Related papers: Some examples of Hecke algebras over 2-dimensional…

We consider the algebra of Hecke correspondences (elementary transformations at a single point) acting on the algebraic K-theory groups of the moduli spaces of stable sheaves on a smooth projective surface S. We derive quadratic relations…

We study the local Hecke algebra $\mathcal{H}_{G}(K)$ for $G = \mathrm{GL}_n$ and $K$ a non-archimedean local field of characteristic zero. We show that for $G = \mathrm{GL}_2$ and any two such fields $K$ and $L$, there is a Morita…

A module over an affine Kac--Moody algebra g^ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical g^-modules of…

The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke…

Let $\mathbf{G}$ be a connected reductive group over a {non-archimedean local field} $F$. Let $K_\mathcal{F}$ be the parahoric subgroup attached to a facet $\mathcal{F}$ in the Bruhat--Tits building of $\mathbf{G}$. The ultimate goal of the…

In this paper, we define a number of closely related isomorphisms. On one side of these isomorphisms sit a number of of algebras generalizing the Hecke and affine Hecke algebras, which we call the "Hecke family"; on the other, we find…

Let $G$ be a split connected reductive group defined over $\mathbb{Z}$. Let $F$ be a locally compact non-Archimedean field with residue characteristic $p$. For a locally compact non-Archimedean field $F'$ that is sufficiently close to $F$,…

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in…

Let $\mathbf{G}$ be a split algebrac group of type $E_n$ defined over a $p$-adic field. This group contains a dual pair $G \times G'$ where one of the groups is of type $G_2$. The minimal representation of $\mathbf{G}$, when restricted to…

We provide a dual version of the Geck--Rouquier Theorem on the center of an Iwahori--Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank $2$, for which the symmetrising trace conjecture is…

We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the…

Let $G$ be a direct product of inner forms of general linear groups over non-archimedean locally compact fields of residue characteristic $p$ and let $K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In this paper we…

Let $G$ be a split reductive group over a finite field $k$. In this note we study the space $V$ of finitely supported functions on the set of isomorphism classes $G$-bundles on the projective line ${\mathbb P}^1$ endowed with a…

This is a survey paper about representation theory and noncommutative geometry of reductive p-adic groups G. The main focus points are: 1. The structure of the Hecke algebra H(G), the Harish-Chandra-Schwartz algebra S(G) and the reduced…

In this work, we study multiplicity-free induced representations of finite groups. We analyze in great detail the structure of the Hecke algebra corresponding to the commutant of an induced representation and then specialize to the…

We define alternating cyclotomic Hecke algebras in higher levels as subalgebras of cyclotomic Hecke algebras under an analogue of Goldman's hash involution. We compute the rank of these algebras and construct a full set of irreducible…

Let $F$ be a non-archimedean local field such that $4|q-1$, with $q$ the order of the residue field of $F$, and let $(M^0,\sigma^0)$ be the depth-zero cuspidal pair for the twisted Levi subgroup $G^0$ of $\mathrm{SL}_8$ arising from…

We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified $p$-adic group $G$ and generalize the formulation to give a description of the Hecke algebra $H_G(V)$ of weight $V$, where $V$ is a lattice in an…

This paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in math.QA/9805134. These new Hecke algebras are associated to triples of the form (A,B,e), where A is an associative algebra…

Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)^s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra…