coarse space has dimension one. In particular \mathcal B has smooth finite-type charts over S. Its geometric generic fiber is \Bun_G(C).

Let B^- be the Borel opposite to B with the same maximal torus T, and define

An object of either stack includes an enhanced flag, that is, a reduction to the indicated unipotent radical. The calculation of [CT, Lemma 8.2] gives

where T changes the two frames on the node gerbe simultaneously. We recall the calculation to verify its compatibility with our semilinear action and with the two different component curves.

On the first branch of the balanced chart, set a=u^{b_*}. In a frame in which inertia is \lambda, a change of frame satisfies

Conjugating to the invariant punctured-disc frame gives g(a)=\lambda(u)^{-1}h(u)\lambda(u). If j=\langle\alpha,\lambda\rangle for a positive root, the positive and negative root coordinates respectively transform as

The toral coordinates are series in a. These formulas identify the full frame-change group with \{g(a)\in G(\Dbar[[a]]):g(0)\in B\}; the assertion over arbitrary test schemes follows by the same root-group factorization in the formal big cell. Evaluation h\mapsto h(0) becomes the torus projection of this Iwahori group. Fixing the gerbe frame therefore gives an enhanced B-flag.

On the second branch the node generator acts by v\mapsto\zeta^{-1}v. Its positively oriented coordinate generator thus sees -\lambda, so the identical calculation gives B^-. On both branches the fixed gerbe torsor has automorphism group T, expressed using the common node generator. Gluing the normalization bundles, with an identification on their gerbes, now gives (5.1). The computation is branchwise and requires no isomorphism between C_1 and C_2.

All these prescriptions use the split group, the group-scheme cocharacter \lambda, and the balanced coordinates chosen over D. They consequently commute with \varphi, including its action on \mu_{b_*}; no individual root of unity has been declared fixed. Thus (5.1) is an equivariant identification. A modification away from the node preserves its gerbe frames and acts on the corresponding factor. This also identifies every successive and multi-leg Hecke correspondence, after pullback to \mathcal A_1^+\times\mathcal A_2^+, with its componentwise counterpart.

5.2 A criterion for nonzero nearby cycles

Nearby cycles can vanish for a nonzero generic complex. The input that excludes this possibility here is the specialization-detection theorem of [CT, Theorem 4.2], with the following precise scope.

Here and below \Psi is geometric nearby cycles, without inertia invariants. The closure in the lemma may be computed after descending its finitely many geometric generic components to a finite extension of the trait. Neither the lemma nor this interpretation requires the complex F itself to descend. These are part of the statement and conventions of [CT, Theorem 4.2].

We will produce the required meeting by extending a bundle carrying a nonzero generic stalk. For G=\SL_n, the uniformization needed for this step has an elementary description. If Y is a smooth projective connected curve over an algebraically closed field and y\in Y, the complement Y\setminus\{y\} is a smooth affine curve. Its coordinate ring is a Dedekind domain. A rank-n projective module over that ring is isomorphic to the sum of a free module of rank n-1 and its determinant. Consequently every \SL_n-bundle is trivial on this complement. A free basis can be adjusted by a unit so that its determinant agrees with the given trivialization. This is the \SL_n case of the affine-curve triviality used in Drinfeld–Simpson uniformization and in [BF, Theorem 1.1]; see also [DS].

In particular, any two \SL_n-bundles on Y differ by a modification at y. Each such modification lies in some finite Schubert bound. In a family, the bounded Hecke space with fixed reference input and fixed smooth section y is proper over the base trait. A geometric generic modification therefore extends after a finite trait extension: its point first descends to a finite generic extension, and the valuative criterion then applies over the corresponding DVR normalization of the complete trait. The output remains unchanged near every marking or node disjoint from y.

5.3 Specializing on the chosen component

Apply Proposition 3.1 to (C,\tau). The generic curve is smooth, projective, and connected of genus at least two; its polarization comes from the projective smoothing. Proposition 4.4 supplies the continuous dense parameter over a finite coefficient field and its equivariance. We obtain a nonzero locally bounded constructible eigencomplex D_\tau on \mathcal B_{\bar\eta}=\Bun_G(C), with its \varphi-structure and coherent equivariant eigenmaps.

The trait S_{b_*} is excellent, complete, and strictly henselian, with algebraically closed characteristic-zero residue field; in particular \ell is invertible. Use L=\mathcal V, the smooth scheme locus of \mathcal C(b_*), a relative curve with L_{\bar\eta}=C and L_s=U_{1,\Dbar}\sqcup U_{2,\Dbar}. At these legs the bounded correspondences have the split affine Grassmannian models in smooth frame charts. Their output maps are representable and proper, and their exact-position maps are smooth over both bundle-and-leg projections. These local models have the normalized constant kernels on their open cells. Modifications there preserve the node type. On the special fiber (5.1), the cotangent description, cone dimension bound, and transverse modifications are precisely the torus-quotient case of [CT, Section 3 and Section 8.2]. The required Lie-theoretic properties hold in characteristic zero. Thus the geometric hypotheses of Proposition 2.4 and Lemma 5.2 hold for this family.

For their eigenvalue hypothesis, use the equivariant finite-torsor tower of Proposition 4.4. For each representation, an invariant lattice has reductions extending lisse over the smooth scheme locus after the extensions S_{b_r}\to S_{b_*}. Their specializations are the component parameter systems, and all tensor maps and semilinear actions are carried by the same tower. Hence

is locally bounded constructible and carries the coherent component eigenstructure and its \varphi-action.

To prove F_s\ne0 on this particular type, first choose a point of \mathcal A_1^+\times\mathcal A_2^+. A smooth chart of \mathcal B through its image, followed by henselian lifting, gives a reference bundle P_0 over S with the prescribed node type. Lift also a point of L_s to a section y of L/S. Choose a bundle P on C/\Omega where D_\tau has nonzero stalk. Such a point exists on a finite-type chart because D_\tau is nonzero and locally bounded constructible.

The Dedekind-domain argument above identifies P and (P_0)_\Omega off y_\Omega. Extend the resulting bounded modification after a finite trait extension. Its output lies in \mathcal B, since its node restriction still equals that of P_0. Take a smooth finite-type chart Z\to\mathcal B through its special value and a residue-field point of the chart above it. If S' is the extended trait, the morphism Z\times_{\mathcal B}S'\to S' is smooth; henselian lifting of that point gives a section, after a further finite extension if necessary. Its geometric generic point has the chosen nonzero stalk. This section witnesses a meeting of the closure in Lemma 5.2 with the special fiber. Thus

Finite extensions in this argument only locate the closure inside the fixed geometric generic fiber; they impose no new descent condition on D_\tau or its semilinear structure. They need not carry \varphi: geometric nearby cycles are unchanged by this finite-trait replacement, and the equivariant object (5.3) was already constructed over S.

Ordinary pullback of F_s to \mathcal A_1^+\times\mathcal A_2^+ is nonzero, because this map is smooth and surjective. Choose a \Dbar-point (a_1,a_2) with nonzero stalk. The bundle and enhanced flag defining a_2 descend to a finite extension D'/D: they are of finite presentation and \Dbar/D is algebraic. Pass to a finite normal extension containing D'. Since \varphi fixes D, its action on this normal extension has finite order. Some power \varphi^m fixes it pointwise, and the descended model then supplies an identification \varphi^{m*}a_2\simeq a_2. We henceforth use this power as generator.

Restricting along \mathcal A_1^+\times\{a_2\} gives a nonzero locally bounded constructible complex F_1^+ on \mathcal A_1^+, with a \varphi^m-structure. The quotient description of the Hecke correspondences and proper base change for each bound give

for every finite leg set I. The same comparisons apply on the successive-modification spaces and on every fusion diagonal. They are natural under semilinear transport, so all these eigenmaps and their coherence constraints are equivariant. Ordinary restriction here is sufficient; no perversity assertion is made for F_1^+.

5.4 Perverse cohomology and removal of the enhancement

To pass from F_1^+ to an ordinary-level perverse eigensheaf, we will take perverse cohomology, push forward along the torsor that forgets the enhancement, and take perverse cohomology once more. The first truncation supplies nilpotent singular support, which controls monodromy along the enhancement torus. Unipotence of that monodromy will ensure that the direct image is nonzero; the second truncation then gives the required perverse object.

The geometric inputs for these steps are [CT, Propositions 6.1 and 7.5]. Their scope is a smooth projective complex curve with one marked point and a split simple simply connected group. Proposition 6.1 applies at either ordinary or enhanced Borel level: for a locally bounded constructible geometric adic complex with a coherent eigenstructure for a Zariski-dense parameter, every perverse cohomology inherits the full eigenstructure and nilpotent singular support, and some perverse cohomology is nonzero if the complex is. Proposition 7.5 removes the enhancement when the boundary monodromy is unipotent. Our group \SL_n meets the group assumptions, and Proposition 4.1 proves both density and the required unipotent boundary monodromy for \sigma_1.

Choose an abstract field isomorphism \Dbar\simeq\C to apply these geometric statements. Such an isomorphism exists: both fields are algebraically closed of characteristic zero and have transcendence degree 2^{\aleph_0} over \Q. We will use their Betti arguments to establish geometric properties, while making all sheaf operations and induced maps in the geometric adic category over \Dbar.

First choose j such that Q={}^pH^j(F_1^+)\ne0. Here is why its eigenstructure retains the semilinear action. For a finite set I, put d=|I|. The proof of [CT, Proposition 6.1, Section 6.3] constructs, on the objects under consideration, the functorial adic comparison

The Betti calculation proves the needed perverse bounds; the comparison itself follows from the universal property of the adic truncation triangles. On the eigenvalue side, put \mathcal L_I=\boxtimes_{i\in I}(V_i)_{\sigma_1}. Since this sheaf is lisse on the smooth d-dimensional leg space, exterior product with \mathcal L_I[d] is perverse exact, so

Applying these comparisons to (5.5) and canceling the common shift [d] gives the relative-normalized eigenmaps for Q.

For a collision \Delta:U_1^J\to U_1^I, the corresponding truncation comparison uses (\id\times\Delta)^*[|J|-|I|] on these locally constant leg families. The comparisons for successive Hecke operations use the total dimension of their retained leg space, counting a shared leg only once. Thus (5.6) carries the entire unit, convolution, permutation, and fusion diagrams. Pullback by \varphi^m preserves the perverse structure and its truncation maps, so these adic comparisons commute with it. No compatibility between a chosen complex realization and \varphi is needed.

Now let

forget the enhancement. This is a T-torsor of relative dimension r=n-1. Set F=q_!Q. We need the intermediate assertion in the proof of [CT, Proposition 7.5]: this particular direct image is nonzero. We recall its mechanism, since a general torus direct image does not preserve nonvanishing.

At enhanced level, a cotangent residue lies in \operatorname{Lie}B. In an adapted local frame write a generically nilpotent Higgs field as \theta=b(t)\,dt/t, with b(t) regular. The positive-degree invariant polynomials vanish on b(t) and hence on the residue b(0); the toral projection of this residue is therefore zero. Thus the enhanced nilpotent cone is the smooth cotangent pullback of the ordinary-level cone and annihilates tangent directions to the T-fibers [CT, Section 3.1]. The microlocal criterion of [CT, Lemma 7.1] now makes the Betti realization of Q monodromic along this torsor: locally on the base it is constructible for a product stratification with the whole torus as second factor. In particular its cohomology sheaves on each fiber are local systems.

For a representation V, let \chi_V be its character on the dual torus \widehat T. The central-sheaf calculation, based on Gaitsgory’s construction [Gai01] and proved in the needed form in [CT, Lemma 7.4 and proof of Proposition 7.5], gives, for every V\in\Rep(\Gd),

The last equality uses unipotence at x_1. On any stalk cohomology of a torsor fiber, the commuting monodromies have joint generalized eigencharacters, viewed as points of the dual torus \widehat T(\E). Character functions of representations span the Weyl-invariant regular functions on \widehat T and separate its Weyl orbits. Equation (5.7) therefore puts each eigencharacter in the orbit of 1, which consists of 1 alone. All enhancement monodromies on these finite-dimensional spaces are unipotent.

Choose a fiber T=q^{-1}(a) where Q_T:=Q|_T is nonzero, and let j_0 be the largest index with \mathcal H^{j_0}(Q_T)\ne0. Its fiber W is a nonzero finite-dimensional representation of \Gamma=X_*(T) with commuting unipotent operators. The augmentation ideal of \E[\Gamma] acts nilpotently on this space, so the coinvariants W_\Gamma are nonzero. Poincaré duality gives

In the compact-support hypercohomology spectral sequence this term has no incoming differential, by maximality of j_0, and no outgoing differential, because H_c^i(T,-)=0 for i>2r. Consequently H_c^{2r+j_0}(T,Q_T)\ne0. Comparison and base change for the adic functor q_! imply F_a\ne0. This argument uses a nilpotence bound only on the chosen finite-dimensional space.

The morphism q has finite type and fixed relative dimension, so F is locally bounded constructible. Away from x_1, Hecke modifications transport an enhanced flag as well as its ordinary flag. Proper base change on a bounded Hecke correspondence and the projection formula therefore give the natural comparison

These comparisons also hold on the successive and collision correspondences. Hence they preserve every coherence constraint. Since q is defined over D, its direct image and these natural comparison maps carry the \varphi^m-structure.

Finally choose a nonzero perverse cohomology M_{\Dbar}={}^pH^a(F). Proposition 6.1 of [CT] applies now at ordinary Borel level, and the same adic truncation argument (5.6) proves equivariance of all the induced eigenmaps. It also gives nilpotent singular support. At ordinary flag level this is \Lambda_{\mathrm{par}}. This completes the proof of Proposition 5.1.

6 Specialization to the finite field

We now have a perverse eigensheaf with an action of \varphi^m on the lifted pointed curve over \Dbar. The arithmetic torsor tower of Proposition 4.1 identifies its eigenvalue after mixed-characteristic specialization with the original arithmetic parameter, including its Frobenius structure.