A Tale of Two Hecke Algebras (original) (raw)
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Formulae relating the Bernstein and Iwahori–Matsumoto presentations of an affine Hecke algebra
Journal of Algebra, 2002
We consider the "antidominant" variants Θ − λ of the elements Θ λ occurring in the Bernstein presentation of an affine Hecke algebra H. We find explicit formulae for Θ − λ in terms of the Iwahori-Matsumoto generators Tw (w ranging over the extended affine Weyl group of the root system R), in the case (i) R is arbitrary and λ is a minuscule coweight, or (ii) R is attached to GLn and λ = me k , where e k is a standard basis vector and m ≥ 1.
The Satake isomorphism for special maximal parahoric Hecke algebras, Represent. Theory 14
2010
Let G denote a connected reductive group over a nonarchimedean local field F. Let K denote a special maximal parahoric subgroup of G(F). We establish a Satake isomorphism for the Hecke algebra HK of K-bi-invariant compactly supported functions on G(F). The key ingredient is a Cartan decomposition describing the double coset space K\G(F)/K. As an application we define a transfer homomorphism t : HK * (G *) → HK (G) where G * is the quasi-split inner form of G. We also describe how our results relate to the treatment of Cartier [Car], where K is replaced by a special maximal compact open subgroup e K ⊂ G(F) and where a Satake isomorphism is established for the Hecke algebra H e K .
2003
Our aim here is to give a fairly self-contained exposition of some basic facts about the Iwahori-Hecke algebra H of a split p-adic group G, including Bernstein's presentation and description of the center, Macdonald's formula, the Casselman-Shalika formula, and the Lusztig-Kato formula.
Infinitesimal Hecke Algebras II
2009
For W a finite (2-)reflection group and B its (generalized) braid group, we determine the Zariski closure of the image of B inside the corresponding Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and generated in the group algebra of W by the reflections of W. We determine its decomposition in simple factors. In case W is a Coxeter group, we prove that the representations involved are unitarizable when the parameters of the representations have modulus 1 and are close to 1. We consequently determine the topological closure in this case.
A generic algebra associated to certain Hecke algebras
2004
We initiate the systematic study of endomorphism algebras of permutation modules and show they are obtainable by a descent from a certain generic Hecke algebra, infinite-dimensional in general, coming from the universal enveloping algebra of gl n (or sl n). The endomorphism algebras and the generic algebras are cellular (in the latter case, of profinite type in the sense of R.M. Green). We give several equivalent descriptions of these algebras, find a number of explicit bases, and describe indexing sets for their irreducible representations.
Frobenius map for the centers of Hecke algebras
Transactions of the American Mathematical Society, 2014
We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.
Center and Representations of Infinitesimal Hecke Algebras of 2
Communications in Algebra®, 2010
In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl2; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl n and sp(2n) for all n. We end by proving an analogue of Duflo's theorem for Hz.
On the structure constants of certain Hecke algebras
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz ON THE STRUCTURE CONSTANTS OF CERTAIN HECKE ALGEBRAS by-Anna Helversen-Pasotto CONTENTS Introduction SI.-Hecke algebras and their structure constants §2.-Examples §3.-The commuting algebra of the Gelfand-Graev representation of the finite group GL(2,F) §4.-Some analogues of Barnes' identity for Gaussian sums over finite fields