Coherent Springer theory and the categorical Deligne-Langlands correspondence

Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields $F$ with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from $K$-theory to Hochschild homology and thereby identify $\mathcal{H}$ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of $\mathcal{H}$-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of $\mathrm{GL}_n(F)$ into coherent sheaves on the stack of Langlands parameters.

Our goals in this paper are to provide a spectral description of the category of representations of the affine Hecke algebra and deduce applications to the local Langlands correspondence. We begin with a quick review of Springer theory and then discuss our main results starting in Section 1.3.
We will work in the setting of derived algebraic geometry over a field k of characteristic zero, as presented in [GR17]. In particular all operations, sheaves, categories etc will be derived unless otherwise noted.
1.1. Springer theory and Hecke algebras. We first review some key points of Springer theory, largely following the perspective of [CG97,Gi98].
Let G denote a complex reductive group with Lie algebra g and Borel B Ă G. We denote by B » G{B the flag variety, N the nilpotent cone, p : r N " T˚B Ñ N the Springer resolution, and Z " r NˆN r N the Steinberg variety. We write r G " GˆG m . The Springer correspondence provides a geometric realization of representations of the Weyl group W of G. The Weyl group is in bijection with the Bruhat double cosets BzG{B " GzpBˆBq, and hence with the conormals to the Schubert varieties, which form the irreducible components of the Steinberg variety Z. In fact the group algebra of the Weyl group can be identified with the top Borel-Moore homology of Z under the convolution product where d " dimpN q " dimp r N q " dimpZq. This realization of W can be converted into a sheaf-theoretic statement. The Springer sheaf S " p˚C Ă N rds P PervpN {Gq is the equivariant perverse sheaf on the nilpotent cone given by the pushforward of the (shifted) constant sheaf on the Springer resolution. Thanks to the definition of Z as the self-fiber-product Z " r NˆN r N , a simple base-change calculation provides an isomorphism Remark 1.5. Our results also allow for an identification of monodromic variants of the affine Hecke category. See Remark 2.29 for details.
The Hochschild homology of categories of coherent sheaves admits a description in the derived algebraic geometry of loop spaces. In particular, we deduce an isomorphism of the affine Hecke algebra with volume forms on the derived loop space to the Steinberg stack, Gq, ω LpZ{ r Gq q. More significantly, the geometry of derived loop spaces provides a natural home for the entire category of H-modules, without fixing central characters.
Definition 1.6. The coherent Springer sheaf S P CohpLpN { r Gqq is the pushforward of the structure sheaf under the loop map Lp : Lp r N { r Gq Ñ LpN { r Gq of the Springer resolution: A priori the coherent Springer sheaf is only a complex of sheaves. However we show, using the theory of traces for monoidal categories in higher algebra, that its Ext algebra is concentrated in degree zero, and is identified with the affine Hecke algebra. This provides the following "coherent Springer correspondence," realizing the representations of the affine Hecke algebra as coherent sheaves.
Theorem 1.7. Let G a reductive algebraic group over with simply connected derived subgroup over an algebraically closed field of characteristic 0.
(1) There is an isomorphism of algebras Gq pSq and all other Ext groups of S vanish.
(3) Let u denote the generator of H ‚ pBS 1 q » krus. Then there is a full embedding of the derived category of modules for the trivial u-deformation Hrus into the u-deformation CohpLpN { r Gqq S 1 .
One consequence of the theorem is an interpretation of the coherent Springer sheaf as a universal family of H-modules.
We also conjecture -and check for SL 2 -that S is actually a coherent sheaf (i.e., lives in the heart of the standard t-structure on coherent sheaves). The vanishing of all nonzero Ext groups of S suggests the existence of a natural "exotic" t-structure for which S is a compact projective object in the heart. For such a t-structure we would then automatically obtain a full embedding of the abelian category H-mod into "exotic" coherent sheaves, where one could expect a geometric description of simple objects. See Section 4.3 for a discussion.
We will explain in Section 1.5 how equivariant localization and Koszul duality patterns in derived algebraic geometry (as developed in [BN13,Ch20a,Ch20b]) provides the precise compatibility between this coherent Springer theory and the usual perverse Springer theory, one parameter at a time.
1.4. Applications to the local Langlands correspondence. Let us consider the derived stack L u q of unipotent Langlands parameters, which parametrizes the unipotent Weil-Deligne representations for a local field F with residue field F q , and whose set of k-points is a variant of the set of Deligne-Langlands parameters in Theorem 1.3 (with semisimplicity of s dropped). Note that the following notions make sense for any q P C, with applications to local Langlands when q is a prime power.
(1) The stack of unipotent Langlands parameters L u q " L q pN {Gq is the derived fixed point stack of multiplication by q P G m on the equivariant formal neighborhood of the nilpotent cone N {G inside g{G (equivalently, the formal neighborhood of the unipotent cone inside G{G). Equivalently, it is the fiber of the loop (or derived inertia) stack of the nilpotent cone over q P G m , Thus informally L q pN {Gq » tg P G, n P N : gng´1 " qnu{G.
(2) The q-coherent Springer sheaf S q P CohpL u q q is the specialization of S to the fiber L u q over q. Equivalently, S q is given by applying the parabolic induction correspondence to the reduced structure sheaf of L u q pT q.
When q is not a root of unity, the a priori derived stack L u q pGq is actually a classical stack by Proposition 3.25 (see also Proposition 4.2 in [H20], Proposition 2.1 in [He20] and Proposition 3.1.5 in [Z20]). Specializing Theorem 1.7 to q P G m we obtain the following: Theorem 1.9. Suppose that q ‰ 1.
(1) There is a full embedding D perf pH q -modq » xS q y Ă CohpL u q q. In particular, if F is a local field with residue field F q (so that q is a prime power), and G _ is a split reductive algebraic group over F with simply connected derived subgroup over an algebraically closed field of characteristic 0, then this gives a full embedding of the principal block of G _ pF q into coherent sheaves on the stack of unipotent Langlands parameters.
(2) This embedding is compatible with parabolic induction, i.e. if P _ " M _ U _ is a parabolic subgroup of G _ , then we have a commutative diagram tunramified principal series of M _ pF qu / / CohpL u q pM qq tunramified principal series of G _ pF qu » DpH q -perfq / / CohpL u q pGqq, where the horizontal maps are full embeddings, the left hand vertical map is the parabolic induction functor i G P from reprentations of M to representations of G, and the right vertical map is obtained by applying the correspondence L u q pM q Ð L u q pP q Ñ L u q pGq. The existence of such an equivalence was conjectured independently by Hellmann in [He20], whose work we learned of at a late stage in the preparation of his paper. Indeed, the above result appears (for G an arbitrary reductive group over F ), as (part of) Conjecture 3.2 of [He20]. Note that the conjecture of Hellmann asserts further that the above correspondence should take the unramified part of the compactly supported Whittaker functions to the structure sheaf on L u q pGq; we do not address this question here. Hellmann's work also gives an alternative characterization of the (q-specialized) coherent Springer sheaf as the Iwahori invariants of a certain family of admissible representations on L u q pGq constructed by Emerton and the third author in [EH14].
A much more general categorical form of the local Langlands correspondence is formulated by Zhu [Z20]. In loc.cit. there is also announced a forthcoming proof by Hemo and Zhu [HZ] of a result closely parallel to ours.
In the case of the general linear group and its Levi subgroups, one can go much further. Namely, in Sections 5 and 6 we combine the local Langlands classification of irreducible representations due to Harris-Taylor and Henniart with the Bushnell-Kutzko theory of types and the ensuing inductive reduction of all representations to the principal block. The result is a spectral description of the entire category of smooth GL n pF q representations. To do so it is imperative to first have a suitable stack of Langlands parameters. These have been studied extensively in mixed characteristic, for instance in [H20] in the case of GL n , or more recently in [BG19,BP19], and [DHKM20] for more general groups. Since in our present context we work over C, the results we need are in general simpler than the results of the above papers, and have not appeared explicitly in the literature in the form we need.
Theorem 1.10. [H20] There is a classical Artin stack locally of finite type X F,GLn , with the following properties: (1) The k-points of X F,GLn are identified with the groupoid of continuous n-dimensional representations of the Weil-Deligne group of F . (2) The formal deformation spaces of Weil-Deligne representations are identified with the formal completions of X F,GLn . (3) The stack L u q pGL n q of unipotent Langlands parameters is a connected component of X F,GLn .
We then deduce a categorical local Langlands correspondence for GL n and its Levi subgroups as follows: Theorem 1.11. For each Levi subgroup M of GL n pF q, there is a full embedding DpM q ãÑ QC ! pX F,M q of the derived category of smooth M -representations into ind-coherent sheaves on the stack of Langlands parameters, uniquely characterized by the following properties: (1) If π is an irreducible cuspidal representation of M , then the image of π under this embedding is the skyscraper sheaf supported at the Langlands parameter associated to π. (2) Let M 1 be a Levi subgroup of G, and let P be a parabolic subgroup of M 1 with Levi subgroup M . There is a commutative diagram of functors: DpM q / / QC ! pX F,M q DpM 1 q / / QC ! pX F,M 1 q in which the horizontal maps are the full embeddings described above, the left-hand vertical map is parabolic induction, and the right-hand vertical map is obtained by applying the correspondence X F,M Ð X F,P Ñ X F,M 1 .
Note that the local Langlands correspondence for cuspidal representations of GL n and its Levis, is an input to the above result.
As with Theorem 1.9 our results here were independently conjectured by Hellmann (see in particular Conjecture 3.2 of [He20]) for more general groups G; these result also fit the general categorical form of the local Langlands correspondence formulated by Zhu [Z20].
1.4.1. Discussion: Categorical Langlands Correspondence. Theorems 1.9 and 1.11 match the expectation in the Langlands program that has emerged in the last couple of years for a strong form of the local Langlands correspondence, in which categories of representations of groups over local fields are identified with categories of coherent sheaves on stacks of Langlands parameters. Such a coherent formulation of the real local Langlands correspondence was discovered in [BN13], while the current paper finds a closely analogous picture in the Deligne-Langlands setting. As this paper was being completed Xinwen Zhu shared the excellent overview [Z20] on this topic and Peter Scholze presented the lecture [Sc20], to which we refer the reader for more details. We only briefly mention three deep recent developments in this general spirit.
The first derives from the work of V. Lafforgue on the global Langlands correspondence over function fields [La18a,La18b]. Lafforgues' construction in Drinfeld's interpretation (cf. [LaZ19, Section 6], [La18b,Remark 8.5] and [Ga16]) predicts the existence of a universal quasicoherent sheaf A X on the stack of representations of π 1 pXq into G corresponding to the cohomology of moduli spaces of shtukas. The theorem of Genestier-Lafforgue [GL18] implies that the category of smooth G _ pF q representations sheafifies over a stack of local Langlands parameters, and the local version A of the Drinfeld-Lafforgue sheaf is expected [Z20] to be a universal G _ pF q-module over the stack of local Langlands parameters. In other words, the fibers A σ are built out of the G _ pF q-representations in the L-packet labelled by σ. The expectation is that the coherent Springer sheaf, which by our results is naturally enriched in H q -modules, is identified with the Iwahori invariants of the local Lafforgue sheaf The second is the theory of categorical traces of Frobenius as developed in [Ga16,Z18,GKRV20]. When applied to a suitably formulated local geometric Langlands correspondence, we obtain an expected equivalence between an automorphic and spectral category. The automorphic category is ShpG _ pF q{ Fr G _ pF qq, the category of Frobenius-twisted adjoint equivariant sheaves on G _ pF q, with orbits given by the Kottwitz set BpG _ q of isomorphism classes of G _ -isocrystals. The spectral category is expected to be a variant of a category QC ! pL G pF qq of ind-coherent sheaves over the stack L G pF q of Langlands parameters into G. The former category contains the categories of representations of G _ pF q and its inner forms as full subcategories, hence we expect a spectral realization in the spirit of Theorems 1.9 and 1.11.
Finally, a major source of inspiration in the mixed characteristic setting is Fargues' conjecture [F16] in the number field setting, which interprets the local Langlands correspondence as a geometric Langlands correspondence. On the automorphic side one considers sheaves on the stack Bun G _ of bundles on the Fargues-Fontaine curve, whose isomorphism classes | Bun G _ | " BpG _ q are given as before by the Kottwitz set of G _ -isocrystals. The trivial bundle has automorphism group G _ pF q, whose representations thus appear inside the automorphic category (more generally one has a decomposition into pieces given by representations of inner forms of Levis of G _ ). On the spectral side we have Langlands parameters, namely G-local systems on the curve. Thus one expects a categorical form of the Fargues conjecture to embed representations of G _ pF q into a category of coherent sheaves over Langlands parameters. Such a refined form was indeed described in [Sc20] as this paper was being completed.
1.5. Compatibility of coherent and perverse Springer theory. In this section we explain how equivariant localization and Koszul duality patterns in derived algebraic geometry (as developed in [BN13,Ch20a,Ch20b]) provide the precise compatibility between this coherent version of the local Langlands correspondence and the more familiar model [Ze81,Lu83,V93] for local Langlands categories with fixed central character via categories of perverse sheaves. This pattern was developed in the context of the real local Langlands correspondence: the work of Adams, Barbasch and Vogan [ABV92,V93] and Soergel's conjecture [S01] describe representations of real groups with fixed infinitesimal character by equivariant perverse sheaves on spaces of Langlands parameters, while [BN13] gives a conjectural description of the full categories of representations in terms of coherent sheaves. Likewise, the solution to the Deligne-Langlands conjecture in [KL87] realizes the irreducible representations of affine Hecke algebras, one central parameter ps, qq at a time, in terms of simple equivariant perverse sheaves (or equivalently D-modules) on a collection of spaces N s,q . On the other hand, Theorem 1.7 provides a uniform description of all representations of H in terms of coherent sheaves on single parameter space.
The underlying mechanism in passing between the coherent sheaves on our algebro-geometric parameter space and perverse sheaves or D-modules on variants of the nilpotent cone is the interpretation of D-modules in the derived algebraic geometry of loop spaces [BN12, BN13, TV11, TV15, P15, Ch20a], a unification of Connes' description of de Rham cohomology as periodic cyclic homology and of the Koszul duality between D-modules and modules for the de Rham complex [BD91,Ka91]. Recall that the loop space, or derived inertia, of a stack X is defined by the mapping space from the circle, or equivalently the (derived) self-intersection of the diagonal LX " MappS 1 , Xq " XˆXˆX X " ∆ X ∆.
For X " SpecpRq affine, the loop space is the spectrum of the (derived) algebra of Hochschild chains HH ‚ pRq " R b RbR R. More generally for any scheme X, we have the (Hochschild-Kostant-Rosenberg) identification LX » T X r´1s " Spec X Sym ‚ X pΩ 1 X r1sq of the loop space with the relative spectrum of (derived) differential forms. Under this identification the loop rotation action of S 1 on LX (Connes' B-differential on the level of Hochschild homology) becomes encoded by the de Rham differential.
Theorem 1.12 (Koszul duality [BN12,TV11,P15]). For X an algebraic space almost of finite type over k a field of characteristic zero, there is a natural equivalence of kppuqq-linear categories When X is a stack, we only have an equivalence between D-modules and S 1 -equivariant sheaves on the formal loop space p LX, i.e. the formal completion of the loop space LpXq at constant loops. The loop space of a smooth global quotient stack LpX{Gq lies over a parameter space LpBGq " G{G, and the equivariant localization patterns in [Ch20a] realize the formal completion (resp. specialization) of LpX{Gq over a semisimple parameter z P G{G as the formal loop space of the classical z-fixed points LpX z {G z q (resp. LpX z q). In particular, in the setting of Deligne-Langlands, specializing at a parameter z recovers the loop space of the fixed point schemes LpN s,q q, and we can pass to D-modules on the correpsonding analytic space via Koszul duality.
In order to formulate the equivalence at completed parameters, we need to renormalize the category of coherent sheaves to include objects such as the structure sheaf or sheaf of distributions on formal completions. This form of Koszul duality is developed by one of the authors in [Ch20b] (see Section 4.2 for the details). We call objects in this category Koszul-perfect sheaves KPerfp p T X r´1sq on the formal odd tangent bundle, and they have the following favorable properties: (1) they are preserved by smooth pullback and proper pushforward in X, (2) for a smooth Artin stack X, Koszul-perfect objects are those which pull back to Koszul-perfect objects along a smooth atlas and (3) for smooth schemes X they are just the coherent complexes.
Theorem 1.13 (Theorem 4.25, [Ch20b]). Let X{G be a global quotient stack and let FDpX{Gq denote the category of filtered renormalized (i.e. ind-coherent) D-modules on the global quotient stack X{G. There is an equivalence of categories Applying this theorem requires choosing, at each parameter, a graded lift of the z-completed (or specialized) coherent Springer sheaf. There is a natural geometric or Hodge graded lift, and using this lift, we establish in Corollary 4.3 that the coherent Springer sheaf is Koszul dual at each parameter to the corresponding perverse Springer sheaves: Corollary 1.14. Fix a semisimple parameter ps, qq P r G, and let dps, qq " dimpN ps,qq q. Then the ps, qq-specialization of the coherent Springer sheaf S is Koszul dual to the ps, qq-Springer sheaf µ z C Ă N ps,qq rdps, qqs, i.e. the pushforward of the (shifted) constant sheaf along ps, qq-fixed points of the Springer resolution.
More precisely, the ps, qq-specialization Sps, qq of S has a Hodge graded lift, which is Koszul dual to the ps, qq-Springer sheaf Sps, qq equipped with its Hodge filtration. Likewise, the Hodge graded lift of the ps, qq-completion Sp x s, qq is naturally isomorphic to the r G ps,qq -equivariant ps, qq-Springer sheaf Sp x s, qq equipped with the Hodge filtration.
1.6. Methods. We now discuss the techniques underlying the proofs of Theorems 1.4 and 1.7 -namely, Bezrukavnikov's Langlands duality for the affine Hecke category and the theory of traces of monoidal dg categories.
1.6.1. Bezrukavnikov's theorem. The Kazhdan-Lusztig theorem (Theorem 1.2) has been famously categorified in the work of Bezrukavnikov [Bez06,Bez16], with numerous applications in representation theory and the local geometric Langlands correspondence.
Theorem 1.15. [Bez16] Let F denote a local field such that the residue field k has |k| " q, and let I Ă GpF q be an Iwahori subgroup. There is an equivalence of monoidal dg categories H q :" DpIzG _ pF q{Iq » CohpZ{Gq intertwining the automorphisms pullback by geometric Frobenius and pullback by multiplication by q.
Remark 1.16. It is natural to expect a graded or mixed version of Theorem 1.15 also holds, identifying H :" CohpZ{ r Gq with the mixed version of the affine Hecke category (as studied in [BY13]). Indeed such a version is needed to directly imply the Kazhdan-Lusztig Theorem 1.2 by passing to Grothendieck groups, rather than its specialization at q " 1. Theorem 1.15 establishes the "princpial block" part of the local geometric Langlands correspondence. Namely, it implies a spectral description of module categories for the affine Hecke category (the geometric counterpart of unramified principal series representations) as suitable sheaves of categories on stacks of Langlands parameters.
We apply Theorem 1.15 in Section 2 to construct a semiorthogonal decomposition of the affine Hecke category. This allows us to calculate its Hochschild and cyclic homology and to establish the comparison with algebraic K-theory.
1.6.2. Trace Decategorifications. To prove Theorem 1.7 we use the relation between the "horizontal" and "vertical" trace decategorifications of a monoidal category, and the calculation of the subtler horizontal trace of the affine Hecke category in [BNP17b].
Let pC,˚q denote a monoidal [dg] category. Then we can take the trace (or Hochschild homology) trpCq " HH˚pCq of the underlying (i.e. ignoring the monoidal structure) dg category C, which forms an associative (or A 8 -)algebra ptrpCq,˚q thanks to the functoriality (specifically the symmetric monoidal structure) of Hochschild homology, as developed in [HSS17,TV15,GKRV20]. This is the naive or "vertical" trace of C. On the other hand, a monoidal dg category has another trace or Hochschild homology TrpC,˚q using the monoidal structure which is itself a dg category -the categorical or "horizontal" trace of pC,˚q. This is the dg category which is the universal receptacle of a trace functor out of the monoidal category C. In particular, the trace of the unit of C defines an object r1 C s P TrpC,˚q -i.e., TrpC,˚q is a pointed (or E 0 -)category 4 . Moreover, as developed in [GKRV20] the categorical trace provides a "delooping" of the naive trace: we have an isomorphism of associative algebras pTrpCq,˚q » End TrpC,˚q prCsq.
In particular taking Hom from the basic object r1 C s defines a functor Hompr1 C s,´q : TrpC,˚q ÝÑ pTrpCq,˚q-mod.
Under suitable compactness assumptions the right adjoint to this functor embeds the "naive" decategorification (the right hand side) as a full subcategory of the "smart" decategorification (the left hand side).
More generally, given a monoidal endofunctor F of pC,˚q, we can replace Hochschild homology (trace of the identity) by trace of the functor F , obtaining two decategorifications (vertical and horizontal) with a similar relation (1.2) Hompr1 C s,´q : TrppC,˚q, F q ÝÑ pTrpC, F q,˚q-mod.
Remark The results of [BNP17b] (based on the technical results of [BNP17a]) provide an affine analog of the results of [BN15,BFO12] for finite Hecke categories and (thanks to Theorem 1.15) a spectral description of the full decategorification of H (statement (1) is directly taken from [BNP17b], and statements (2)-(4) follow immediately from the same techniques).
Theorem 1.21. Let G be a reductive group over a field of characteristic 0.
(2) The same assertion holds with G replaced by r G " GˆG m .
(3) The trace of multiplication by q P G m acting on the monoidal category pCohpZ{Gq,˚q is identified as TrppCohpZ{Gq,˚q, q˚q " CohpL u q " L q pN {Gqq. (4) The distinguished object r1 C s in each of these trace decategorifications is given by the coherent Springer sheaf S. Hence the endomorphisms of the coherent Springer sheaf recover the affine Hecke algebra (the vertical trace, as in Theorem 1.7), and the natural functor in Equation 3.1 is identified with HompS q ,´q : CohpL u q q ÝÑ H q -mod.
In other words, we identify the entire category of coherent sheaves on the stack of unipotent Langlands parameters as the categorical trace of the affine Hecke category. Inside we find the unramified principal series as modules for the naive trace (the Springer block). Just as the decategorification of the finite Hecke category (Example 1.18) knows all unipotent representations of Chevalley groups, the decategorification CohpL u q q of the affine Hecke category contains in particular all unipotent representations of G _ pF q -i.e., the complete L-packets of unramified principal series representations -thanks to Lusztig's remarkable Langlands duality for unipotent representations [Lu95]: The irreducible unipotent representations of G _ pF q are in bijection with G-conjugacy classes of triples ps, n, χq with s, n q-commuting as in Theorem 1.3 and χ an arbitrary G-equivariant local system on the orbit of ps, nq.
It would be extremely interesting to understand Theorem 1.22 using trace decategorification of Bezrukavnikov's Theorem 1.15. In particular we expect the full category of unipotent representations to be embedded in QC ! pL u q q as well as its cyclic deformation QC ! pL u q q S 1 .
1.7. Assumptions and notation. We work throughout over a field k of characteristic zero. We will sometimes work in the specific case of k " Q (e.g. in Section 2.2), and our main results require in addition that the field is algebraically closed (see Remark 2.25). This requirement that k is algebraically closed is also used in Section 4 in order to apply equivariant localization. All functors and categories are dg derived unless noted otherwise. All (co)chain complexes are cohomologically indexed, even if refered to as a chain complex. We abusively use HH to denote the Hochschild chain complex rather than its homology groups, and use H ‚ pHHq to denote the latter (and similarly for its cyclic variants HC, HP ).
1.7.1. Categories. Let A be a Noetherian dg algebra. We let A -mod denote the dg derived category of A-modules, A -perf denote the subcategory of perfect complexes, and A -coh denote the subcategory of coherent objects, i.e. cohomologically bounded complexes with coherent cohomology over π 0 pAq " H 0 pAq. Let C denote a symmetric monoidal dg category, and A P AlgpCq an algebra object. We denote by A -mod C the category of A-module objects in C. We denote the compact objects in a stable 8-category C by C ω , i.e. the objects X P C for which Hom C pX,´q commutes with all infinite direct sums (i.e. at least the countable cardinal ω). Let C be a stable k-linear 8-category (or a k-linear triangulated category or a pretriangulated dg category). These come in two primary flavors, "big" and "small": dgCat k is the 8-category of presentable stable k-linear 8-categories (with colimit-preserving functors), and dgcat k is the 8-category of small idempotent-complete stable k-linear 8-categories (with exact functors). Both dgCat k and dgcat k are symmetric monoidal 8-categories under the Lurie tensor product, with units Vect k " k -mod and Perf k " k -perf " k -coh the dg categories of chain complexes of k-vector spaces and perfect chain complexes, respectively. We have a symmetric monoidal ind-completion functor: Ind : dgcat k Ñ dgCat k .
It defines an equivalence between dgcat k and the subcategory of dgCat k defined by compactly generated categories and compact functors (functors preserving compact objects, or equivalently, possessing colimit preserving right adjoints).
Assume that C is either small or that it is compactly generated, and let X P C be an object, which we require to be compact in the latter case. We denote by xXy the subcategory (classicaly or weakly) generated by X. 1.7.2. Representation theory. Unless otherwise noted, G denotes a reductive group over a field k of characteristic 0 (which morally is the dual group G _ of a p-adic group), with Borel B and torus T Ă B with universal Cartan H and (finite) universal Weyl group W f . The extended affine Weyl group is denoted W a :" X ‚ pT q¸W f . In line with the assumptions of the main theorems of [KL87] [CG97], we will eventually assume that G has simply connected derived subgroup.
We denote by ReppGq " QCpBGq the derived category of rational representations of G.
We will often be interested in equivariance with respect to the trivial extension of G, which we denote 5 r G " GˆG m . Likewise, g " LiepGq, b " LiepBq, et cetera. Let B " G{B denote the flag variety, N denote the formal neighborhood 6 of the nilpotent cone of g or the unipotent cone of G (identified by the exponential map), r N " T˚G{B the Springer resolution, and r g the Grothendieck-Springer resolution. We let r G denote the group-theoretic Grothendieck-Springer resolution. Let Z " r Nˆg r N denote the derived Steinberg scheme, and Z 1 " r Nˆg r g denote the non-reduced Steinberg scheme. The classical scheme of the derived Steinberg variety π 0 pZq and the reduced scheme to the non-reduced Steinberg scheme pZ 1 q red are naturally isomorphic to the classical Steinberg variety.
The mixed affine Hecke category 7 is defined H :" CohpZ{ r Gq, and the affine Hecke algebra is denoted H. We define the coherent Springer sheaf to be 1.7.3. Algebraic geometry. We work in the setting of derived algebraic geometry over a field k of characteristic zero, in the setting presented in [GR17]. Namely, this is a version of algebraic geometry in which functors of (discrete) categories from rings to sets are replaced by prestacks, functors of (8-)categories from connective commutative dg k-algebras to simplicial sets. Examples of prestacks are given by both classical schemes and stacks and topological spaces (or rather the corresponding simplicial sets of singuar chains) such as S 1 , considered as constant functors. We will only be concerned with QCA (derived) stacks as in [DG13], i.e., quasicompact stacks with affine diagonal (in fact only with quotients of schemes by affine group-schemes), and use the term stack to refer to such an object.
A stack X carries a symmetric monoidal 8-category (i.e., a commutative algebra object in dgCat k ) QCpXq of quasicoherent sheaves, defined by right Kan extension from the case of representable functors X " SpecpRq which are assigned QCpSpec Rq " R-mod. For all stacks we will encounter (and more generally for perfect stacks in the sense of [BFN10]), we have QCpXq » IndpPerfpXqq, i.e., quasicoherent sheaves are compactly generated and the compact objects are perfect complexes (PerfpXq P dgCat k forms a small symmetric monoidal dg category). We can also consider the category QC ! pXq " IndpCohpXqq P dgCat k of ind-coherent sheaves, whose theory is developed in detail in the book [GR17] (see also the earlier [Ga13]). The category QC ! pXq (under our assumption that X is QCA) is compactly generated by CohpXq, the objects which are coherent after smooth pullback to a scheme (see Theorem 3.3.5 of [DG13]). For 5 We explain this choice of notation. In the usual convention (opposite to ours), G denotes a group on the "automorphic" side of Langlands and L G is used to denote its Langlands dual on the "spectral" side. It was proposed in [BuG14] [Ber20] to replace G with a (possibly nontrivial) central extension of G by Gm, denoted r G, whose Langlands dual would be denoted C G. When G is adjoint (therefore L G simply connected), the center is trivial and therefore r G " GˆGm is a trivial extension, and C G " L GˆGm. Note that in our work is mostly on the spectral side so we depart from this convention in using G to denote a group on the spectral side rather than L G for convenience. We note there is an inherent asymmetry since taking Langlands duals flips the ordering in the short exact sequence 1 Ñ Gm Ñ G Ñ r G Ñ 1. 6 This unusual choice is made to avoid cumbersome notation, since it is this formal neighborhood which will appear most often. When referring to the usual (reduced) nilpotent cone we will write N red .
7 As opposed to the affine Hecke category CohpZ{Gq.
smooth X, the notions of coherent and perfect, hence ind-coherent and quasicoherent, sheaves are equivalent. A crucial formalism developed in detail in [GR17] is the functoriality of QC ! . Namely for a map p : X Ñ Y of stacks, we have colimit-preserving functors of pushforward p˚: QC ! pXq Ñ QC ! pY q and exceptional pullback p ! : QC ! pY q Ñ QC ! pXq, which form an adjoint pair pp˚, p ! q for p proper. These functors satisfy a strong form of base change, which makes QC ! a functor -in fact a symmetric monoidal functor 8 -out of the category of correspondences of stacks (the strongest form of this result is [GR17, Theorem III.3.5.4.3, III.3.6.3]).
For X a stack, we depart from some conventions by using Ω 1 X to denote the cotangent complex of X. We will never consider the sheaf of Kähler differentials from classical algebraic geometry except in the smooth case where they coincide. We denote by T X its O X -linear dual.
1.8. Acknowledgments. We would like to thank Xinwen Zhu for very enlightening conversations on the topic of categorical traces, the Drinfeld-Lafforgue sheaf and its relation to the coherent Springer sheaf and for sharing with us an early draft of his paper [Z20], and Pramod Achar for discussions of purity and Tate-ness properties in Springer theory. We would also like to thank Sam Raskin for suggestions related to renormalized categories of sheaves on formal odd tangent bundles, and Gurbir Dhillon for helpful discussions.

Hochschild homology of the affine Hecke category
In this section we calculate the Hochschild and cyclic homology of the affine Hecke category. In particular we prove that the Chern character from K-theory factors through an isomorphism between K 0 and Hochschild homology. For this we use Bezrukavnikov's Langlands duality for the affine Hecke category to construct a semiorthogonal decomposition on the equivariant derived category of the Steinberg variety with simple components, from which the calculation of localizing invariants is immediate.
The results of Subsection 2.1.1 apply for any field k of characteristic zero. The results of Subsections 2.1.2 and 2.2 specifically apply to the case k " Q . In Corollary 2.20 we will pass to Hochschild homology, where statements will hold for any field of characteristic zero. Finally, in Subsection 2.4 we will use a theorem of Ginzburg-Kazhdan-Lusztig which further requires k to be algebraically closed.
2.1. Background. We first review some standard notions regarding Hochschild homology and equivariant -adic sheaves that we need for our arguments.
2.1.1. Hochschild homology and traces. An extended discussion of the notions of this subsection can be found in [GKRV20], [BN19] and [Ch20a]. We recall the notion of a dualizable object X of a symmetric monoidal 8-category C b with monoidal unit 1 b .
Definition 2.1. The object X is dualizable if there exists an object X _ and coevaluation and evaluation morphisms Dualizability is a property rather than an additional structure on X: by Proposition 4.6.1.10 in [L18], the space of dualizing structures on X is contractible. The trace of an endomorphism f P End C pXq of a dualizable object is defined by The trace of the identity f " id X is called the dimension dimpXq.
We are interested in the case when X is an algebra object in the symmetric monoidal 8category C b , and the resulting algebra structure on traces. To formulate this, we note that traces are canonically symmetric monoidal with respect to the monoidal structure in C b and composition in End Cb p1 b q. In addition, we require a natural functoriality enjoyed by the abstract construction of traces in the higher-categorical setting, see [TV15,HSS17,GKRV20] (see also [BN19] for an informal discussion). Namely the dimension of an object is covariantly functorial under right-dualizable morphisms.
Definition 2.2. A morphism of pairs pF, ψq : pX, f q Ñ pY, gq is a right-dualizable morphism F : X Ñ Y (i.e. has a right adjoint G) along with a commuting structure ψ : F˝f g˝F » . Given a morphism of pairs as above, we can define trpF, ψq : trpX, f q Ñ trpY, gq as in Definition 3.24 of [BN19].
Thus, the trace construction enhances to a symmetric monoidal functor from the 8-category of endomorphisms of dualizable objects in C b to endomorphisms of the unit 1 b , see [HSS17,2], [TV15, 2.5] and [GKRV20, 3] for details. In particular, if X is an algebra object in C b and f is an algebra endofunctor, then trpX, f q is an algebra object in endomorphisms of 1 b .
In this paper, we only consider the case C b " dgCat k , the 8-category of cocomplete klinear dg categories, with morphisms given by left adjoint (i.e. cocomplete) functors. We now specialize to this case.
Example 2.3. Any compactly generated dg category C " IndpC ω q P dgCat k is dualizable, with dual given by taking the opposite of compact generators C _ " IndpC ω,op q. Thus we may speak of its dimension and traces of its endofunctors, which are endomorphisms of the unit, i.e.

chain complexes
End dgCat k pVect k q » Vect k .
Furthermore, note that a morphism of pairs is a functor that has a continuous right adjoint, or equivalently for compactly generated categories, a functor which preserves compact objects.
Definition 2.4. The Hochschild homology of a dualizable (for instance, compactly generated) dg category C P dgCat k is its dimension More generally, the Hochschild homology of C with coefficients in a continuous endofunctor F is HHpC, F q " trpC, F q P Vect k .
Let A be an dg algebra object in ReppGq for a reductive group G. We describe an explicit algebraic model for the Hochschild homology of the category of A-modules over ReppGq due to Block and Getzler [BlGe94].
Definition 2.5. Denoting the coaction map corresponding to the ReppGq-structure on the algebra A by c : A Ñ A b krGs, the Block-Getzler complex C ‚ pA, Gq is the mixed complex associated to the following cyclic vector space. We define the n-simplices by C´npA, Gq " pA bn`1 b krGsq G with face and degeneracy maps d i pa 0 b¨¨¨b a n b f q " a 0 b¨¨¨b a i a i`1 b¨¨¨b a n b f i " 0, . . . , n´1, d n pa 0 b¨¨¨b a n b f q " cpa n qa 0 b¨¨¨b f, and cyclic structure tpa 0 b¨¨¨b a n b f q " cpa n q b a 0 b¨¨¨b a n´1 b f, s n`1 pa 0 b¨¨¨b a n b f q " 1 b a 0 b¨¨¨b a n b f.
We define the G-enhanced Block-Getzler complex r C ‚ pA, Gq to be complex defined in the same way but without taking G-invariants, i.e. with C´npA, Gq " A bn`1 b krGs. Note that unlike the Block-Getzer complex, the enhanced version is not a cyclic object in vector spaces. 9 For g P Gpkq, we define C ‚ g pA, Gq :" r C ‚ pA, Gq b krGs krGs{I g where I g is the ideal cutting out g P G.
Essentially by definition, we see that the complex C ‚ g pAq computes HHpA -mod, g˚q where gi s the autofunctor of A -mod induced by the automorphism of A via action by g considered as a G-representation.
Finally, we will use the universal S 1 -equivariant trace map from connective K-theory to Hochschild homology constructed in [BGT13]. To avoid overloading the word "trace" we refer to this map as the Chern character.
Definition 2.7. For any small k-linear dg-category C, the connective K-theory spectrum KpCq is the connective K-theory of the corresponding Waldhausen category defined in Section 5.2 of [Ke06]. The universal cyclic Chern character 10 is the map ch : KpCq Ñ HN pCq. We use the same notation to refer to the Chern character ch : KpCq Ñ HHpCq.
Remark 2.8. By functoriality of the Chern character and using the lax monoidal structure of K-theory, we see that for a monoidal category C the Chern character defines a map of algebras.
Often in applications to geometric representation theory, we are only interested in (or able to) compute the Grothendieck group K 0 . In order to compare K 0 with Hochschild homology, we require certain vanishing conditions to hold. We say that C has a 0-truncated Chern character if we have a factorization

KpCq
HN pCq K 0 pCq ch By S 1 -equivariance of the Chern character, it is equivalent to check that the Chern character to HHpCq has the same factorization. It is clear that if HHpCq or HN pCq is coconnective, then C has a 0-truncated Chern character.
2.1.2. Equivariant -adic sheaves, weights, and Tate type. In this subsection we review some standard notions concerning weights and the -adic cohomology of BG. In this section and the following one, we fix a prime power q " p r and a prime ‰ p, and will work with -adic sheaves F on F q -schemes X. All schemes and sheaves on them that arise are defined over F q , i.e., X will come with a geometric Frobenius automorphism Fr and F with a Fr-equivariant (Weil) structure, which will be left implicit. Fix a square root of q in Q , thereby defining a notion of half Tate twist (this choice can be avoided by judicious use of extended groups as in [BuG14,Z17,Ber20]). For F P ShpXq where X is over F q , we will denote the Tate twist by Fpn{2q for n P Z. For a scheme X over F q with a group action G, we denote by ShpX{Gq " Sh G pXq the bounded derived category of finite G-equivariant Q -sheaves on X (see Section 1.3 of [BY13] and [BL94]). In this context, the cohomology of a sheaf H ‚ pX,´q will be understood to meanétale cohomology.
Following the Appendix of [Ga00], this notion can be extended to G-equivariant ind-schemes, where G is a pro-affine algebraic ind-group acting in a sufficiently finite way. We say a Gaction on X is nice if the following two properties hold: (1) every closed subscheme Z Ă X is contained in a closed G-stable subscheme Z 1 Ă X such that the action of G on Z 1 factors through an quotient of G which is affine algebraic, and (2) G contains a pro-unipotent subgroup of finite codimension, i.e. if G " lim nÑ8 G n , then there is an n such that kerpG Ñ G n q is a projective limit of unipotent affine algebraic groups. If G is a pro-affine group scheme acting nicely on X, and X " colim iÑ8 X i with affine quotient G i acting on X i , then we define 11 Sh G pXq " colim iÑ8 Sh Gi pX i q. 10 We use this terminology to avoid overloading the word "trace." 11 This definition is independent of the choice of presentation, since by [BL94] Theorem 3.4.1(ii) if G i Ñ G j is a surjection with unipotent kernel, then Sh G j pY q Ñ Sh G i pY q is an equivalence for any Y on which G j acts. See also Section A.4 of [Ga00].
Finally, we need a notion of Frobenius weights acting on a Q -vector space V , which for us will beétale cohomology groups. We will generally only be concerned with the weak notion of weights and will omit the adjective "weak" for brevity.
Definition 2.9. Let V be a finite-dimensional Q -vector space equipped with an endomorphism F , and fix a prime power q " p r . We say V is strongly pure of weight n if every eigenvalue of F is equal to q n{2 . We say V is weakly pure of weight n if every eigenvalue of F is equal to ζq n{2 for varying roots of unity ζ P Q . If V is a (cohomologically) graded vector space with finite-dimensional homogeneous parts V k , then we say V is strongly (resp. weakly) pure of weight n if V k is strongly (resp. weakly) pure of weight n`k.
Finally we recall the -adic cohomology ring of BG, whose description we repeat for convenience following [Vi15] (in the Hodge-theory context).
Proposition 2.10. Let G be a pro-affine group scheme with split reductive quotient over k. Then, H ‚ pBG, Q q is polynomial, generated in even degrees, and pure of weight 0. In particular, H 2k pBG, Q q has weight 2k.
Proof. First, since G is pro-affine, there is a reductive (finite type) algebraic group G 0 such that the kernel kerpG Ñ G 0 q is pro-unipotent. By Theorem 3.4.1(ii) in [BL94] we may assume that G is reductive (and finite type).
It is a standard calculation that H ‚ pG m , Q q " H 0 pG m , Q q ' H 1 pG m , Q q with H 0 of weight 0 and H 1 of weight 2. By Corollary 10.4 of [LO08], H ‚ pBG m , Q q » Q rus where u has cohomological degree |u| " 2 and weight 2. In particular, by the Kunneth formula (Theorem 11.4 in op. cit.) we have that for a split torus T , H ‚ pBT ; Q q is pure of weight 0 and polynomial in even degrees. Thus, the claim is true when G " T is a torus. Now, assume T is a split torus inside a reductive group G, and B is a Borel subgroup with 2.2. Automorphic and spectral realizations of the affine Hecke category. We follow the set-up of Bezrukavnikov in [Bez16]. Let F " F q pptqq and O " F q rrtss. Let G be the a fixed reductive algebraic group with simply connected derived subgroup (e.g. the group from the discussion in the introduction). The dual group G _ has adjoint type derived subgroup. We choose q large enough so that the dual group G _ is split. We denote G :" G _ pF q to be its dual group with coefficients in F , which we consider as an ind-group scheme over F q , and its subgroup G 0 :" G _ pOq, a pro-affine group scheme over F q . The Iwahori subgroup of G is I :" G 0ˆG_ pFqq B _ pF q q, which inherits its structure as a closed subgroup and is therefore also a pro-affine group. We let I 0 :" G 0ˆG_ pFqq U _ pF q q denote its pro-unipotent radical.
We are interested in the affine flag variety Fl " G{I, an ind-proper ind-scheme constructed in the Appendix of [Ga00]. It carries a left action of I whose orbits are of finite type and naturally indexed 12 by the affine Weyl group W a . For w P W , we denote by Fl w the corresponding orbit. Denote by j w : Fl w ãÑ Fl the inclusion of the corresponding I-orbit. Let : W a Ñ Z ě0 denote the length function on the affine Weyl group.
On the automorphic side, we will consider equivariant Q -sheaves on Fl. On the spectral side, the stacks that appear are defined over Q .
Theorem 2.11. [Theorem 1, Lemma 43 [Bez16]] There are equivalences of categories Φ and Φ 1 and a commutative diagram 2 Note it is the extended affine Weyl group of G and the affine Weyl group of G.
where π : IzFl Ñ I 0 zFl is the quotient map and i : Z 1 {G ãÑ Z{G is the inclusion. Moreover the functors admits the following natural structures: ‚ Φ is naturally an equivalence of monoidal categories, and ‚ Φ and Φ 1 intertwine the action of Frobenius on Sh I pFlq (resp. Sh I 0 pFlq) with the action of q P G m on Z{G (resp. Z 1 {G).
Note that the Frobenius property of Φ appears as Proposition 53 in [Bez16]. We point out certain distinguished sheaves in Sh I pFlq and Sh I 0 pFlq (computed explicitly for G " SL 2 , P GL 2 in Examples 2.2.3-5 in [NY19]): (a) Let λ P X˚pT q Ă W a be a character of the maximal torus of G, considered as an element of the affine Weyl group of the dual group. The Wakimoto sheaves J λ are defined as follows. When λ is dominant, we take J λ " j λ,˚Q Fl λ rx2ρ, λys. When λ is antidominant, we take J λ " j λ,! Q Fl λ rx2ρ,´λys. In general, writing λ " λ 1´λ2 , we define J λ " J λ1˚J´λ2 , which is independent of choices due to Corollary 1 in Section 3.2 of [AB09]. (b) For any w P W a , we define the corresponding costandard (resp. standard ) object by ∇ w :" j w,˚Q Fl w r pwqs (resp. ∆ w :" j w,! Q Fl w r pwqs). They are monoidal inverses by Lemma 8 in Section 3.2 of [AB09]. By Lemma 4 of [Bez16], we have ∇ w˚∇w 1 " ∇ ww 1 (and likewise for standard objects) when pwq` pw 1 q " pww 1 q. If λ P X˚pT q is dominant, then the Wakimoto is costandard J λ " ∇ λ and if λ is antidominant, the Wakimoto is standard J λ " ∆ λ . (c) Let w 0 P W f Ă W a be the longest element of the finite Weyl group. The antispherical projector or big tilting sheaf Ξ P Sh I 0 pFlq is defined to be the tilting extension of the constant sheaf Q Fl w 0 off Fl w0 to Fl, as in Proposition 11 and Section 5 of [Bez16]. Note that this object does not descend to Sh I pFlq.
We abusively use the same notation to denote sheaves in Sh I 0 pFlq; note that π˚∆ w » ∆ w and π˚∇ w » ∇ w by base change. All sheaves above are perverse sheaves, since the inclusion maps of strata are affine.
For our applications, we need to work not with Z{G but with Z{ r G. The following proposition is the key technical argument we need to construct the semiorthogonal decomposition of CohpZ{ r Gq and hence deduce results on its homological invariants -a graded lift of standards and costandards under Bezrukavnikov's theorem. It is conjectured in [Bez16] that the equivalences in Theorem 2.11 should have mixed versions, relating a mixed form of the Iwahori-equivariant category of Fl with a G m -equivariant version of CohpZ{Gq, i.e. CohpZ{ r Gq, which would immediately give us the desired result. In particular, see Example 57 in [Bez16] for an expectation of what the sheaves Φp∆ w q are explicitly and note that they have G m -equivariant lifts.
Proposition 2.12. The objects Φp∇ w q, Φp∆ w q P CohpZ{Gq have lifts to objects in CohpZ{ r Gq for all w P W a .
Proof. We will prove the statements for the standard objects; the analogous statement for costandards follows by a similar argument. Wakimoto sheaves are sent to twists of the diagonal ΦpJ λ q » O ∆ pλq by Section 4.1.1 of [Bez16], which evidently have G m -equivariant lifts. Convolution is evidently G m -equivariant, so the convolution of two sheaves with G m -lifts also has a G m -lift. Assiming that the standard objects corresponding to finite reflections have G m -lifts, by Lemma 4 of [Bez16] we can write the standard for the affine reflection as a convolution of Wakimoto sheaves and standard objects for finite reflections. Thus, we have reduced to showing that all standard objects Φp∆ w q have G m -lifts for w a simple finite reflection.
By Corollary 42 of [Bez16] Φ 1 has the favorable property that Z 1 is a classical (non-reduced) scheme, and that it restricts to a map on abelian categories on Perv U _ pG _ {B _ q Ă Perv I 0 pFlq taking values in CohpZ 1 {Gq ♥ (though it is not surjective). In particular, by Proposition 26 and Lemma 28 in [Bez16] it takes the tilting sheaf Ξ to O Z 1 {G , which manifestly has a G m -lift. Note that G m -lifts for the Φ 1 p∆ w q P CohpZ 1 {Gq for w P W f induce G m -ifts for the Φp∆ w q P CohpZ{Gq. Since Z is a derived scheme, a priori an object of CohpZ{Gq may contain more structure than its image in CohpZ 1 {Gq under i˚. But since Φ 1 p∆ w q » i˚Φp∆ w q are in the heart and i˚is exact, we have that Φp∆ w q P CohpZ{Gq ♥ . In particular, the image of CohpZ{Gq ♥ under i˚is a subcategory of CohpZ 1 {Gq ♥ and thus a G m -lift on Φ 1 p∆ w q induces a G m -lift on Φp∆ w q. Thus, we have reduced to showing that the finite simple standard objects Φ 1 p∆ w q P CohpZ 1 {Gq have G m -lifts.
By Lemma 4.4.11 in [BY13], Ξ is a successive extension of standard objects ∆ w p pwq{2q for w P W f (the finite Weyl group). Since Ξ is a projective cover of δ e " ∆ e " ∇ e , we have a morphism Ξ δ e which Φ 1 takes to the quotient map O Z 1 O Ă N . This is evidently a G mequivariant map between G m -equivariant sheaves, and in particular the kernel has a graded lift. Let K " kerpΞ δ e q. By loc. cit. there is a standard object ∆ w p pwq{2q and a Frobenius-equivariant surjection K ∆ w p pwq{2q. This implies that the kernel K 1 " kerpK ∆ w p pwq{2qq is a Frobeniusequivariant subobject of K. On the spectral side, using Proposition 53 in op. cit., this means that Φ 1 pK 1 q Ă Φ 1 pKq is a q-equivariant subobject with quotient Φ 1 p∆ w p pwq{2qq. We wish to show that the quotient has a G m -equivariant lift, which amounts to showing that Φ 1 pK 1 q is a G m -equivariant subobject.
Since ΦpKq is already endowed with a G m -equivariant structure, q-equivariance for a subobject of a G m -equivariant object is property, not an additional structure. We claim that for q not a root of unity, any q-closed subsheaf of a G m -equivariant sheaf on a quotient stack must be G m -closed as well (i.e. the isomorphism defining the G m -equivariant structure restricts to the subsheaf). Assuming this claim, we find that Φ 1 p∆ w q has a G m -equivariant lift for w P W f , completing the proof.
We now justify the claim. First, if F is a sheaf on a quotient stack X{G with a G m -action, we can forget the G-equivariance (i.e. base change to the standard atlas X Ñ X{G). Now, by reducing to an open affine G m -closed cover of X, we can assume X is affine. On an affine scheme X " SpecpAq, the G m -action gives the structure of a Z-grading on A, and a submodule of a graded A-module M 1 Ă M is q-equivariant if it is a sum of q-eigenspaces, and G m -equivariant if it is a sum of homogeneous submodules. The claim follows from the observation that any m P M 1 can only have eigenvalues q n for n P Z, which are distinct, so the q-eigenspaces entirely determine the G m -weights.
2.3. Semiorthogonal decomposition. In this section, we describe an "Iwahori-Matsumoto" semiorthogonal decomposition of the category CohpZ{ r Gq, arising from the stratification of the affine flag variety Fl on the automorphic side of Bezrukavnikov's equivalence Theorem 2.11 and the lifting result in Proposition 2.12. This will, in turn, induce a direct sum decomposition on Hochschild homology. First, let us establish terminology.
Definition 2.13. Let tS n u nPN denote a collection of full subcategories of a dg category C. We say that tS n u defines a semiorthogonal decomposition of C if Ť nPN S n generates C, and if Hom ‚ C pX n , X m q » 0 for X i P S i and n ą m.
The following result is standard.
Proposition 2.14. Let G be a pro-affine group scheme acting nicely on an ind-scheme X.
Assume that the stabilizer of each orbit is connected. Let I be an indexing set for the G-orbits X i under the closure relation, i.e. X n Ă X m when m ě n, and let j n : X n ãÑ X denote the inclusion. Then, xj n! Q Xn y defines a semiorthogonal decomposition of Sh G pXq, where the ordering is given by any choice of extension of the partial order to a total order.
Proof. We note that each orbit is equivariantly BH where H is the stabilizer, and ShpBHq is generated by the constant sheaf Q when H is connected.
Corollary 2.15. Fix a Bruhat ordering of the affine Weyl group W a . The standard objects x∇ w y give a semiorthogonal decomposition of Sh G pFlq.
Remark 2.16. The objects j n˚Q Xn define a semiorthogonal decomposition in the reverse order.
In particular, the costandard objects also give a semiorthogonal decomposition, in the reverse order.
By Theorem 2.11, we obtain a semiorthogonal decomposition of CohpZ{Gq. We would like to lift it to a decomposition of CohpZ{ r Gq. We do so by combining the G m -equivariant lifts of the objects Φp∆ w q from Proposition 2.12 with the following results, which we will apply in the case C " CohpZ{ r Gq, C 1 " CohpZ{Gq and H " G m " Spec krz, z´1s.
Lemma 2.17. Let k be a field of characteristic zero, H a group-scheme over k, C a pretriangulated ReppHq-linear category, and let F : C Ñ C 1 " C b ReppHq Vect k denote the forgetful functor. Let E P C be a compact object such that F pEq is a generator for C 1 . Then E is a ReppHq-generator of C, i.e., C is equivalent to modules in ReppHq for the internal endomorphism algebra A " End ReppHq pEq op P AlgpReppHqq.
Proof. The lemma is an application of the rigidity of ReppHq and the Barr-Beck-Lurie monadicity theorem. Explicitly, recall (e.g. in Definition 9.1.2 of [GR17]) rigidity implies that the functor act E : ReppHq Ñ C given by action on E has a ReppHq-linear continuous right adjoint Ψ " Hom ReppHq pE,´q, which takes E to the internal endomorphism algebra (which represents the corresponding monad Ψ˝act E on ReppHq). Note that F has a continuous right adjoint G : C 1 Ñ C given by the tensoring with the regular representation, and hence preserves compact objects. Since F pEq is a compact generator for C 1 , the functor Ψ 1 p´q " Hom C 1 pF pEq,´q : is an equivalence, giving us the commuting square of left adjoint functors: where F 1 is also the forgetful functor. Applying Barr-Beck to the functors F, F 1 and their right adjoints, the monads in C 1 and A -mod are identified under the equivalence Φ 1 and therefore Φ is an equivalence.
Corollary 2.18. Let k be a field of characteristic zero, H a group-scheme over k, C be a pretriangulated ReppHq-linear dg-category, and let F : C Ñ C 1 " C b ReppHq Vect k denote the forgetful functor. Let tE n P C | n P Nu be a linearly ordered set of objects such that xF pE n qy defines a semiorthogonal decomposition in C 1 . Denote by A n " End C 1 pE n q op the algebras in ReppHq from the previous lemma. Then, we have HHpCq » à α HHpA n -mod ReppHq q.
Proof. Let C 1 n :" xF pE n qy be the category generated by F pE n q, and let C n be the preimage under the forgetful functor. We have a semiorthogonal decomposition of C by the categories C n . Hochschild homology is a localizing invariant in the sense of [BGT13], and in particular takes semiorthogonal decompositions to direct sums (this can also easily be seen directly via the dg model for Hochschild homology). Thus we have an equivalence HHpCq " à nPZ HHpC n q.
Next, we specialize to the case C " CohpZ{ r Gq, C 1 " CohpZ{Gq and H " G m " Spec krz, z´1s, and compute the endomorphisms algebras of the generators ∇ w in our semiorthogonal decomposition as ReppG m q-algebras.
Proposition 2.19. Let E w denote the G m -lifts of Φp∆ w q constructed in Proposition 2.12, and A w " End CohpZ{ r Gq pE w q. We have a quasi-isomorphism A w » Sym Q hr´2s where hr´2s is the universal Cartan shifted into cohomological degree 2 with G m -weight 1. In particular, A w is formal.
Proof. Recall that the pullback along multiplication by q corresponds under Φ to the Frobenius automorphism, i.e. Frobenius acts on the nth homogeneous graded piece of T w! by multiplication by q n . Since q is not a root of unity, we can determine G m -weights by (necessarily integral) Frobenius weights as in the proof of Proposition 2.12.
Further, since Φ is an equivalence of categories we can compute A w on the automorphic side. The unit map F Ñ j ! j ! F is an equivalence for j a locally closed immersion, so that Since Fl w is an I-orbit, letting I w denote its stabilizer for a choice of base point in Fl w , we find that A w » C ‚ pBI w ; Q q is the equivariant cohomology chain complex for BI w with Qcoefficients under the cup product. The reductive quotient (i.e. by the pro-unipotent radical) of I w is T , so A w » C ‚ pBT ; Q q. By Proposition 2.10, the Frobenius weight is equal to the cohomological degree, and the Frobenius weight is equal to twice the G m -weight, proving the claim regarding G m -weights.
Finally, we need to show formality of A w as an algebra. By purity, any cohomological degree 2n class in C ‚ pBT ; Q q has weight 2n. By a standard weight-degree shearing argument, this implies formality.
We now apply Corollary 2.18 to the set-up in the above proposition. We will see that since Hochschild homology is insensitive to field extensions and all our stacks of interest are defined over Q, the following results hold for any field k of characteristic 0 (i.e. not just k " Q ).
Corollary 2.20. Let k be any field of characteristic 0. The isomorphism from above induces an isomorphism of krz, z´1s-modules HHpHq " kW a b k krz, z´1s.
In particular, HHpHq is cohomologically concentrated in degree zero. Therefore, the natural trace map KpHq Ñ HHpHq factors through K 0 pHq. Furthermore, the map K 0 pHq b Z k Ñ HHpHq is an equivalence, and H satisfies Hochschild-to-cyclic degeneration, i.e. HP pHq » HHpHqrruss.
Proof. Fix a Bruhat order on W a , extended to a total order. Let us first prove the case k " Q . Applying Corollary 2.18 in the case C " H " CohpZ{ r Gq, C; " CohpZ{Gq, and H " G m , we have a canonical equivalence where A " Sym ‚ Q h˚r´2s » A w is the algebra from Proposition 2.19 (which does not depend on w P W a ).
Note that A is canonically defined over any characteristic 0 field k. The Hochschild homology of this category is computed by the Block-Getzler complex C ‚ pA, G m q, which has terms pA bn`1 b krz, z´1sq Gm . Since z has G m -weight 0, there is an isomorphism pA bn`1 b krz, z´1sq Gm » pA bn`1 q Gm b krz, z´1s and we observe that pA bn`1 q Gm " k since each A is generated over k by positive weights. Thus, the natural map C ‚ pk, G m q Ñ C ‚ pA, G m q is a quasi-isomorphism, so the first claim claim follows. Factorization through K 0 follows since the Hochschild homology is coconnective.
To show that the map K 0 pA -mod ReppGmq q b Z k Ñ HHpA -mod ReppGmq q is an equivalence, first note that since HHpA -mod ReppGmq q is concentrated in degree zero, the Chern character factors through K 0 , i.e. we have a commuting diagram for each summand The claim follows from the observation that K 0 pReppG m qq Ñ K 0 pA -perf ReppGmq q is an equivalence, since the free object A » krrtss has no retracts. Next, to prove the equivalence for general fields k, note that all stacks in question are welldefined over Q, and thus any field of characteristic 0. Let Q Ă K be a field extension. By the change of rings formula in Hochschild homology, we have a canonical equivalence Now, suppose that k Ă Q is a field extension. To conclude the result for k, we need to show that the k-subspaces coincide under the equivalence; this follows from the calculation of HHpA -perf ReppGmq q via the Block-Getzler complex. Thus, the equivalence is preserved by field extension and restricts to subfields, and thus holds for any field k of characteristic 0.
We also have the following result for the non-G m -equivariant version.
Corollary 2.21. The map of algebras KpCohpZ{Gqq Ñ HHpCohpZ{Gqq factors through K 0 and we have an isomorphism as dg k-modules HHpCohpZ{Gqq » kW a b k Sym ‚ k ph˚r´1s, h˚r´2sq. Furthermore, the Connes B-differential is given by the extending identity map h˚r´2s Ñ h˚r´1s, so that applying the Tate construction we have an isomorphism of modules Proof. Essentially the same as the previous corollary, along with a direct calculation of the Hochschild homology of the formal dg ring HHpSh T pptqq " HHpkrhr´2ss -modq.
Remark 2.22. Note that while this implies HHpCohpZ{Gqq is formal as a module, we do not know that it is formal as an algebra.
2.4. Hochschild and cyclic homology of the affine Hecke category. Recall that H " CohpZ{ r Gq denotes the affine Hecke category, H denotes the affine Hecke algebra, kW a denotes the group ring of the extended affine Weyl group, and that r G " GˆG m . We will assume that G is a reductive algebraic group with simply connected derived subgroup throughout the section. We begin by quoting the following celebrated theorem by Ginzburg, Kazhdan and Lusztig.
Theorem 2.23 (Ginzburg-Kazhdan-Lusztig). Let k be an algebraically closed field of characteristic 0, and assume that G has simply connected derived subgroup. Then there is an equivalence of associative algebras H Ñ K 0 pHq b Z k, compatibly with an identification of the center with K 0 pRepp r Gqq b Z k. Likewise, there is an equivalence of associative algebras kW a » K 0 pCohpZ{Gqq b Z k with center K 0 pReppGqq.
Proof. The only difference between our statement and that in [KL87] [CG97] is their Steinberg stack is the classical stack π 0 pZq{ r G, which has no derived structure. On the other hand, we are interested in Z{ r G which has better formal properties. The statement follows from the fact that the Grothendieck group is insensitive to derived structure, i.e. the ideal sheaf for the embedding π 0 pZq{ r G ãÑ Z{ r G acts nilpotently on any coherent complex. Finally, note that while the statement of Theorem 3.5 of [KL87] and Theorem 7.2.5 in [CG97] are made for k " C, the proofs do not employ topological methods and apply to any algebraically closed field of characteristic zero.
We combine the above theorem with Corollary 2.20 to arrive at the following main theorem.
Theorem 2.24. Assume that G has simply connected derived subgroup over an algebraically closed field k of characteristic 0. There is an equivalence of algebras, and an identification of the center: H HHpHq That the map is an isomorphism is a combination of Theorem 1.2 and Corollary 2.20; that the map is a map of algebras follows by functoriality of the Chern character map from K-theory to Hochschild homology, whence it preserves convolution algebra structure.
Remark 2.25. We only require the field k to be algebraically closed, and the group G to have simply connected derived subgroup, in order to apply the main theorem of [KL87] [CG97].
The following may also be of interest, and is the analogue to Corollary 2.21. Note that in this case, the map to Hochschild homology is not an equivalence, though it does induce an equivalence on HH 0 and on periodic cyclic homology HP .
Corollary 2.26. With the assumptions above, there is a commuting diagram of algebras: Taking the Tate construction, there is an equivalence of kppuqq-algebras, and an identification of the center: kW a ppuqq HP pCohpZ{Gqq krGs G ppuqq HP pReppGqq.

» »
Proof. Note that we HHpCohpZ{Gqq is coconnective, so the Chern character from KpCohpZ{Gqq factors through K 0 pCohpZ{Gqq b Z k " kW a . Thus we have a map of algebras kW a Ñ HHpCohpZ{Gqq which induces an equivalence on H 0 . Next, note that the subcategory Sh I pFlq generated by the monoidal unit (skyscraper sheaf), which is closed under the monoidal structure, is in the center of CohpZ{Gq, so that the subalgebra Sym ‚ k phr´1s ' hr´2sq Ă HHpCohpZ{Gqq is central. Thus we have a map HHpkrhr´2ssq -mod Ñ HHpCohpZ{Gqq. Thus we have a map of algebras out of the tensor product 2.4.1. Trace of scaling by q. Let q : Z{G Ñ Z{G be the scaling by q P G m map. In this section we compute the trace of the functor q˚on the category C 1 " CohpZ{Gq. First, we observe that if F is an endofunctor of a category C 1 and E P C 1 , then a F -equivariant structure on E induces an automorphism of the dg algebra A " End C 1 pEq.
Proposition 2.27. Let q ‰ 1 and let A w denote the algebras from Proposition 2.19. Then, HHpA w , q˚q " k.
Proof. First, observe that the functor q˚induces the automorphism on the algebra A w » Sym k h˚r´2s coming from a shifted version of the q-scaling map on h (in particular, h˚has weight´1). The claim is a direct calculation using the complex C q pA w , G m q from Definition 2.5 via Koszul resolutions: C q pA w , G m q is the derived tensor product A w b L AwbAw A w where A w is the diagonal bimodule for one factor and is twisted by q˚on the other factor.
Rather than a direct calculation, we give a geometric argument. First, note that q˚preserves the G m -weights of A w » Sym ‚ k h˚r´2s (i.e. since q P G m is central). We apply a Tate shearing (i.e. sending bidegree pa, bq to pa´2b, bq) to the algebra Sym k h˚r´2s to obtain the algebra Ophq " Sym ‚ k h˚. Note that HHpPerfphq, q˚q " Oph q q, i.e. functions on the derived fixed points of action by q. When q ‰ 1 we have h q " t0u, so HHpPerfphq, q˚q " k. Undoing the shearing, we find that the natural map HHpA w , q˚q Ñ HHpk, q˚q is an equivalence.
Corollary 2.28. Let H q denote the specialization of the affine Hecke algebra at q P G m . If q ‰ 1, we have an equivalence of algebras Proof. The calculation in Proposition 2.27 shows that specialization at q P G m induces an equivalence on Block-Getzler complexes inducing an equivalence HHpCohpZ{ r Gqq b krz,z´1s k q » HHpCohpZ{Gq, q˚q, since the trace of an endofunctor F on a category C takes semiorthogonal decompositions preserved by F to direct sums. Consequently, under the identification of algebras HHpCohpZ{ r Gqq » H, specialization at q defines an equivalence HHpCohpZ{Gq, q˚q » H q .
Remark 2.29. Our methods also allow for an identification of the following monodromic variants of the affine Hecke category introduced in [Bez16] (where Z^is the formal completion of r gˆg r g along Z): The only difference in the above cases are the choice of generating object E w on orbits (w P W a ) and the derived endormorphism algebra A w . For Z 1 , the generating object is the constant sheaf on orbits, and for Z^it is the constant sheaf on orbits in the universal H-torsor (which generates sheaves with unipotent H-monodromy).

The affine Hecke algebra and the coherent Springer sheaf
We have seen in Theorem 2.24 that the affine Hecke algebra H is identified with the Hochschild homology of the affine Hecke category H " CohpZ{ r Gq. In this section we use tools from derived algebraic geometry to explain why this is a useful realization. Namely, the geometric realization of Hochschild homology via derived loop spaces implies a realization of the affine Hecke algebra as endomorphisms of a coherent sheaf on the loop space of the Steinberg variety, the coherent Springer sheaf, and hence a localization description of its category of modules as a category of coherent sheaves. We also explain the role of this realization from the perspective of the theory of categorical traces.
3.1. Traces of monoidal categories. In this section we present the two different trace decategorifications for a monoidal category and their relation, following [GKRV20].
Definition 3.1. There are two notions of Hochschild homology or trace of a presentable k-linear 8-category C P Pr L k . Let pC,˚q denote a E 1 -monoidal dg category such that the multiplication functor˚: C b C Ñ C preserves compact objects and the monoidal unit is compact, and F a monoidal endofunctor.
‚ The naive or vertical trace (or Hochschild homology) trpC, F q " HHpC, F q of the underlying dg category C, described in Section 2.1.1, has the additional structure of an associative (or E 1 -)algebra pHHpCq,˚q by functoriality. It is a decategorification assigning a chain complex to a (k-linear 8-)category. This chain complex has an S 1 -action, i.e. has the structure of a mixed complex. ‚ The 2-categorical or horizontal trace 13 (or categorical Hochschild homology) assigns to an E 1 -symmetric monoidal category pC,˚q and a monoidal endofunctor F the dg category When F " id C is the identity functor, we often omit it from the notation and write TrpC,˚q. This is the tautological receptacle for characters of C-module categories. This category carries an S 1 -action along with a universal (2-categorical) class map r´s : C Ñ TrppC,˚q, F q.
In particular, the regular representation C itself defines an object rCs P TrpC,˚q, i.e. TrpC,˚q is a pointed (or E 0 -)category. For further discussion, see Section 5 of [BFN10] and Sections 3.7 and 3.8 of [GKRV20].
Moreover, the categorical trace provides a "delooping" of the naive trace. To make this relation precise, we first recall the notion of a rigid monoidal category (see Definition 9.1.2 in [GR17] and Lemma 9.1.5).
Definition 3.2. Let A be a compactly generated stable monoidal 8-category, with multiplication µ : A b A Ñ A. We say A is rigid if the monoidal unit is compact, µ preserves compact objects, and if every compact object of A admits a left and right (monoidal) dual.
The following is Theorem 3.8.5 of [GKRV20].
Proposition 3.3. Assume that C is compactly generated and rigid. Then, HHpC, F q " End TrpC,F q prCsq.
Furthermore, if rC, F s is a compact object, then this functor restricts to compact objects pHHpC, F q,˚q -perf ãÑ TrpF, C,˚qq ω .
3.1.1. Traces in geometric settings. The geometric avatar for Hochschild homology is the derived loop space, see [BN19, BN12] for extended discussions. Recall for a derived stack X its loop space LX is defined to be the derived mapping stack from the circle, or more concretely the derived self-intersection of the diagonal LX " MappS 1 , Xq » XˆXˆX X.
For example for X a scheme we have LX » T X r´1s the total space of the shifted tangent complex to X, while for X " pt {G we have LX " G{G » Loc G pS 1 q. For a general stack the loop space is a combination of the shifted tangent complex with the inertia stack. Note the parallel between the loop space, which is the self-intersection of the diagonal (the identity self-correspondence from X) and Hochschild homology (the trace of the identity on a category). As a result the push-pull functoriality of categories of sheaves under correspondences implies an immediate relation between their Hochschild homology and loop spaces. Since QC is functorial under˚-pullbacks and QC ! under !-pullbacks, this produces the following answers, both of which hold in particular for QCA geometric stacks (see Corollary 4.2.2 of [DG13] and also [BN19]): In other words the Hochschild homology of QCpXq (respectively QC ! pXq) is given by functions (respectively volume forms) on the derived loop space. For X " SpecpRq an affine scheme this recovers the Hochschild-Kostant-Rosenberg identification of Hochschild homology of R-mod with differentials on R, HHpR-modq " OpLXq " OpT X r´1sq " Sym ‚ pTRr1sq " Ω´‚pRq.
More generally, if f : X Ñ X is a self-map, then we have HHpQCpXq, f˚q » ΓpX f , O X f q and HHpQC ! pXq, f ! q » ΓpX f , ω X f q where X f are the derived fixed points of the self-map.
Example 3.4 (Quasicoherent sheaves under tensor product). Let X be a perfect stack in the sense of [BFN10]. Then, QCpXq has a monoidal structure via tensor product of sheaves. We have that HHpQCpXqq " OpLXq, which is an algebra object via the shuffle product, and the universal trace QCpXq Ñ TrpQCpXqq " QCpLXq given by pullback along evaluation at the identity. Furthermore, the monoidal unit is O X P QCpXq with trace rO X s " O LX P QCpLXq. Finally, we have OpLXq -mod » xO LX y Ă QCpLXq where the fully faithful inclusion is an equivalence if X is affine.

Convolution patterns in Hochschild homology.
Convolution patterns in Borel-Moore homology and algebraic K-theory play a central role in the results of [CG97]. We now describe a similar pattern which appears in Hochschild homology.
We will work with the following general setup: ‚ f : X Ñ Y is a proper morphism of smooth stacks, and Z " XˆY X. In this setup, the category QC ! pZq carries a monoidal structure under convolution 14 , and thanks to the smoothness of X (hence finite Tor-dimension of the diagonal of X) and the properness of f , this structure preserves CohpZq. Indeed by Theorem 1.1.3 in [BNP17a], there is an equivalence of monoidal categories pCohpXˆY Xq,˚q » pFun ex PerfpY q pCohpXq, CohpXqq,˝q. Moreover, we will argue in Theorem 3.10 that pQC ! pZq,˚q is rigid monoidal. The monoidal unit is the dualizing sheaf of the relative diagonal ι : X Ñ XˆY X, ω ∆ :" ι˚ω X .
Recall (Section 2.1.1) that Hochschild homology of CohpZq (or equivalently of its large variant QC ! pZq by Remark 2.2.11 of [Ch20a]) for a stack Z is given geometrically by volume forms on the loop space HHpCohpZqq » Γ LZ pω LZ q. Thus the vertical trace of the monoidal category CohpZq defines an algebra structure on Γpω LZ q.
We want to relate this convolution structure on sheaves to a decategorified version, involving volume forms on the corresponding loop spaces. Thus we consider the loop map Lf : LX Ñ LY to f , whose self-fiber product is LZ » LX » LY LX. Note that Lf is itself a proper map of quasismooth derived stacks (see [AG14] for the notion of quasi-smoothness and related results). In particular, ω LX is coherent (a compact object in QC ! pLXq) and Lf˚preserves coherence. We thus define our main object of interest: Definition 3.5. The coherent Springer sheaf is defined to be S :" pLf q˚ω LX » pLf q˚O LX P CohpLY q.
The latter isomorphism follows since the loop space of smooth stacks are naturally Calabi-Yau, which we establish in the following lemma.
Lemma 3.6. Let X be a smooth geometric stack (i.e. an Artin stack with affine diagonal). There is a canonical equivalence ω LX » O LX .
Proof. Let p : LX Ñ X be the evaluation map and ∆ : X Ñ XˆX the diagonal. Both are affine by assumption. We will produce a map p˚O X Ñ p ! ω X , which we will then show is an equivalence. Equivalently, we need to produce a map O X Ñ p˚p ! ω X . The diagonal ∆ is quasi-smooth, and we have a natural equivalence ∆ ! p´q " ∆˚p´q b X ω´1 X where ω X is the (shifted) dualizing bundle. Thus, applying base change and the projection formula, we find that p˚p ! ω X » ∆˚∆˚O X » p˚p˚O X . We define the map O X Ñ p˚p ! ω X » p˚p˚O X to be the unit map of the adjunction. Since p is affine, a map of sheaves on LX is an equivalence if and only it is after application of p˚, so the map p˚O X Ñ p ! ω X is an equivalence as claimed.
Remark 3.7 (Convolution of volume forms and endomorphisms of S.). If we sheafify over LY , we can identify this algebra structure more concretely as convolution of volume forms on LZ: LZ " LXˆL Y LX has the structure of proper monoid in stacks over LY , from which one deduces the structure of algebra object in pQC ! pLY q, b ! q on the pushforward of ω LZ . One can also use proper descent for Lf : LX Ñ LY to identify this sheaf of algebras with the internal endomorphism sheaf of S -an analog, in the setting of derived categories of coherent sheaves on derived stacks, of the standard proof (see e.g. [CG97]) that self-Ext of the Springer sheaf is identified with Borel-Moore homology of Z. It would be interesting to see how these arguments globalize over LY to give the isomorphism Γpω LZ q » EndpSq of Theorem 3.10.
3.2.1. Horizontal trace of Hecke categories. Now recall that Proposition 3.3 identifies the vertical trace HHpCohpZqq,˚q as the endomorphisms of the distinguished object in the horizontal trace HHpCohpZq,˚q, the Hochschild homology category of the monoidal category CohpZq, under the assumption that this object is compact. To apply this we need to use the description of this horizontal trace from [BNP17b]. We will use the following correspondence: 15 Note that the below theorem refers to a certain singular support locus Λ X{Y ; it will not appear in our story and we refer the reader to [AG14] [BNP17b] for details. Note that the surjectivity condition is not needed; it is subsumed by the singular support condition. TrpQC ! pZq,˚q » QC ! Λ X{Y pLY q, with the universal trace given by T r " π˚δ ! : QC ! pXˆY Xq Ñ QC ! Λ X{Y pLY q.
Next we identify the coherent Springer sheaf as the trace of the unit (which is a compact object of the trace category): Lemma 3.9. There is a natural equivalence π˚δ ! ω ∆ » S " Lf˚ω LX in CohpLY q.
Proof. The calculation of δ ! ω ∆ " δ ! ι˚ω X arises via base change along the diagram LX XˆXˆY X X XˆY X ι and the statement follows.
We now deduce the main structural relation behind this paper 15 The intermediate term can be thought of as a path in X whose endpoints lie in the same fiber over Y .
Theorem 3.10. Let f : X Ñ Y be as above. The vertical trace of the Hecke category pCohpZq,˚q is identified as an algebra with the endomorphisms of the coherent Springer sheaf, compatibly with the natural S 1 -actions (from the cyclic trace and loop rotation, respectively).
Proof. To apply Theorem 3.8.5 in [GKRV20] we need to verify that QC ! pZq is rigid monoidal. Standard arguments show that integral transforms arising via coherent sheaves preserve compact objects; this statement is also contained within Theorem 1.1.3 in [BNP17a]; one further immediately observes that the monoidal unit ∆˚ω X is a compact object, i.e. coherent, since the diagonal is a closed embedding. It remains to verify that the right and left duals of coherent sheaves K P QC ! pZq are again coherent. Using loc. cit., it suffices to show that the right and left adjoints of the corresponding integral transform F K : QCpXq Ñ QCpXq preserve compact objects. We note that since the projection maps p : Z Ñ X are quasi-smooth, the functors p ! and p˚differ by a shifted line bundle. By Lemma 3.0.8 in op. cit. we can consider equivalently either the˚or !-transforms up to twisting by Grothendieck duality. For convenience we will consider the˚-transform. To see the claim, note that we can write the˚-integral transform F K as a composition: p˚´bK pW e claim that the right adjoint preserves compact objects. The claim for the left adjoint follows similarly by replacing p˚with a twist of p ! with a shifted line bundle. The right adjoints define a sequence of functors We now justify the claim that´b K _ is the right adjoint to´b K, which uses the fact that K is a compact object in ind-coehrent sheaves. It is a standard fact that the functor´b K : QCpZq Ñ QCpZq has right adjoint given by Hom Z pK,´q. We renormalize this adjunction as follows. Let F P QCpZq and G " colim G i P QC ! pZq where G i P CohpZq. Then, we have since K is compact in QC ! pZq, Now, by the usual adjunction in QC, we have Finally, we verify that K _ is coherent, which establishes rigidity. Note that the Grothendieck dual DpKq " Hom Z pK, ω Z q is coherent. Since Z is quasi-smooth, ω Z is a line bundle, so we have DpKq " Hom Z pK, O Z q b Z ω Z " K _ b Z ω Z , and K _ is coherent.

3.2.2.
Application to the affine Hecke algebra. We now specialize the discussion of Section 3.2 to our Springer theory setting. We take X " r N { r G, Y " g{ r G and the Springer resolution 16 . We note the following convenient presentation of the stacks Lp r N { r Gq and LpN { r Gq. 16 We can also restrict attention to the formal neighborhood of the nilpotent cone in g, which we will abusively denote N as well -the singular nilpotent cone itself will not play a role.
Remark 3.11. We realize LpN { r Gq as the formal completion of Lpg{ r Gq Ñ g{ r G over the nilpotent cone. By Proposition 2.1.8 of [Ch20a], we can write Lpg{ r Gq as the pullback here the bottom right map is given by subtraction in g, a is the action map, p the projection, and ∆ the diagonal. Explicitly, the map gˆr G Ñ g is given by px, g, qq Þ Ñ q´1Ad g pxq´x. There is a similar description for Lp r N { r Gq " Lpn{ r Bq.
Our main result is the following combination of Theorems 2.24 and 3.10: Theorem 3.12. The dg algebra of endomorphisms of the coherent Springer sheaf is concentrated in degree zero and is identified with the affine Hecke algebra, Gq pSq » H. In particular, S generates a full embedding, the Deligne-Langlands functor: Remark 3.13. We comment briefly on the absence of a singular support condition. There are two versions of the unipotent Steinberg variety leading to two versions of the unipotent affine Hecke algebra: our version Z " r Nˆg r N and another Z^which is the completion of the Lie algebra Steinberg r gˆg r g along the nilpotent elements. Theorem 4.4.1 of [BNP17b] shows the trace TrpCohpZ^{ r Gqq has a nilpotent singular support condition. We now argue that this singular support condition does not appear for TrpCohpZ{ r Gqq. As in the proof of Theorem 4.4.1 of [BNP17b], we find that the singular locus of LpN { r Gq at a pair η " pn, z " pg, qqq where gng´1 " qn is the set Gqq η " tpv, tq P g ' k | Ad g pvq " v, ad n pvq " tvu.
A calculation shows that the singular support locus is given by: Note that since n is nilpotent, we must have t " 0. The argument in loc. cit. (see also the remark after Proposition 3.1.1 in [Gi12], i.e. for any t P k, n, v generate a solvable Lie subalgebra) establishes that for any nilpotent n and v P g which commute, there exists a Borel containing both. This gives the claim. We note that it is the Lie algebra of the Borel b that appears in the above condition rather than its nilradical n since Example 3.14. The Deligne-Langlands functor is not expected to be an equivalence before applying the Tate construction, even for GL n where we do not expect cuspidal parameters. Taking G " GL 1 , the category H -mod has a compact generator, whereas CohpLpN { r Gqq contains a factor of CohpBGL 1 q and therefore does not. Put another way, CohpLpN { r Gqq S 1 is not a constant u-deformation but the subcategory generated by the Springer sheaf is. A very computable toy example where this occurs is CohpLpBT qq S 1 (see Example 4.1.4 in [Ch20a]).
It is natural to conjecture that the coherent Springer sheaf is in fact a sheaf -i.e., lives in the heart of the dg category CohpLpg{ r Gqq. We prove this in the case G " GL 2 , SL 2 in Proposition 3.27. Remark 3.16. A variant of conjecture 3.15 was answered in the affirmative in Corollary 4.4.6 of [Gi12]. Namely, in loc. cit. it is proven that the Lie algebra version of our coherent Springer sheaf without the q-deformation has vanishing higher cohomology.
Remark 3.17. When r G acts on r N by finitely many orbits, then Lp r N { r Gq has trivial derived structure, and the conjecture is implied by the vanishing of higher cohomology of a classical scheme The G-orbits in the Springer resolution are known to be finite exactly in types A 1 , A 2 , A 3 , A 4 , B 2 by [Kas90].
We discuss the relation of the Deligne-Langlands correspondence and t-structures in more detail in Section 4.3.
3.3. Specializing q. For number theory applications, we will be interested in specializing at q a prime power. There are the algebraic specializations of the affine Hecke algebra, which has no derived structure since H is flat over krz, z´1s.
A potentially different algebra arises when specializing geometrically, i.e. taking endomorphisms of a q-specialized Springer sheaf. We introduce the following unmixed version of the affine Hecke algebra, which is obtained by taking G-equivariant endomorphisms of the Springer sheaf without taking G m -invariants. Recall that LpN { r Gq parametrizes triples pn, g, qq P gˆGˆG m (with n in the formal neighborhood of the nilpotent cone) satisfying gng´1 " qn, up to the action of r G " GˆG m . The base-changed version LpN { r GqˆB Gm pt parametrizes the same triples but only up to the action of G.
Definition 3.19. We define the unmixed affine Hecke algebra and its specialization by GqˆB Gm pt pSq, H un q :" H un b L krz,z´1s krz, z´1s{xz´qy. The algebra H un has the additional structure of a G m -representation, i.e. a weight grading 17 . Note the weight 0 part of H un is H, and that H un 1 " HHpCohpZ{Gqq " kW a rhr´1s ' hr´2ss (which differs from H 1 " kW a ).
Remark 3.20. Note that Conjecture 3.15 would imply in particular that H un is coconnective.
We now identify two corollaries of Theorem 2.24 when we specialize at q P G m . Note that there are two versions of q-specialization. In one version we specialize to the constant loops BG m Ă LpBG m q, i.e. retaining G m -equivariance; in this case, the resulting Springer subcategory is identified with H q -modules, i.e. the usual Iwahori-Hecke algebra from the representation theory of p-adic groups (where the field has residue field F q ). In another version, we specialize along the map pt Ñ LpBG m q, i.e. we forget the G m -equivariance; in this case the category is naturally identified with H un q -modules. Definition 3.21. We will pull the coherent Springer sheaf back along a base change of the mapsq : We will only state definitions for the second version, but all definitions will make sense for the first, which we denote by replacing q withq. We denote the "inclusion" 18 of q-specialized loops by ι q : L q pN { r Gq Ñ LpN { r Gq, and the coherent q-Springer sheaf by S q :" ιq S (sometimes denoted S q˚) . We also define a !-variant, which we denote S q! :" ι ! q S. Note that since ι q is 17 The grading differs from the cohomological grading of graded Hecke algebras; in particular these algebras are not Lusztig's graded Hecke algebras. 18 Note that ιq is a closed immersion, but ιq is not since it involves a factor which is the smooth map pt Ñ BGm.
quasi-smooth 19 , S q! and S q˚d iffer by tensoring with a shifted line bundle (see Proposition 7.4.3 of [AG14]).
Proposition 3.22. The coherent q-Springer sheaf S q is an object of CohpL q pN { r Gqq. The same is true for the !-versions and theq-versions.
Gq denote the q-specialization of the Springer resolution. Note that the mapι q : Gq Ñ LpG m q is quasi-smooth since it is absolutely quasi-smooth and LpG m q is smooth; therefore L q p r N { r Gq Ñ tqu is quasi-smooth since quasi-smoothness is preserved by derived base change. Therefore, we can apply Proposition 7.2.2(c) of [AG14]. In particular, Gq is perfect (and coherent, since Lp r N { r Gq is quasismooth, since r N { r G is smooth), so its singular support lies in the zero section, and the support condition of loc. cit. is satisfied. Finally, proper pushforward along Lµ q preserves coherence.
We first deal with the q-specialization where the G m -action is forgotten. In this version, by Koszul duality (see Section 4.3), the Springer block affine character sheaves at semisimple parameter s are strongly G s -equivariant D-modules on ps, qq-fixed points. We point out that H un q » H q when q ‰ 1. where DL q is fully faithful and identifies the free module with S q , i.e. has essential image xS q y.
There is also a version of the above diagram with a functor DL q! with essential image xS q! y.
Proof. Consider the forgetful functor for the natural map of algebras induced by applying Hom LpN { r Gq pS,´q to the unit S Ñ ι q,˚ιq S " S q and applying the pιq ι q,˚q adjunction. By Proposition 3.22, S q is a compact object. In particular, Hom LpN  This proves the claim. The commuting of the diagram follows by taking right adjoints. The claim regarding DL q! follows since S q! differs from S q by a shifted line bundle, and so they have the same endomorphism algebra. Finally, note that by base change, we have an identification Gq ; and that the map L q µ : L q p r N { r Gq Ñ L q pN { r Gq is obtained via Lµ by 19 In particular, it is a composition of a base change along ιq : tqu ãÑ Gm, which is quasi-smooth, and pt Ñ BGm, which is smooth.
taking derived q-fixed points, i.e. for any r G-scheme X we have a natural identification L q pX{ r Gq X{G X{G X{GˆX{G Γq ∆ Thus, by Proposition 5.3 of [BN19], H un q " HHpCohpZ{Gq, q˚q, where q : Z{G Ñ Z{G is the scaling by q-map for fixed q P G m . In particular, due to Corollary 2.28, there is a natural isomorphism of algebras H un q » H q when q ‰ 1. We also consider the case where we specialize at q P G m but the G m -action is remembered. We omit the proof since it is essentially contained in the previous one. In this version, the Springer block affine character sheaves at parameters are D-modules which have the additional requirement that they are weakly G m -equivariant (see Remark 4.26).
Proposition 3.24. We have a natural equivalence End LqpN { r Gq pSqq » H q . Further, there is a commuting diagram of functors where DLq is fully faithful and identifies the free module with Sq, i.e. has essential image xSqy.
There is also a version of the above diagram with a functor DLq ! with essential image xSq ! y.
Finally, we record the following mild generalization and direct consequence of Proposition 4.2 in [H20] and Proposition 2.1 in [He20]. We will not use it, but find it of interest. It was also proven for q a prime power in Proposition 3.1.5 of [Z20].
Proposition 3.25. If q is not a root of unity, then L q pN { r Gq is a classical stack, i.e. has trivial derived structure and is supported at the nilpotent cone.
Proof. Note that the description in Remark 3.11 has a similar version for the q-specialized loop space. Recall that N is defined to be the formal neighborhood of the nilpotent cone in g{ r G. We will argue that L q pg{ r Gq is set-theoretically supported over the nilpotent cone (i.e. taking formal completions does nothing); vanishing of derived structure then follows by loc. cit. and in view of Remark 2.2(b) of op. cit.. The map g Ñ h{{W is GˆG m -equivariant, with G m acting on h{{W by differing weights and G acting trivially. If q is not a root of unity, then the q-fixed points of h{{W are just zero, so the pg, qq-fixed points of g lie in the nilpotent cone for any g P G.
Remark 3.26. It is necessary to exclude roots of unity; when G " SL 2 , the weight of h{{W is 2, so the argument fails for q "˘1. When G " SL 2 , the weights of h{{W are 2 and 3, so the argument fails for q "˘1 and any cubic root of unity.
3.4. The case G " SL 2 . Let G " SL 2 . Since r G acts on both N and r N by finitely many orbits, the derived loop spaces LpN { r Gq and Lp r N { r Gq are classical stacks. Recall that N is a formal completion; if the reader would rather do so, they may replace N with g, which is also acted on by finitely many orbits. We prove Conjecture 3.15 for G " SL 2 .
Proof. We give a proof for G " SL 2 ; the case of G " GL 2 is the same. In view of Remark 3.17, it suffices to forget equivariance and show vanishing of higher cohomology. Since X :" Lp r N { r GqˆB r G pt is a closed subscheme of gˆG{BˆG, and dimpG{Bq " 1, we know that RΓ i pX,´q " 0 for i ą 1. To verify vanishing for i " 1, let i : X ãÑ r Nˆr G be the closed immersion. We have a short exact sequence of sheaves: leading to a long exact sequence with vanishing H 2 terms (for the above reason). Thus, it suffices to show that H 1 p r Nˆr G, O Ă Nˆr G q. By the projection formula, we have H 1 p r Example 3.28 (Looped Springer resolution geometry). We describe the geometry of the looped Springer resolution. Though this example is well-known, we reproduce it for the reader's convenience. Let Aps, nq denote the component group of the double stabilizer group, i.e. the component group of tg P G | gng´1 " n, gs " sgu. Let A 1 node " Spec krx, ys{xy denote the affine nodal curve, and the normalization of a scheme by p´q ν .
n Aps, nq q s "ˆλ Remark 3.29. Note that in the above example, the component groups all become trivial if we take the stabilizer inside r G rather than G. In particular, by the Koszul duality described in Section 4.2, it is necessary to specialize at tqu or tqu{G m (rather than complete) to obtain a parameter space that can see unipotent cuspidal representations outside of the category generated by the coherent Springer sheaf (see [Lu95]).

The coherent Springer sheaf at parameters
Completing or specializing the coherent Springer sheaf at semisimple parameters recovers classical Springer sheaves in the constructible or D-module context. This process happens in two steps: first we apply an equivariant localization pattern described in [Ch20a] to pass between the stack of unipotent Langlands parameters LpN { r Gq to a completed or specialized version at a semisimple parameter z " ps, qq, and second we apply a Koszul duality equivalence of categories between S 1 -equivariant sheaves at this parameter and a certain category of filtered D-modules. All results in this section take place over an algebraically closed field k of characteristic 0. 4.1. Equivariant localization of derived loop spaces. We now describe equivariant localization patterns in derived loop spaces. See Section 3 of [Ch20a] for an extended discussion, as well as Section 2 of op. cit. and Section 4 of [BN12] for a discussion of derived loop spaces. We fix a reductive group G (over an algebraically closed field k of characteristic zero). Let LpBGq " G{G ÝÑ G{{G denote the "characteristic polynomial" map from the quotient stack of G by conjugation to the affine quotient, i.e., to the variety parametrizing semisimple conjugacy classes. Let z P G be a semisimple element with centralizer G z . We denote by O z » BG z Ă G{G its equivariant conjugacy class and rzs P G{{G its class in the affine quotient.
For a G-variety X we have the maps The (left) loop map parametrizes fixed points of elements of G -i.e., for g P G the fiber of LpX{Gq X g LpX{Gq tgu G{G over g : pt Ñ G{G is the derived fixed point scheme X g , i.e. the derived fiber product Γg ∆ This allows us to define variants of the fixed points according to the Jordan decomposition in G. In particular we are interested in loops whose semisimple part 20 is conjugate to z.
Definition 4.1. The z-unipotent loop space of X, denoted L u z pX{Gq, is the completion of LpX{Gq along the inverse image of the saturation rzs P G{{G. The z-formal loop space p L z pX{Gq is the completion of LpX{Gq along the orbit O z and the z-specialized loop space L 1 z pX{Gq is the (derived) fiber of LpX{Gq over O z .
We will state the equivariant localization theorem of [Ch20a], which is a form of Jordan decomposition for loops, describing loops in the quotient stack X{G with given semisimple part z in terms of unipotent loops on the quotient stack X z {G z of a slight modification of the z-fixed points of the classical (underived) fixed points by the centralizer of z (using a natural map X z {G z ãÑ X{G z Ñ X{G). We now describe this modification X z in the setting of complete intersections.
Definition 4.2. Let z P G be a semisimple closed point. Recall that the classical z-fixed points of a G-variety can be expressed as the underlying classical scheme π 0 pX z q of the derived fixed points.
(1) A G-variety X is said to be a G-complete intersection if X is given as a fiber product X » YˆZ W in the category of G-varieties, with Y, Z and W smooth.
(2) The modified z-fixed points X z for a G-complete intersection is the (derived) fiber product of the classical fixed points X z :" π 0 pY z qˆπ 0 pZ z q π 0 pW z q.
In particular we have (derived) G z -equivariant containments We consider X z with its induced structure as a G z variety with a trivialized 21 action of z.
Remark 4.3. Note that for X a smooth G-scheme, we have that X z " π 0 pX z q is smooth. For X quasismooth, we have that X z is quasismooth, and in particular may have nontrivial derived structure.
Remark 4.4. As a consequence of the next theorem and the fact that formal loop spaces commute with fiber products, one can recover the derived fixed points as the derived loop space of the modified fixed points X z » LpX z q.
20 Note that the preimage of rzs P G{{G in G{G is the closed substack of group elements whose semisimple part in the Jordan decomposition is conjugate to z. 21 A z-trivialization of a G-scheme Y is a G 1 :" G{Zz-action on Y along with an identification Y {G » Y {G 1ˆB G 1 BG. These choices are canonical if Y is a classical scheme; since the X z we consider are built functorially from classical ones, there will always be a canonical choice which we suppress throughout the exposition.
Theorem 4.5 (Equivariant localization for derived loop spaces). For X a G-complete intersection, the unipotent z-localization map u z : L u z pX z {G z q Ñ L u z pX{Gq is an S 1 -equivariant equivalence.
Proof. This is Theorem A in [Ch20a], along with the observation that derived loop spaces commute with fiber products. For a precise definition of the unipotent z-localization map, see Definition 3.1.6 of [Ch20a].
Remark 4.6. Note that it follows that the corresponding localization maps on formal and specialized loopsˆ z pX{Gq are also equivalences in this setting.
We will also need the following version of localization where we specialize at one component and complete at another. We omit the proof as it is a straightforward combination of the above.
Definition 4.7. Let GˆH be a reductive group acting on a smooth scheme X. Let pg, hq P GˆH be semisimple. Define Proposition 4.8. In the above set-up, there is an S 1 -equivariant equivalence of derived stacks Proof. The only issue is the question of S 1 -equivariance; this is discussed in the next subsection and is resolved by Proposition 4.14.
4.1.1. Central shifting. Let Z " ZpHq be the center of a group prestack H. For any H-space Y the action of Z on Y commutes with the action of H, hence defines an action on the quotient Y {H, which we denote by shifting Passing to loop spaces, the shifting action identifies 22 the fiber of LpY {Hq over 1 and over z.
For example in the setting of Theorem 4.5, taking Y " X z and H " G z with its central element z P ZpG z q, we get equivalences of stacks by shifting by z. The left identification is however not S 1 -equivariant for the loop rotation; we need to twist the loop rotation on one side.
Definition 4.9. We have a group structure on the classifying stack BZ of the center and a group homomorphism BZ Ñ AutpBHq induced by the trivialization of the conjugation action of Z on H. In particular fixing z P Z we obtain a twisting by z action of S 1 " BZ on BH, which we denote σpzq. This structure generalizes to H-spaces Y that are equipped with a trivialization of the action of z (extending the case Y " pt above). Namely, the trivialization of the z-action produces a lift of the twisting S 1 -action on Y {H Ñ BH which we also denote by σpzq.
Remark 4.10. Letting H 1 " H{Zz, the twisting S 1 -action on Y {H can also be described using the identification Y {H » Y {H 1ˆB H 1 BH and noting that the fiber product diagram is S 1equivariant, where we let S 1 act trivially on Y {H 1 and BH 1 , and via the z-twisting S 1 -action on BH.
We can combine the twisting and shifting S 1 -actions as follows. Note that the loops to the ztwisting action Lpσpzqq naturally commutes with the loop rotation S 1 -action on LpBHq » H{H, which we denote ρ.
Definition 4.11. We define ρpzq to be the diagonal to the S 1ˆS1 -action ρˆLpσpzqq.
Thus we have the following Jordan decomposition result: shifting by z intertwines ρ with the twisted version ρpzq " ρ˝Lpσpzqq.
Corollary 4.12. For X a G-complete intersection, the shifted localization map defines an equivalence s u z : L u pX z {G z q L u z pX{Gq » which is S 1 -equivariant with respect to ρpzq on the source and ρ on the target, and likewise for the shifts of the completed and specialized localization maps s ẑ and s 1 z . 4.1.2. Neutral blocks. In order to apply Koszul duality (as in Section 5 of [BN12]), we are interested in identifying a subcategory of various categories of sheaves on derived loop spaces over semisimple parameter z on which the z-twisting is trivial, so that the twisted rotation is equal to the untwisted rotation. This is useful since the ρ circle action on unipotent loop spaces factors through an action of BG a , but the twisted ρpzq action does not (since it has nontrivial semisimple part). This problem is an obstacle to applying the Koszul duality described in [BN12] to obtain an identification of Cohp p L z pX{Gqq S 1 with some kind of category of D-modules. We avoid this obstacle by focusing only on the z-trivial block. For this we give a categorical interpretation of the geometric z-twisting S 1 -action σpzq discussed above.
Definition 4.13. Let H be an affine algebraic group, z P H central and C a category over ReppHq. A z-trivialization of C is an identification of the action of z on C with the identity functor. 23 The category C H of equivariant objects then acquires an automorphism of the identity functor (i.e., S 1 -action) as the ratio of the z-trivialization and the equivariance structure for z. We define the subcategory C H z Ă C H of z-trivial objects to be the full subcategory on which this automorphism is trivial, i.e., on which the equivariance agrees with the z-trivialization.
We can apply this categorical notion to the categories of sheaves Perf, Coh, QC, and QC ! on a scheme Y with trivialization of the z-action. In particular the z-twisting action on the z-trivial subcategory of equivariant sheaves in each case is trivial. Further, all sheaves on the z-specialized loop space are z-trivial, so z-triviality is only relevant for Koszul duality for stacks.
Proof. It is more or less immediate to see that the z-twisting action σpzq acts trivially on the z-trivial block of any ReppGq-category C. Furthermore, we observe that there is a canonical identification Lpσ X{G pzqq " σ LpX{Gq pzq, and the claim follows. To see that z-triviality is an empty condition on the specialized loop space, note that the twisting σpzq acts trivially on the identity e P LpBGq, and therefore trivially on the base change teuˆL pBGq LpX{Gq.
We can see via examples that z-triviality is not an empty condition for formal loop spaces.
Example 4.15. Consider Example 4.1.6 from [Ch20a], i.e. take the z-twisted loop rotation action on LpBT q " TˆBT . Let Λ be the character lattice of T , so that OpT q is spanned by t λ for λ P Λ. We have PerfpLpBT qqq " à λPΛ PerfpT q b ReppT q and therefore where PreMF is defined in [P11]. The z-trivial subcategory corresponds to the subcategory of ReppT q of representations on which z P T acts trivially, i.e. if T 1 " T {Zz, then Perfp p LpBT qq ρpzq z " à λPΛ PreMFp p T , 1´t λ pzqt λ q b ReppT 1 q.
Theorem 4.25 (Koszul duality for loop spaces of quotient stacks). Let X{G be a smooth Remark 4.27. We observe that in the case of the coherent Springer sheaf, the equivalence of Theorem 2.24 is an equivalence before taking S 1 -equivariant objects. Thus, in our setting we may actually use the easier graded Koszul duality which corresponds on the D-modules side to passing to the associated graded of a filtered D-module. However, we discuss the full theory for completeness.
The following example of a category of filtered D-modules is parallel to Example 4.16.
Example 4.28. Take G " G m and X " pt and fix an isomorphism C ‚ pG m ; kq » kr s; then the category FD ω pXq splits as a direct sum by isotypic component of the underlying G mrepresentation: FD ω pXq " à nPZ FD ω pXq n .
We have that FD ω pXq n " krt, ts -coh with |t| " 0 and | t| "´1 (in particular, p tq 2 " 0) and internal differential dp tq " nt. When n " 0, this dg algebra is a graded version of the usual shifted dual numbers kr ts, and when n ‰ 0, it is quasi-isomorphic to k, i.e.
Note that if we forget the filtration, only the trivial isotypic summand survives.
4.3. The coherent Springer sheaf at parameters. We can now construct a variety of localization functors between the category of unipotent Langlands parameters QC ! V pLpN { r Gqq and categories of D-modules. Note there are variants of the statements where we complete in one factor and specialize at another, as well as weakly equivariant variants; we will leave their statements to the interested reader.
We begin in a general setting, considering subcategories of the category Coh Λ pLpX{Gqq S 1 generated by a sheaf xSy satisfying a z-triviality condition. Since Koszul duality requires us to consider an additional G m -equivariant structure, we will need to choose a graded lift of the z-localization of S. In general there may be many choices, and choices cannot always be made globally.
Before we proceed, let us review our notation conventions. The sheaf S P CohpLpX{Gqq S 1 is an S 1 -equivariant sheaf on the global derived loop space. For semisimple parameters z, we define its z-completion by Spp zq and its z-specialization by Spzq. We denote graded lifts by r Spp zq and r Spzq. The corresponding filtered complex of D-modules under Koszul duality are denoted r Spp zq and r Spzq. Forgetting the filtration, we obtain D-modules Spp zq and Spzq In the following, for : "^, 1 , we let f : z : L : pX z {G z q Ñ L : z pX{Gq Ñ LpX{Gq be the composition of the "shift by z" map with the equivariant localization map; f : z is S 1 -equivariant where S 1 acts via ρpzq on the source and ρ on the target. We let the undecorated f z : X z {G z Ñ LpX{Gq be the pre-composition with the inclusion of constant loops. Let Ć BG a " BG a¸Gm , let Ć Tate be the G m -equivariant Tate construction, and recall the notation from Definition 4.17. Let C be a category over ReppG m q and F : C Ñ C 1 the forgetful functor; a graded lift of an object X P C 1 is an object r X P C along with an equivalence F p r Xq » X.
Proposition 4.29. Let X{G be a smooth quasiprojective quotient stack by a reductive group, and Λ Ă SingpLpX{Gqq a singular support condition, with restriction Λ z " f ! z Λ. Let S P Coh Λ pLpX{Gqq S 1 be such that Spp zq :" p f ! z S is in the z-trivial block and choose a graded lift r Spp zq. Then, there is a commuting diagram xSy xSpp zqy x r Spp zqy xSpp zqy Remark 4.30. The functor Coh Λ pLpX{Gqq S 1 Ñ KPerf Λz p p L z pX{Gqq S 1 z is only well-defined on the subcategory xSy due to the z-triviality requirement.
There is a version for specialization as well, which we state; it has the additional feature that there is a functor from the category of D-modules to the category of coherent sheaves on the derived loop space. Recall that the z-triviality condition vanishes under specialization via Proposition 4.14, and that for smooth schemes X, KPerfpT X r´1sq " CohpT X r´1sq.
Proposition 4.31. Let X{G be a smooth quasiprojective quotient stack by a reductive group, and Λ Ă SingpLpX{Gqq a singular support condition, with restriction Λ z " f ! z Λ. Let S P Coh Λ pLpX{Gqq S 1 and choose a graded lift r Spzq. Then, there is a commuting diagram xSy xSpzqy x r Spzqy xSpzqy We now consider a more specific context where the sheaf S is of geometric origin: let µ : r X Ñ X be a G-equivariant proper map of smooth G-schemes, and let S :" Lµ˚O Lp Ă X{Gq " Lµ˚ω Lp Ă X{G . We first verify the z-triviality condition required in the above results.
Lemma 4.32. Let z P G be semisimple. The sheaf Spp zq is z-trivial for every semisimple z P r G (and likewise for Spzq).
Proof. Follows by equivariant locaization and base change, i.e. Spp zq is the pushforward of ω p Lp Ă X z {G z q , which is z-trivial since z acts trivially on r X z and z is central in G z .
In adddition to z-triviality being automatic in this setting, there is a canonical choice for graded lifts when S " Lµ˚O LpX{Gq .
Definition 4.33. Let S " Lµ˚O LpX{Gq . For any z P G semisimple, there is a geometric (or However, note that the cohomological grading on H is Tate sheared; H p pX, Ω q X q is in cohomological degree p´q rather than p`q (and in weight´q). In particular, if in addition, X has Tate type (i.e. the above groups vanish unless p " q), then H is an algebra concentrated in degree 0. In this setting, the free object H P pH -modq ♥ in the heart of the standard t-structure on H -mod, but corresponds to the object O LX P CohpLXq which is not in the heart of the standard t-structure on CohpLXq. Passing through Koszul duality and using the geometric graded lift, O LX corresponds to the D-module O X equipped with the usual order filtration; note that O X P DpXq ♥ is in the heart on the Koszul dual side.

Blocks, semisimple types, and affine Hecke algebras
We now turn to arithmetic applications of our results. Let F be a p-adic field, with residue field F q , and let G _ denote a connected, split, reductive group over F . We henceforth assume that G is the Langlands dual group to G _ . Then the derived category DpG _ q of smooth complex representations of G _ pF q admits a decomposition into blocks. The so-called principal block of DpG _ q (that is, the block containing the trivial representation) is naturally equivalent to the category of H q -modules, where H q now denotes the affine Hecke algebra associated to G _ , with parameter q. Proposition 3.23 then gives a fully faithful embedding from this principal block Gq has a natural interpretation in terms of Langlands parameters for G _ pF q. Recall that a Langlands parameter for G _ is a pair pρ, N q, where ρ : W F Ñ GpCq is a homomorphism with open kernel, and N is a nilpotent element of Lie G such that, for all σ in the inertia group I K of W K , one has AdpρpFr n σqqpN q " q n N, where Fr denotes a Frobenius element of W F . On the other hand, the underlying stack of L q can be regarded as the moduli stack of pairs ps, N q, where s P GpCq, N P Lie G, and AdpsqpN q " qN , up to G-conjugacy. To such a pair we can attach the Langlands parameter pρ, N q, where ρ is the unramified representation of W F taking Fr to s. Such a Langlands parameter is called unipotent, and this construction identifies Gq with the moduli stack of unipotent Langlands parameters, modulo G-conjugacy. 27 We thus obtain a fully faithful embedding from the principal block of DpG _ q into the category of ind-coherent sheaves on the moduli stack of unipotent Langlands parameters. It is natural to ask if this extends to an embedding of all of DpG _ q into a category of sheaves on the moduli stack of all Langlands parameters. We will show that, at least when G _ " GL n over F , this is indeed the case.
Our argument proceeds by reducing to the principal block. On the representation theory side, this reduction is a consequence of the Bushnell-Kutzko theory of types and covers, which we now recall.
5.1. Supercuspidal support. Let P _ be a parabolic subgroup of G _ " GL n , with Levi M _ and unipotent radical U _ , and let π be a smooth complex representation of M _ . Recall that the parabolic induction i G _ P _ π is obtained by inflating π to a representation of P _ , twisting by the square root of the modulus character of P _ , and inducing to G _ . The parabolic induction functor i G _ P _ has a natural left adjoint, the parabolic restriction r P _ G _ . Definition 5.1. A complex representation π of G _ is supercuspidal if, for all proper parabolic subgroups P _ of G _ , the parabolic restriction r P _ G _ π vanishes. Definition 5.2. Let π be an irreducible supercuspidal representation of M _ . An irreducible complex representation Π has supercuspidal support pM _ , πq if Π is isomorphic to a subquotient of i G _ P _ π. Given Π, the pair pM _ , πq is well-defined up to conjugacy. 27 Strictly speaking, a Langlands parameter is a pair pρ, N q as above in which ρ is semisimple. When building a moduli space of Langlands parameters we must drop this condition, however, as the space of semisimple parameters is not a well-behaved geometric object. In particular the locus in Lq consisting of pairs ps, N q in which s is semisimple is neither closed nor open in Lq.
Let M _ 0 be the smallest subgroup of M _ containing every compact open subgroup; then M _ {M _ 0 is free abelian, of rank equal to the dimension of the center of M _ . Definition 5.4. The pairs pM _ , πq and pL _ , π 1 q are inertially equivalent if there exists an element g of G _ such that pM _ q g " L _ and π g " π 1 b χ for some unramified character χ of L _ .
Fix a pair pM _ , πq, with π an irreducible supercuspidal representation of M _ . Following Bernstein-Deligne [BD84], we let DpG _ q rM _ ,πs denote the full subcategory of DpG _ q whose objects Π have the property that every irreducible subquotient of Π has supercuspidal support inertially equivalent to pM _ , πq. Then Bernstein-Deligne show: Theorem 5.5. The full subcategory DpG _ q rM _ ,πs is a block of DpG _ q.

Types and Hecke algebras.
Recall that a type for G _ is a pair pK, τ q, where K is a compact open subgroup of G _ and τ is an irreducible complex representation of U . Attached to a type we have its Hecke algebra The main result of [BK99] describes an arbitrary block of DpG _ q as a category of modules for a certain tensor product of Hecke algebras, via the theory of G _ -covers.
Let L _ be a Levi subgroup of G _ and let π be a supercuspidal representation of L _ . Let H be the subgroup of HompL _ {L _ 0 , Cˆq consisting of unramified characters χ such that π b χ is isomorphic to π. Then the irreducibles in DpL _ q rL _ ,πs are in bijection with HompL _ {L _ 0 , Cˆq{H. Moreover, there is an equivalence of categories: We may rephrase this equivalence in terms of types and Hecke algebras as follows: first, we may choose a maximal distinguished cuspidal type pK L _ , τ L _ q contained in π. One then has a natural support-preserving isomorphism of HpL _ , K L _ , τ L _ q with CrL _ {L _ 0 s H . Under this isomorphism, the equivalence sends a CrL _ {L _ 0 s H -module to its tensor product, over HpL _ , K L _ , τ L _ q, with the compact induction c-Ind L _ K L _ τ L _ . We then have the following, which is a composite of results of [BK99]: Theorem 5.6. Let rL _ , πs and the cuspidal type pK L _ , τ L _ q be as above, and let P _ be a parabolic subgroup of G _ with Levi L _ . There exists a Levi subgroup pL : q _ of G _ , and types pK : , τ : q of L : _ and pK, τ q of G _ with the following properties: (1) The type pK : , τ : q is a "simple type" of pL : q _ in the sense of [BK99].
(4) Suppose pL : q _ decomposes as a product of direct factors pL : q _ i , with each pL : q _ i isomorphic to GL ni for some n i . Let L _ i be the projection of L to pL : q _ i , and let π i be the projection of π to L _ i . Let H i denote the group of unramified characters χ of pL : q _ i such that π b χ is isomorphic to π, and let r i denote the order of H i . Then n i factors as r i m i , for some positive integer m i , and there is a natural isomorphism (that depends on the choice of π): where H q r i pm i q denotes the affine Hecke algebra associated to GL mi with parameter q ri .
These constructions are naturally compatible with parabolic induction, in the following sense: let M _ be a Levi of G _ containing L _ , and let Q _ denote the parabolic M _ P _ . Then Theorem 5.6 gives us an M _ -cover pK M _ , τ M _ q of pK L _ , τ L _ q and a G _ -cover pK, τ q of pK L _ , τ L _ q, as well as maps:

We then have:
Theorem 5.7. There exists a unique map: Moreover, for any V P DpM _ q, we have an isomorphism of HpG _ , K, τ q-modules: The fundamental (and motivating) example for this is when L _ is the standard maximal torus T _ of G _ , the parabolic P _ is the standard Borel of G _ , and π is the trivial character of T _ . In this setting K L _ is the maximal compact subgroup T _ 0 of T _ , and τ L _ is the trivial character. Moreover pL : q _ " G _ , the subgroup K of G _ is the Iwahori subgroup, and τ is the trivial representation of K. The Hecke algebra HpL _ , K L _ , 1q is then naturally isomorphic to CrT _ {T _ 0 s, and if X ‚ denotes the cocharacter group of T _ , we may identify this with CrX ‚ s. We then have a commutative diagram: Ó Ó H q -HpG _ , K, 1q in which the left-hand vertical map is the standard inclusion of CrX ‚ s in H q , and the right-hand vertical map is T P _ .
More generally, if M _ is a Levi subgroup of G _ and Q _ is the standard parabolic with Levi M _ , then K _ M is the Iwahori subgroup K X M _ of M _ , and the map is uniquely determined by the following properties: KwK. This picture is compatible with the general situation in the following sense. Suppose for simplicity that pL : q _ " G _ . Then L _ is a product of m copies of GL n m for some divisor m of n, and (after an unramified twist) we may assume that π has the form π bm 0 . There is an extension E{F of degree n m and ramification index r, and an embedding of G _ E " GL m pEq in G _ , such that the intersection of L _ with G _ E is the standard maximal torus T _ E of G _ E . Let X ‚ denote the cocharacter group of this torus, and let M _ E be the intersection of M _ with G _ E . The choice of π then gives rise to an isomorphism of CrX ‚ s with HpL _ , K L _ , τ L _ q, such that for each coharacter λ the image of λ is supported on the double coset K L _ λp E qK L _ , and such that the induced action of X ‚ on the Hecke module attached to π is trivial.
We then have: Theorem 5.8. The isomorphism of H q r pmq with HpG _ , K, τ q fits into a commutative diagram: where G _ E " GL m pEq, I E is the standard Iwahori subgroup of G _ E , and we identify T _ E with the standard maximal torus of G _ E . Thus when rL _ , πs is "simple" (that is, when M _ " G _ ), we have a natural reduction of DpG _ q rL _ ,πs to the principal block of DpG _ E q, in a manner compatible with parabolic induction. In general we obtain a reduction of DpG _ q rL _ ,πs to a tensor product of such principal blocks.

Moduli of Langlands parameters for GL n
We now turn to the study of moduli spaces of Langlands parameters. These have been studied extensively in mixed characteristic, for instance in [H20] in the case of GL n , or more recently in [BG19,BP19], and [DHKM20] for more general groups. Since in our present context we work over C, the results we need are in general simpler than the results of the above papers, and have not appeared explicitly in the literature in the form we need.
6.1. The moduli stacks X ν F,G . We first consider these moduli spaces as underived stacks; we will later consider certain derived structures on them. As in the previous section, we take G " GL n , considered as the Langlands dual of G _ " GL n pF q.
Let I be an open normal subgroup of I F . Then there is a scheme X I F,G parameterizing pairs pρ, N q, where ρ : W F {I Ñ GL n is a homomorphism, and N is a nilpotent n by n matrix such that for all σ P I F , Ad ρpFr n σqpN q " q n N . For any ν : I F {I Ñ GL n pCq, we may consider the subscheme X ν F,G of X I F,G corresponding to pairs pρ, N q such that the restriction of ρ to I F is conjugate to ν; it is easy to see that X ν F,G is both open and closed in X I F,G . We will say that a Langlands parameter is of "type ν" if it lies in X ν F,G . Note that when ν " 1 is the trivial representation, the quotient stack X 1 F,G {G is isomorphic to the underlying underived stack of L q pN { r Gq, as we remarked in the previous section. We will show that in fact, for ν arbitrary, the stack X ν F,G {G is isomorphic (again as underived stacks) to a product of stacks of the form L q r i pN i { r G i q, in a manner that exactly parallels the type-theoretic reductions of the previous section. This will allow us to transfer the structures we have built up on L q r i pN i { r G i q to stacks of the form X ν F,G {G for arbitrary ν. Our approach very closely parallels the construction of [H20], sections 7 and 8, with the exception that we are able to work with the full inertia group I F , whereas the integral -adic setting of [H20] requires one to work with the prime-to-inertia instead.
For any irreducible complex representation η of I F , let W η be the finite index subgroup of W F consisting of all w P W F such that η w is isomorphic to η. Then η extends to a representation of W η , although not uniquely. We denote by E η the fixed field of W η , and by r η the degree of E η over F . Also let d η denote the dimension of η.
Fix, for each W F -orbit of irreducible representations of I F {I, a representative η of that orbit and an extensionη of η to a representation of W η . This choice of extension defines, for any C-algebra R, and any ρ : W F {I Ñ GL n pRq, a natural action of W η {I F on the space Hom I F pη, ρq, such that the injection:η b Hom I F pη, ρq Ñ ρ is W η -equivariant. Frobenius reciprocity then gives an injection: Ind W F Wη pη b Hom I F pη, ρqq Ñ ρ. The image of this injection is the sum of the I F -subrepresentations of ρ isomorphic to a W Fconjugate of η. We thus have a direct sum decomposition: where τ runs over a set of representatives for the W F -orbits of irreducible representations of I F {I. Moreover, the map N is I F -equivariant, and thus induces, for each η, a nilpotent endomorphism N η of Hom I F pη, ρq. If Fr η is a Frobenius element of W η , we have Fr η N η Fr´1 η " q rη N η . Let n η pρq be the dimension of the space Hom I F pη, ρq. (Since n η pρq only depends on the type ν of ρ, we may also write this as n η pνq.) A choice of R-basis for Hom I F pη, ρq then gives a homomorphism: ρ η : W F {I F Ñ GL ni pRq and realizes N η as a nilpotent element of M ni pRq such that pρ η , N η q is an R-point of X 1 Eη,GL nη pρq . We thus define: Definition 6.1. A pseudo-framing of a Langlands parameter pρ, N q over R is a choice, for all η such that n η pρq is nonzero, of an R-basis for Hom I F pη, ρq.
LetX ν F,G be the moduli scheme parameterizing parameters pρ, N q of type ν together with a pseudo-framing, and let G ν be the product, over all η such that n η pνq is nonzero, of the groups GL nη . Then G ν acts onX ν F,G via "change of pseudo-framing", and this action makesX ν F,G into a G ν -torsor over X ν F,G . On the other hand, given an R-point pρ, N q ofX ν F,G , the pseudo-framing gives, for each η, an R-point pρ η , N η q of X 1 Eη,GL nη pνq . We thus obtain a natural map: Eη,GL nη pνq and the conjugation action of G onX ν F,G makesX ν F,G into G-torsor over the product on the right hand side.
We thus obtain natural isomorphisms of quotient stacks: Note that the composite isomorphism depends on the choice, for each η, of an extensionη of η to W F . 6.2. The ν-Springer sheaves. As remarked above, we now have natural identifications, for each η, of the stack X 1 Eη,GL nη pνq { GL nηpνq with L q rη pN nηpνq { r G nηpνq q, where r G nηpνq is the group GL nηpνqˆGm and N nηpνq is the nilpotent cone of GL nηpνq . This gives a natural derived structure on X 1 Eη,GL nη pνq { GL nηpνq , and we can transfer these derived structures across the isomorphism: Eη,GL nη pνq¸{ G ν of the previous subsection. Similarly, the product, over η, of the sheaves S q rη on the moduli stack X 1 Eη,GL nη pνq { GL nηpνq gives rise to a sheaf on X ν F,G {G that we denote by S ν ; we call this the Springer sheaf of type ν or the ν-Springer sheaf.
The endomorphisms of the ν-Springer sheaf are a tensor product Â η H q rη pn η pνqq; we henceforth denote this Hecke algebra by H ν . We thus obtain a fully faithful embedding of the category of H ν -modules in the category of quasi-coherent sheaves on X ν F,G {G. However, since our identifications depend, ultimately, on our choices ofη, this embedding will also depend on these choices. (By contrast, the sheaf S ν itself is, at least up to isomorphism, independent of the choices ofη.) We can remove this dependence by rephrasing this embedding in terms of smooth representations of G _ , via the type theory of the previous section.
Let L _ ν be the standard Levi of G _ corresponding to block diagonal matrices whose blocks consist, for each η, of n η pνq blocks of size r η d η . Let π 0 η be the cuspidal representation of GL rηdη corresponding to Ind W F Wηη under the local Langlands correspondence, and let π ν be the cuspidal representation: π ν :" â η pπ 0 η q bnηpνq of L _ ν . Then representations in the block DpG _ q rL _ ν ,πν s correspond, via local Langlands, to Langlands parameters for G of type ν.
For each η, we can find a cuspidal type pK η , τ η q in GL rηdη for π 0 η . From this we can form the type pK Lν , τ Lν q in L _ ν , by setting K Lν " ś η K nηpνq η and τ Lν " Â η τ bnηpνq η . This type is associated to the block rL _ ν , π ν s in DpL _ ν q. Let P _ be the standard parabolic of G _ with Levi L _ , and let pP 1 q _ denote the opposite parabloc. The theory of section 5 then gives us a Levi subgroup pL : q _ of G _ containing L _ ν , an pL : q _ -cover pK : ν , τ : ν q of pK Lν , τ Lν q, and a G _ -cover pK ν , τ ν q of pK : ν , τ : ν q. These covers depend on a choice of parabolic with Levi L _ ; we choose our covers to be the ones associated to the opposite parabolic pP 1 q _ . In particular we obtain a map T pP 1 q _ : HpL _ ν , K L _ ν , τ L _ ν q Ñ HpG _ , K ν , τ ν q that is compatible with the parabolic induction functor i G _ P _ on DpL _ ν q in the sense of Theorem 5.6.
One verifies, by compatibility of local Langlands with unramified twists, that for each η the group of unramified characters χ of GL rηdη such that π 0 η bχ is isomorphic to π 0 η is r η . Thus there is an isomorphism of the Hecke algebra HpG _ , K ν , τ ν q with H ν . Moreover, the composition: is independent of the choices ofη. (This essentially boils down to the compatibility of the local Langlands correspondence with unramified twists and parabolic induction.) Since DpG _ q rL _ ν ,πν s is canonically equivalent to the category of HpG _ , K ν , τ ν q-modules, and this equivalence associates the representations c-Ind G _ Kν τ ν to the free HpG _ , K ν , τ ν q-module of rank one, we have shown: Theorem 6.2. For each ν there is a natural fully faithful functor: LL G,ν : DpG _ q rL _ ν ,πν s ãÑ QC ! pX ν F,G q that takes the generator c-Ind G _ Kν τ ν to S ν . 6.3. A direct construction of S ν . In this section we give a more intrinsic construction of S ν . Fix a particular ν, and let L denote the Langlands dual of L _ ν ; we identify L with the standard block diagonal Levi of G containing n η pνq blocks of size r η d η , ordered according to some fixed ordering of the η appearing in ν. Let ν 1 : I F Ñ L be the representation of I F on L whose projection to each block of L of type η is the sum of the W F -conjugates of η. We then have a moduli space X ν 1 F,L parameterizing Langlands parameters pρ, N q for L that are of type ν 1 . Let P be the standard (block upper triangular) parabolic of G containing L ν . We then also have a moduli space X ν 1 F,P parameterizing Langlands parameters pρ, N q for G that factor through P , and whose projection to L ν is of type ν 1 . The inclusion of P in G, and the projection of P onto L induce maps: ι P : X ν 1 F,P {P Ñ X ν F,G {G and π P : X ν 1 F,P {P Ñ X ν 1 F,L {L. We then have: Theorem 6.3. There are natural isomorphisms: S ν -pι P q˚O ν 1 F,P -pι P q˚πP O ν 1 F,L , where O ν 1 F,P and O ν 1 F,L denote the structure sheaves on X ν 1 F,P {P and X ν 1 F,L {L, respectively.
Proof. Let L : be the standard Levi of G that is block diagonal of block sizes n η pνqr η d η (embedded in G using the same ordering of the η that was used to define the embedding of L, so that all the blocks of L corresponding to η are mapped to the single block of L : corresponding to η). Let Q be the standard block upper triangular parabolic of G with Levi L : , and let ν 2 be the composition of ν 1 with the inclusion of L in L : . We then have spaces X ν 2 F,L : and X ν 2 F,Q , where the former parameterizes pairs pρ, N q for L : that are of type ν 2 , and the latter parameterizes pairs pρ, N q for G that factor through Q and whose projection to L : is of type ν 2 . We may also consider the space X ν 1 F,P XL : , which parameterizes pairs pρ, N q for L : that factor through P X L : and whose projection to L is of type ν 1 . We then have a natural cartesian diagram: X ν 1 F,P {P Ñ X ν 1 F,P XL : {P X L : Ó Ó X ν 2 F,Q {Q Ñ X ν 2 F,L : {L : from which we conclude that pι P q˚πP O ν 1 F,L is isomorphic to pι Q q˚πQpιP XL : q˚π P XL : O ν 1 F,L , where π Q , ι Q are the maps from X ν 2 F,Q {Q to X ν F,G {G and X ν 2 F,L : {L : , respectively, and π P XL : , ι P XL : are the maps from X ν 2 F,P XL : {pP X L : q to X ν 1 F,L {L and X ν 2 F,L : {L : , respectively. On the other hand, let B η and T η denote the standard Borel subgroup and maximal torus of GL nηpνq , for each η. We then have a commutative diagram: where the bottom two vertical maps on the left are the identity. It follows that the iterated pull-push pι Q q˚πQpι P XL : q˚πP XL : O ν 1 F,L corresponds, under the bottom isomorphism, to S ν , as the latter is simply the pushforward to ś η L q rη pN nηpνq {G nηpνq q of the structure sheaf on ś η L q rη pN Bη {Bq.
6.4. Compatibility with parabolic induction. As in the previous subsection, we fix a particular ν and let L _ ν , L and P be as above. Let Q be a standard Levi subgroup of G whose standard Levi subgroup M contains L _ ν , and let M _ and Q _ be the corresponding dual subgroups of G _ . Let ν 1 be the inertial type I F Ñ L ν constructed in the previous subsection, and let ν 2 be the composition of ν 1 with the inclusion of L ν in M .
We have a diagram: F,G {G in which the square is cartesian, the left-hand horizontal maps are π P XM and π Q , and the lefthand vertical maps are ι P XM and ι Q . Denote the upper right horizontal map by π P,P XM and the upper right vertical map by ι P,Q . Theorem 6.3 shows that S ν is isomorphic to the pushforward to X ν F,G {G of the structure sheaf on X ν 1 F,P {P , and the corresponding sheaf S ν,M on X ν 2 F,M is the pushforward to X ν 2 F,M {M of the structure sheaf on X ν 1 F,P XM {pP X M q. The above diagram then gives us a natural isomorphism: S ν -pι Q q˚πQS ν,M .
Via functoriality and this isomorphism one obtains an embedding of EndpS ν,M q in EndpS ν q.
Recall that we have identified these endomorphism rings with certain Hecke algebras via type theory. In particular, we have the type pK Lν , τ Lν q of L _ ν , an M _ -cover pK M _ , τ M _ q coming from the parabolic pP 1 q _ X M _ opposite P _ X M _ , and a G _ -cover pK, τ q coming from the parabolic pP 1 q _ opposite P _ . Theorem 5.7 then gives us a map: T pQ 1 q _ : HpM _ , K M _ , τ M _ q Ñ HpG _ , K, τ q. Lemma 6.4. We have a commutative diagram: HpM _ , K M _ , τ M _ q -EndpS ν,M q Ó Ó HpG _ , K, τ q -EndpS ν q in which the left-hand vertical map is T pQ 1 q _ and the right hand map is induced by the isomorphism of S ν with pι Q q˚πQS ν,M .
Proof. The machinery of the previous subsection, together with the compatibility of the general case with the Iwahori case in section 5 allow us to reduce to the case where ν " 1. In this case the claim reduces to the compatibility of the Ginsburg-Kazhdan-Lusztig interpretation of the affine Hecke algebra as K 0 of the Steinberg variety with parabolic induction, and may be checked directly.
As a consequence, we deduce: Theorem 6.5. We have a commutative diagram of functors: DpM _ q rLν ,τν s Ñ QC ! pX ν F,M q Ó Ó DpG _ q rLν ,τν s Ñ QC ! pX ν F,G q in which the horizontal functors are the fully faithful functors LL M,ν and LL G,ν , the left-hand functor is parabolic induction i G _ Q _ , and the right-hand-functor is pι Q q˚πQ. Proof. We have isomorphisms: from which the result follows.