scholarly journals Central elements in affine mod p Hecke algebras via perverse -sheaves

2021 ◽  
Vol 157 (10) ◽  
pp. 2215-2241
Author(s):  
Robert Cass
Keyword(s):  
Finite Field ◽  
Reductive Group ◽  
Hecke Algebra ◽  
Convolution Product ◽  
Local Models ◽  
Frobenius Splitting ◽  
Rational Singularities ◽  
Nearby Cycles ◽  
Central Elements ◽  
Affine Flag Variety

Abstract Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text {der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb {F}_p$ -sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse $\mathbb {F}_p$ -sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly $F$ -regular, and hence they are $F$ -rational and have pseudo-rational singularities.

scholarly journals Local models for Weil-restricted groups

2016 ◽  
Vol 152 (12) ◽  
pp. 2563-2601 ◽  
Author(s):  
Brandon Levin
Keyword(s):  
Hecke Algebra ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Local Models ◽  
Central Function ◽  
Nearby Cycles ◽  
Weil Restriction ◽  
Trace Of Frobenius ◽  
Invent Math

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

scholarly journals On a conjecture of Pappas and Rapoport about the standard local model for GL_d

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Type A ◽  
Positive Answer ◽  
Local Model ◽  
Schubert Varieties ◽  
Shimura Varieties ◽  
Local Models ◽  
Full Generality ◽  
Frobenius Splitting ◽  
Affine Grassmannians ◽  
Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

Distribution Algebras on p-adic Groups and Lie Algebras

2011 ◽  
Vol 63 (5) ◽  
pp. 1137-1160 ◽  
Author(s):  
Allen Moy
Keyword(s):  
Lie Algebras ◽  
Identity Element ◽  
Hecke Algebra ◽  
Symmetric Bilinear Form ◽  
Convolution Product ◽  
Complex Vector ◽  
Enveloping Algebra ◽  
Invariant Distributions ◽  
Reductive Lie Algebra ◽  
The Family

Abstract When F is a p-adic field, and is the group of F-rational points of a connected algebraic F-group, the complex vector space of compactly supported locally constant distributions on G has a natural convolution product that makes it into a ℂ-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for p-adic groups of the enveloping algebra of a Lie group. However, has drawbacks such as the lack of an identity element, and the process is not a functor. Bernstein introduced an enlargement . The algebra consists of the distributions that are left essentially compact. We show that the process is a functor. If is a morphism of p-adic groups, let be the morphism of ℂ-algebras. We identify the kernel of in terms of Ker. In the setting of p-adic Lie algebras, with g a reductive Lie algebra, m a Levi, and the natural projection, we show that maps G-invariant distributions on to NG(m)-invariant distributions on m. Finally, we exhibit a natural family of G-invariant essentially compact distributions on g associated with a G-invariant non-degenerate symmetric bilinear form on g and in the case of SL(2) show how certain members of the family can be moved to the group.

scholarly journals ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
GEORGE LUSZTIG ◽  
ZHIWEI YUN
Keyword(s):  
Finite Field ◽  
Reductive Group ◽  
The Other ◽  
Torus Action ◽  
Similar Relationship ◽  
Unipotent Representations ◽  
Character Sheaves ◽  
Trivial Monodromy ◽  
Unipotent Character ◽  
Endoscopic Group

For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$ , after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$ .

scholarly journals Duality for representations of a reductive group over a finite field

1982 ◽  
Vol 74 (1) ◽  
pp. 284-291 ◽  
Author(s):  
Pierre Deligne ◽  
George Lusztig
Keyword(s):  
Finite Field ◽  
Reductive Group

scholarly journals On Twisted Gelfand Pairs Through Commutativity of a Hecke Algebra

2019 ◽  
Author(s):  
Yotam I Hendel
Keyword(s):  
Reductive Group ◽  
Hecke Algebra ◽  
Irreducible Representations ◽  
Gelfand Pairs ◽  
Locally Compact ◽  
Totally Disconnected ◽  
Cuspidal Representations

Abstract For a locally compact, totally disconnected group $G$, a subgroup $H$, and a character $\chi :H \to \mathbb{C}^{\times }$ we define a Hecke algebra ${\mathcal{H}}_\chi$ and explore the connection between commutativity of ${\mathcal{H}}_\chi$ and the $\chi$-Gelfand property of $(G,H)$, that is, the property $\dim _{\mathbb{C}} (\rho ^*)^{(H,\chi ^{-1})} \leq 1$ for every $\rho \in \textrm{Irr}(G)$, the irreducible representations of $G$. We show that the conditions of the Gelfand–Kazhdan criterion imply commutativity of ${\mathcal{H}}_\chi$ and verify in several simple cases that commutativity of ${\mathcal{H}}_\chi$ is equivalent to the $\chi$-Gelfand property of $(G,H)$. We then show that if $G$ is a connected reductive group over a $p$-adic field $F$, and $G/H$ is $F$-spherical, then the cuspidal part of ${\mathcal{H}}_\chi$ is commutative if and only if $(G,H)$ satisfies the $\chi$-Gelfand property with respect to all cuspidal representations ${\rho \in \textrm{Irr}(G)}$. We conclude by showing that if $(G,H)$ satisfies the $\chi$-Gelfand property with respect to all irreducible $(H\backslash G,\chi ^{-1})$-tempered representations of $G$ then ${\mathcal{H}}_\chi$ is commutative.

scholarly journals Automorphic vector bundles on the stack of G-zips

2021 ◽  
Vol 9 ◽  
Author(s):  
Naoki Imai ◽  
Jean-Stefan Koskivirta
Keyword(s):  
Vector Bundle ◽  
Finite Field ◽  
Vector Bundles ◽  
Reductive Group ◽  
Levi Subgroup ◽  
Algebraic Subgroup ◽  
Equivalence Of Categories

Abstract For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

scholarly journals Algebraic and analytic Dirac induction for graded affine Hecke algebras

2013 ◽  
Vol 13 (3) ◽  
pp. 447-486 ◽  
Author(s):  
Dan Ciubotaru ◽  
Eric M. Opdam ◽  
Peter E. Trapa
Keyword(s):  
Weyl Group ◽  
Discrete Series ◽  
Hecke Algebras ◽  
Reductive Group ◽  
Dirac Operators ◽  
Hecke Algebra ◽  
Semisimple Lie Groups ◽  
Affine Hecke Algebra ◽  
Affine Hecke Algebras ◽  
Dirac Induction

AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

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