Related papers: Macdonald's formula for Kac-Moody groups over loca…
We define the spherical Hecke algebra H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The…
Let G be a split Kac-Moody group over a non-archimedean local field. We define a completion of the Iwahori-Hecke algebra of G. We determine its center and prove that it is isomorphic to the spherical Hecke algebra of G using the Satake…
We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the…
This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper…
Let $G$ be a split Kac-Moody group over a non-Archimedean local field, and let $\mathcal{H}$ be the Iwahori-Hecke algebra of $G$. In this paper, we construct a completed Iwahori-Hecke algebra $\widehat{\mathcal{H}}$ and prove that it…
In this paper we consider the Hecke algebra $\mathcal {H}$ associated to an extended affine Weyl group of type $\widetilde{B_2}$. We give some interesting formulas on $C_{rt}S_{\lambda}$, which imply some relations between the…
Let V be a finite dimensional representation of the connected complex reductive group H. Denote by G the derived subgroup of H and assume that the categorical quotient of V by G is one dimensional. In this situation there exists a…
In this paper we give a geometric construction of Cherednik's double affine Hecke algebra. We construct the algebra as the equivariant $K$-theory of the Lagrangian subvariety of the cotangent variety of the square of the flag variety of…
We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We…
In this paper we give an elementary proof of certain finiteness results about affine Kac-Moody groups over a local non-archimedian field K. Our results imply those proven earlier by Braverman-Kazhdan, Braverman-Finkelberg-Kazhdan and…
The Kac-Moody affine Hecke algebra $\mathcal{H}$ was first constructed as the Iwahori-Hecke algebra of a $p$-adic Kac-Moody group by work of Braverman, Kazhdan, and Patnaik, and by work of Bardy-Panse, Gaussent, and Rousseau. Since…
Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the…
There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [S\'em. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald…
In a previous paper J.-G. Luque and the author (Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the symmetric group $\mathcal{S}_{N}$. The…
Given a reductive group G, Kostant and Kumar defined a nil Hecke algebra that may be viewed as a degenerate version of the double affine nil Hecke algebra introduced by Cherednik. In this paper, we construct an isomorphism of the spherical…
We introduce a path-theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher level generalisations over fields of arbitrary characteristic. Our first main result is a…
In this paper we define a two-variable, generic Hecke algebra, H, for each complex reflection group G(b,1,n). The algebra H specializes to the group algebra of G(b,1,n) and also to an endomorphism algebra of a representation of GL(n,q)…
We prove a version of the Gindikin-Karpelevich formula for untwisted affine Kac-Moody groups over a local field of positive characteristic. The proof is geometric and it is based on the results of [1] about intersection cohomology of…
We consider the Iwahori-Hecke algebra associated to an almost split Kac-Moody group $G$ (affine or not) over a nonarchimedean local field $K$. It has a canonical double-coset basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ indexed by a…
The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication.…