Invariants, Shimura Variety, and Hecke Algebra

2013 ◽  
pp. 83-144
Author(s):  
Haruzo Hida
Keyword(s):  
Hecke Algebra ◽  
Shimura Variety

scholarly journals Vanishing theorems for the mod p cohomology of some simple Shimura varieties

2020 ◽  
Vol 8 ◽  
Author(s):  
Teruhisa Koshikawa
Keyword(s):  
Additional Assumption ◽  
Hecke Algebra ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Vanishing Theorems ◽  
Low Dimensions ◽  
Eisenstein Ideal ◽  
Mod P

Abstract We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

scholarly journals SUR LA TORSION DANS LA COHOMOLOGIE DES VARIÉTÉS DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR

2017 ◽  
Vol 18 (3) ◽  
pp. 499-517 ◽  
Author(s):  
Pascal Boyer
Keyword(s):  
Maximal Ideal ◽  
Cohomology Class ◽  
Hecke Algebra ◽  
Galois Representation ◽  
Local System ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Torsion Freeness ◽  
Satake Parameters ◽  
Torsion Free

(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) When the level at $l$ of a Shimura variety of Kottwitz–Harris–Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb{Z}}_{l}$-local system isn’t in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak{m}$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak{m}$ itself or on the Galois representation $\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), we prove that the torsion doesn’t give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.

scholarly journals Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 779
Author(s):  
Charles F. Dunkl
Keyword(s):  
Preceding Paper ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Special Points ◽  
Two Parameters

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

REPRESENTATIONS OF THE HECKE ALGEBRA FOR GL4(q)

2005 ◽  
Vol 04 (06) ◽  
pp. 631-644
Author(s):  
KENICHI SHINODA ◽  
ILKNUR TULUNAY
Keyword(s):  
Hecke Algebra ◽  
Standard Basis

In this article, we explicitly calculated the values of the representations of the Hecke algebra [Formula: see text], associated with a Gelfand–Graev character of GL 4(q), at some of the standard basis elements.

scholarly journals Hecke Algebra Actions on the Coinvariant Algebra

2000 ◽  
Vol 233 (2) ◽  
pp. 594-613 ◽  
Author(s):  
Ron M. Adin ◽  
Alexander Postnikov ◽  
Yuval Roichman
Keyword(s):  
Hecke Algebra ◽  
Coinvariant Algebra

Corrigendum to the article On the supersingular locus of the GU(2,2) Shimura variety

2021 ◽  
Vol 15 (1) ◽  
pp. 307-308
Author(s):  
Benjamin Howard ◽  
Georgios Pappas
Keyword(s):  
Shimura Variety

scholarly journals The supersingular locus of the Shimura variety of GU(1,n−1) II

2010 ◽  
Vol 184 (3) ◽  
pp. 591-627 ◽  
Author(s):  
Inken Vollaard ◽  
Torsten Wedhorn
Keyword(s):  
Shimura Variety
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