scholarly journals Construction of central elements in the affine Hecke algebra via nearby cycles

2001 ◽  
Vol 144 (2) ◽  
pp. 253-280 ◽  
Author(s):  
D. Gaitsgory
Keyword(s):  
Hecke Algebra ◽  
Affine Hecke Algebra ◽  
Nearby Cycles ◽  
Central Elements

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 the centre of the cyclotomic Hecke algebra of G(m, 1, 2)

2012 ◽  
Vol 55 (2) ◽  
pp. 497-506 ◽  
Author(s):  
Kevin McGerty
Keyword(s):  
Hecke Algebra ◽  
Affine Hecke Algebra ◽  
Cyclotomic Hecke Algebra

AbstractWe compute the centre of the cyclotomic Hecke algebra attached to G(m, 1, 2) and show that if q ≠ 1, it is equal to the image of the centre of the affine Hecke algebra Haff2. We also briefly discuss what is known about the relation between the centre of an arbitrary cyclotomic Hecke algebra and the centre of the affine Hecke algebra of type A.

Duality between GL(n,R), GL(n,Qp), and the degenerate affine Hecke algebra for gl(n)

2012 ◽  
Vol 134 (1) ◽  
pp. 141-170 ◽  
Author(s):  
Dan Ciubotaru ◽  
Peter E. Trapa
Keyword(s):  
Hecke Algebra ◽  
Affine Hecke Algebra

Integrability, Fusion, and Duality in the Elliptic Ruijsenaars Model

1997 ◽  
Vol 12 (11) ◽  
pp. 751-761 ◽  
Author(s):  
Kazuhiro Hikami ◽  
Yasushi Komori
Keyword(s):  
Difference Operator ◽  
Hecke Algebra ◽  
Root Systems ◽  
Baxter Equation ◽  
Reflection Equation ◽  
Fusion Procedure ◽  
Affine Hecke Algebra ◽  
Difference Analog ◽  
Type D ◽  
Ruijsenaars Models

The generalized elliptic Ruijsenaars models, which are regarded as a difference analog of the Calogero–Sutherland–Moser models associated with the classical root systems are studied. The integrability and the duality using the fusion procedure of operator-valued solutions of the Yang–Baxter equation and the reflection equation are shown. In particular a new integrable difference operator of type-D is proposed. The trigonometric models are also considered in terms of the representation of the affine Hecke algebra.

scholarly journals Elliptic Double Affine Hecke Algebras

2020 ◽  
Author(s):  
Eric M. Rains ◽  
Keyword(s):  
Hilbert Scheme ◽  
Hecke Algebra ◽  
Rational Surface ◽  
Symmetric Power ◽  
Affine Hecke Algebra ◽  
Double Affine Hecke Algebra ◽  
Affine Weyl Group ◽  
Double Affine Hecke Algebras ◽  
Hilbert Scheme Of Points ◽  
Affine Hecke Algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C<sub>n</sub> version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

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