Showing papers on "Reductive group published in 2021"

Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale

Abstract: We associate to every irreducible representation of a reductive group over a local field of equal characteristics a local Langlands parameter up to semisimplification and prove the compatibility with the global parameterization constructed by the second author. Our method involves stacks of restricted shtukas (which are analogues of truncated Barsotti-Tate groups) and nearby cycles over arbitrary bases.

Quantitative reduction theory and unlikely intersections

Abstract: We prove quantitative versions of Borel and Harish-Chandra’s theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila–Zannier strategy to the Zilber–Pink conjecture for the moduli space of principally polarised abelian surfaces. Building on our previous paper, we prove this conjecture under a Galois orbits hypothesis. Finally, we establish the Galois orbits hypothesis for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions.

Types for tame p-adic groups

Abstract: Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we show that every smooth irreducible complex representation of G(k) contains an $\mathfrak{s}$-type of the form constructed by Kim and Yu and that every irreducible supercuspidal representation arises from Yu's construction. This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on p. By contrast, our bound on p is explicit and tight, and our result holds in positive characteristic as well. Moreover, our approach is more explicit in extracting an input for Yu's construction from a given representation.

The Grothendieck–Serre conjecture over valuation rings

Abstract: In this article, we establish the Grothendieck–Serre conjecture over valuation rings: for a reductive group scheme G over a valuation ring V with fraction field K, a G-torsor over V is trivial if it is trivial over K. This result is predicted by the original Grothendieck–Serre conjecture and the resolution of singularities. The novelty of our proof lies in overcoming subtleties brought by general nondiscrete valuation rings. By using flasque resolutions and inducting with local cohomology, we prove a non-Noetherian counterpart of Colliot-Thelene–Sansuc's case of tori. Then, taking advantage of techniques in algebraization, we obtain the passage to the Henselian rank one case. Finally, we induct on Levi subgroups and use the integrality of rational points of anisotropic groups to reduce to the semisimple anisotropic case, in which we appeal to properties of parahoric subgroups in Bruhat–Tits theory to conclude. In the last section, by using properties of reflexive sheaves, we also prove a variant of Nisnevich's purity conjecture.

Parabolic eigenvarieties via overconvergent cohomology

Abstract: Let $$\mathcal {G}$$ be a connected reductive group over $$\mathbf {Q}$$ such that $$G = \mathcal {G}/\mathbf {Q}_p$$ is quasi-split, and let $$Q \subset G$$ be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to Q, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When Q is a Borel, we recover the usual theory of overconvergent cohomology, and our classicality theorem gives a stronger slope bound than in the existing literature. We use our theory to construct Q-parabolic eigenvarieties, which parametrise p-adic families of systems of Hecke eigenvalues that are finite slope at Q, but that allow infinite slope away from Q.

Nilpotent Elements and Reductive Subgroups Over a Local Field

Abstract: Let ${\mathcal {K}}$ be a local field – i.e. the field of fractions of a complete DVR ${\mathcal {A}}$ whose residue field k has characteristic p > 0 – and let G be a connected, absolutely simple algebraic ${\mathcal {K}}$ -group G which splits over an unramified extension of ${\mathcal {K}}$ . We study the rational nilpotent orbits of G– i.e. the orbits of $G({\mathcal {K}})$ in the nilpotent elements of $\text {Lie}(G)({\mathcal {K}})$ – under the assumption p > 2h − 2 where h is the Coxeter number of G. A reductive group M over ${\mathcal {K}}$ is unramified if there is a reductive model ${{\mathscr{M}}}$ over ${\mathcal {A}}$ for which $M = {{\mathscr{M}}}_{{\mathcal {K}}}$ . Our main result shows for any nilpotent element X1 ∈Lie(G) that there is an unramified, reductive ${\mathcal {K}}$ -subgroup M which contains a maximal torus of G and for which X1 ∈Lie(M) is geometrically distinguished. The proof uses a variation on a result of DeBacker relating the nilpotent orbits of G with the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated with G.

Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

Abstract: We study irreducible odd mod p Galois representations ρ¯:Gal(F‾∕F)→G(F‾p), for F a totally real number field and G a general reductive group. For p≫G,F0, we show that any ρ¯ that lifts locally, and at places above p to de Rham and Hodge–Tate regular representations, has a geometric p-adic lift. We also prove non-geometric lifting results without any oddness assumption.

Total Positivity in Springer Fibres

Abstract: Let u be a unipotent element in the totally positive part of a complex reductive group We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold

On the structure of some $p$-adic period domains

Abstract: We study the geometry of the p-adic analogues of the complex analytic period spaces first introduced by Griffiths. More precisely, we prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter µ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G, µ) is fully Hodge-Newton decomposable.

Tame Tori in p-Adic Groups and Good Semisimple Elements

Abstract: Let G be a reductive group over a non-archimedean local field k. We provide necessary conditions and sufficient conditions for all tori of G to split over a tamely ramified extension of k. We then show the existence of good semisimple elements in every Moy-Prasad filtration coset of the group G(k) and its Lie algebra, assuming the above sufficient conditions are met.

On Parabolic Restriction of Perverse Sheaves

Abstract: We prove exactness of parabolic restriction and induction functors for conjugation equivariant sheaves on a reductive group generalizing a well known result of Lusztig who established this property for character sheaves. We propose a conjectural (but known for character sheaves) t-exactness property of the Harish-Chandra transform and provide an evidence for that conjecture. We also present two applications generalizing some results of Gabber and Loeser on perverse sheaves on an algebraic torus to an arbitrary reductive group.

Associated varieties for real reductive groups

Abstract: We give an algorithm to compute the associated variety of a Harish- Chandra module for a real reductive group $G({\mathbb R})$. The algorithm is implemented in the atlas software package.

Cohomological representations of parahoric subgroups

Abstract: We generalize a cohomological construction of representations due to Lusztig from the hyperspecial case to arbitrary parahoric subgroups of a reductive group over a local field, which splits over an unramified extension. We compute the character of these representations on certain very regular elements.

Automorphic L-invariants for reductive groups

Abstract: Let $G$ be a reductive group over a number field $F$, which is split at a finite place $\mathfrak{p}$ of $F$, and let $\pi$ be a cuspidal automorphic representation of $G$, which is cohomological with respect to the trivial coefficient system and Steinberg at $\mathfrak{p}$. We use the cohomology of $\mathfrak{p}$-arithmetic subgroups of $G$ to attach automorphic $\mathcal{L}$-invariants to $\pi$. This generalizes a construction of Darmon (respectively Spies), who considered the case $G=GL_2$ over the rationals (respectively over a totally real number field). These $\mathcal{L}$-invariants depend a priori on a choice of degree of cohomology, in which the representation $\pi$ occurs. We show that they are independent of this choice provided that the $\pi$-isotypical part of cohomology is cyclic over Venkatesh's derived Hecke algebra. Further, we show that automorphic $\mathcal{L}$-invariants can be detected by completed cohomology. Combined with a local-global compatibility result of Ding it follows that for certain representations of definite unitary groups the automorphic $\mathcal{L}$-invariants are equal to the Fontaine-Mazur $\mathcal{L}$-invariants of the associated Galois representation.

Minimal Newton Strata in Iwahori Double Cosets

Abstract: The set of Newton strata in a given Iwahori double coset in the loop group of a reductive group G is indexed by a finite subset of the set B(G) of Frobenius-conjugacy classes. For unramified $G$ we show that it has a unique minimal element and determine this element. Under a regularity assumption we also compute the dimension of the corresponding Newton stratum. We derive corresponding results for affine Deligne-Lusztig varieties.

Annihilator varieties of distinguished modules of reductive Lie algebras

Abstract: We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let be a complex algebraic reductive group and be a spherical algebraic subgroup. Let denote the Lie algebras of and , and let denote the orthogonal complement to in . A -module is called -distinguished if it admits a nonzero -invariant functional. We show that the maximal -orbit in the annihilator variety of any irreducible -distinguished -module intersects . This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

Cordial elements and dimensions of affine Deligne–Lusztig varieties

Abstract: The affine Deligne–Lusztig variety in the affine flag variety of a reductive group depends on two parameters: the -conjugacy class and the element w in the Iwahori–Weyl group of . In this paper, for any given -conjugacy class , we determine the nonemptiness pattern and the dimension formula of for most .

Linearly Reductive Quotient Singularities

Abstract: We study isolated quotient singularities by finite and linearly reductive group schemes (lrq singularities for short) and show that they satisfy many, but not all, of the known properties of finite quotient singularities in characteristic zero: (1) From the lrq singularity we can recover the group scheme and the quotient presentation. (2) We establish canonical lifts to characteristic zero, which leads to a bijection between lrq singularities and certain characteristic zero counterparts. (3) We classify subgroup schemes of ${\mathbf{GL}}_d$ and ${\mathbf{SL}}_d$ that correspond to lrq singularities. For $d=2$, this generalises results of Klein, Brieskorn, and Hashimoto. Also, our classification is closely related to the spherical space form problem. (4) F-regular (resp. F-regular and Gorenstein) surface singularities are precisely the lrq singularities by finite and linearly reductive subgroup schemes of ${\mathbf{GL}}_2$ (resp. ${\mathbf{SL}}_2$). This generalises results of Klein and Du Val. (5) Lrq singularities in dimension $\geq 4$ are infinitesimally rigid. We classify lrq singularities in dimension $3$ that are not infinitesimally rigid and compute their deformation spaces. This generalises Schlessinger's rigidity theorem to positive and mixed characteristic. Finally, we study Riemenschneider's conjecture in this context, that is, whether lrq singularities deform to lrq singularities.

Parity sheaves and Smith theory

Abstract: Let $p$ be a prime number and let $X$ be a complex algebraic variety with an action of $\mathbb{Z}/p\mathbb{Z}$. We develop the theory of parity complexes in a certain $2$-periodic localization of the equivariant constructible derived category $D^b_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_p)$. Under certain assumptions, we use this to define a functor from the category of parity sheaves on $X$ to the category of parity sheaves on the fixed-point locus $X^{\mathbb{Z}/p\mathbb{Z}}$. This may be thought of as a categorification of Smith theory. When $X$ is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.

Partial Hasse invariants for Shimura varieties of Hodge-type

Abstract: For a connected reductive group $G$ over a finite field, we define partial Hasse invariants on the stack of $G$-zip flags. We obtain similar sections on the flag space of Shimura varieties of Hodge-type. They are mod $p$ automorphic forms which cut out a single codimension one stratum. We study their properties and show that such invariants admit a natural factorization through higher rank automorphic vector bundles. We define the socle of an automorphic vector bundle, and show that partial Hasse invariants lie in this socle.

A Hecke algebra isomorphism over close local fields

Abstract: Let $G$ be a split connected reductive group over $\mathbb{Z}$. Let $F$ be a non-archimedean local field. With $K_m: = Ker(G(\mathfrak{O}_F) \rightarrow G(\mathfrak{O}_F/\mathfrak{p}_F^m))$, Kazhdan proved that for a field $F'$sufficiently close local field to $F$, the Hecke algebras $\mathcal{H}(G(F),K_m)$ and $\mathcal{H}(G(F'),K_m')$ are isomorphic, where $K_m'$ denotes the corresponding object over $F'$. In this article, we generalize this result to general connected reductive groups.

On the 1-Euler Characteristic of the Variety of Maximal Tori in a Reductive Group

Abstract: We show that for a reductive group $G$ over a field $k$ the $\\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in $G$ is an invertible element of the Grothendieck–Witt ring ${\\textrm{GW}}(k)$, settling the weak form of a conjecture by Fabien Morel. As an application we obtain a generalized splitting principle that allows one to reduce the structure group of a Nisnevich locally trivial $G$-torsor to the normalizer of a maximal torus.

Paley-wiener Theorems for a $p$-adic Spherical Variety

Abstract: Let SpXq be the Schwartz space of compactly supported smooth functions on the p-adic points of a spherical variety X , and let C pXq be the topological space of Harish-Chandra Schwartz functions. Under assumptions on the spherical variety, which are satisfied at least when it is symmetric, we prove Paley-Wiener theorems for the two spaces, characterizing them in terms of their spectral transforms. As a corollary, we get a relative analog of the (smooth or tempered) Bernstein center – rings of multipliers for SpXq and C pXq. When X “ a reductive group, our theorem for C pXq specializes to the well-known theorem of Harish-Chandra, and our theorem for SpXq corresponds to a first step – enough to recover the structure of the Bernstein center – towards the well-known theorem of Bernstein and Heiermann. CONTENTS

On the Moy-Prasad filtration

Abstract: Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Prasad filtration representations, which extend earlier results by Reeder and Yu (which required p to be large) and by Romano and the author (which required G to be absolutely simple and split). This yields new supercuspidal representations. We also treat reductive groups G that are not necessarily split over a tamely ramified field extension.

Restricting supercuspidal representations via a restriction of data

Abstract: Let $F$ be a non-archimedean local field of residual characteristic $p$. Let $\mathbb{G}$ be a reductive group defined over $F$ which splits over a tamely ramified extension and set $G=\mathbb{G}(F)$. We assume that $p$ does not divide the order of the Weyl group of $\mathbb{G}$. Given a closed connected $F$-subgroup $\mathbb{H}$ that contains the derived subgroup of $\mathbb{G}$, we study the restriction to $H$ of an irreducible supercuspidal representation $\pi=\pi_G(\Psi)$ of $G$, where $\Psi$ is a $G$-datum as per the J.K. Yu Construction. We provide a full description of $\pi|_H$ into irreducible components, with multiplicity, via a restriction of data which constructs $H$-data from $\Psi$. Analogously, we define a restriction of Kim-Yu types to study the restriction of irreducible representations of $G$ which are not supercuspidal.

Revisiting the moduli space of semistable g-bundles over elliptic curves

Abstract: We show that the moduli space of semistable G-bundles on an elliptic curve for a reductive group G is isomorphic to a power of the elliptic curve modulo a certain Weyl group which depend on the topological type of the bundle. This generalizes a result of Laszlo to arbitrary connected components and recovers the global description of the moduli space due to Friedman-Morgan-Witten and Schweigert. The proof is entirely in the realm of algebraic geometry and works in arbitrary characteristic.

Central extensions by $${\mathbf {K}}_2$$ K 2 and factorization line bundles

Abstract: Let X be a smooth, geometrically connected curve over a perfect field k. Given a connected, reductive group G, we prove that central extensions of G by the sheaf $${\mathbf {K}}_2$$ on the big Zariski site of X, studied in Brylinski–Deligne [5], are equivalent to factorization line bundles on the Beilinson–Drinfeld affine Grassmannian $$\text{ Gr}_G$$ . Our result affirms a conjecture of Gaitsgory–Lysenko [13] and classifies factorization line bundles on $$\text{ Gr}_G$$ .