Construction of central elements in the affine Hecke algebra via nearby cycles
type: Item
instance of: scholarly article
title: Construction of central elements in the affine Hecke algebra via nearby cycles
zbMATH Open document ID: 1072.14055
author: Dennis Gaitsgory
published in: Inventiones Mathematicae
publication date: 2002-08-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9912074
review text: Let \(G\) be a connected reductive group over a finite field \(\mathbb{F}_q\). The corresponding group \(G(K)\) over the local field \(K= \mathbb{F}_q((t))\) contains \(G({\mathcal O})\) as a maximal compact subgroup, where \({\mathcal O}:= \mathbb{F}_qt\), and the latter group gives rise to the Hecke algebra \(H\) of compactly supported bi-\(G({\mathcal O})\)-invariant functions \(f: G(K)\to\overline{\mathbb{Q}}_\ell\), equipped with the usual convolution product. It is known that the Hecke algebra \(H\) is commutative, whereas the Hecke algebra \(H_I\) associated with the Iwahori subgroup \(I\subset G({\mathcal O})\) is noncommutative. Moreover, if \(G\) is a split group, then \(H\) is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of the Langlands dual group \(\check{G}\). The existence of such an isomorphism has been established by I. Satake many years ago. On the other hand, a result by \textit{I. N. Bernstein} [in: Représentations des groupes réductifs sur un corps local, 1--32 (1984; Zbl 0599.22016)] states that the representation ring of the Langlands dual group \(\check{G}\) is also isomorphic to the center of the Hecke algebra \(H_I\). Therefore there is a well-defined isomorphism \(\pi: Z(H_I)\widetilde\to H\), the so-called Satake-Bernstein isomorphism, and the main goal of the paper under review is to describe explicitely the inverse of this isomorphism. This is done by giving geometric interpretations of both \(H\) and \(H_I\) in terms of certain perverse sheaves. More precisely, the author considers the quotients \(Gr:= G(K)/G({\mathcal O})\) and \(Fl:= G(K)/I\) as group ind-schemes over \(\mathbb{F}_q\) and studies their associated categories \(P_{G({\mathcal O})}(Gr)\) and \(P_I(Fl)\) of equivariant perverse sheaves (on \(Gr\)) or \(I\)-equivariant perverse sheaves (on \(Fl\)), respectively. The better part of the paper is then devoted to the construction of a functor \(Z: P_{G({\mathcal O})}(Gr)\to P_I…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 1655624
zbMATH Keywords: group schemes
zbMATH Keywords: Hecke algebras
zbMATH Keywords: Grassmannians
zbMATH Keywords: flag manifolds
zbMATH Keywords: preserve sheaves
zbMATH Keywords: reductive groups
zbMATH Keywords: finite ground fields
zbMATH Keywords: local ground fields
MaRDI profile type: Publication
OpenAlex ID: W2021615957
arXiv ID: math/9912074
DOI: 10.1007/S002220100122
description: scientific article; zbMATH DE number 1655624