Related papers: The mirabolic Hecke algebra

Using norms, the second author constructed a basis for the centre of the Hecke algebra of the symmetric group over $\Q[\xi]$ in 1990. An integral "minimal" basis was later given by the first author in 1999, following work of Geck and…

Let $R$ be a commutative ring with identity and let $V$ be a free $R$-module of rank $n$ for some $n\in\mathbb{N}$. Fixing an $R$-basis $\mathcal{E}$ of $V$, the symmetric group $\mathfrak{S}_n$ acts on $V$ by permuting $\mathcal{E}$ and…

We describe the center of the Hecke algebra of a type attached to a Bernstein block under some hypothesis. When $\bf G$ is a connected reductive group over non-archimedean local field $F$ that splits over a tamely ramified extension of $F$…

A general method for computing irreducible representations of Weyl groups and Iwahori-Hecke algebras was introduced earlier by the first author. The representations of the algebras of types $A_n$, $B_n$, $D_n$ and $G_2$ were computed and it…

The Lascoux-Leclerc-Thibon-Ariki theory asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie…

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)…

Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our…

We use Bernstein's presentation of the Iwahori-Matsumoto Hecke algebra to obtain a simple proof of the Satake isomorphism and, in the same stroke, compute the center of the Iwahori-Matsumoto Hecke algebra.

This article gives a fairly self-contained treatment of the basic facts about the Iwahori-Hecke algebra of a split p-adic group, including Bernstein's presentation, Macdonald's formula, the Casselman-Shalika formula, and the Lusztig-Kato…

Consider a connected reductive algebraic group $ G $ and a symmetric subgroup $ K $. Let $ \mathfrak{X} = K/B_K \times G/P $ be a double flag variety of finite type, where $ B_K $ is a Borel subgroup of $ K $, and $ P $ a parabolic subgroup…

We survey some of the known results on the relation between the homology of the {\em full} Hecke algebra of a reductive $p$-adic group $G$, and the representation theory of $G$. Let us denote by $\CIc(G)$ the full Hecke algebra of $G$ and…

The Peter-Weyl idempotent $e_{\mathcal{P}}$ of a parahoric subgroup ${\mathcal{P}}$ is the sum of the idempotents of irreducible representations of $\mathcal{P}$ which have a nonzero Iwahori fixed vector. The convolution algebra associated…

We show that the Hall algebra of the category of coherent sheaves on an elliptic curve (or, equivalently, the algebra of unramified automorphic forms for GL(n) for all n) is equal to the stable limit of spherical double affine Hecke…

The (Iwahori-)Hecke algebra in the title is a $q$-deformation $\sH$ of the group algebra of a finite Weyl group $W$. The algebra $\sH$ has a natural enlargement to an endomorphism algebra $\sA=\End_\sH(\sT)$ where $\sT$ is a $q$-permutation…

Every double coset in $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i \end{pmatrix} (i>1)\}$ if…

Let the symmetric group $\mathfrak{S}_n$ act on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The coinvariant algebra is the graded $\mathfrak{S}_n$-module $R_n :=…

We use the Hecke algebras of affine symmetric groups and their associated Schur algebras to construct a new algebra through a basis, and a set of generators and explicit multiplication formulas of basis elements by generators. We prove that…

The fundumental invariant of the Hecke algebra $H_{n}(q)$ is the $q$-deformed class-sum of transpositions of the symmetric group $S_{n}$. Irreducible representations of $H_{n}(q)$, for generic $q$, are shown to be completely characterized…

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 find new presentations of the modified Ariki-Koike algebra (known also as Shoji's algebra) $\mathcal H_{n,r}$ over an integral domain $R$ associated with a set of parameters $q,u_1,\ldots,u_r$ in $R$. It turns out that the algebra…