The generating function for elements of the Bethe subalgebra of Hecke algebra is constructed as Sklyanin's transfer-matrix operator for Hecke chain. We show that in a special classical limit q → 1 the Hamiltonians of the Gaudin model can be derived from the transfer-matrix operator of Hecke chain. We consruct a non-local analogue of the Gaudin Hamiltonians for the case of Hecke algebras.

Cells of Coxeter groups are certain equivalence classes defined by the Kazhdan-Lusztig type basis of the associated Hecke algebra with a set of parameters. In this paper, we prove that, for finite Coxeter groups, cells arising from a multiparameter Hecke algebra are determined by those arising from a Hecke algebra with parameters which are powers of a single parameter.

We start by briefly explaining key results on the representation theory of semisimple Lie algebras in large enough positive characteristic. After that, we start a new topic: the complex representation theory of finite groups of Lie type and Hecke algebras. Today we consider the most basic group: G := GLn(Fq). We introduce the Hecke algebra as the endomorphism algebra of the G-module C[B \ G]. W...

We use the results on the minimal basis of the centre of an Iwahori–Hecke algebra from our earlier work, as well as some additional results on the minimal basis, to describe the image and kernel of the Brauer homomorphism for Iwahori–Hecke algebras defined by L. Jones (Jones, L. Centres of Generic Hecke Algebras; Ph.D. Thesis; University of Virginia, 1987.).

We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix) influence the algebraic structure of the corresponding Reflection Equation (RE) algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the matrix of generators of the RE algebra related to a finite rank even Hecke symmetry are derived.

We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Z...

We construct solutions to Sklyanin's reeection equation in the case in which the bulk Yang-Baxter solution is of Hecke algebra type. Each solution constitutes an extension of the Hecke algebra together with a spectral parameter dependent boundary operator, K(). We solve for the deening relations of the extension, and for the spectral parameter dependence of the boundary operator. Finding concre...

Hecke operators play an important role in the theory of automorphic forms, and automorphic forms are closely linked to various cohomology groups. This paper is mostly a survey of Hecke operators acting on certain types of cohomology groups. The class of cohomology on which Hecke operators are introduced includes the group cohomology of discrete subgroups of a semisimple Lie group, the de Rham c...

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V ). This paper classifies all the algebras obtained by applying Drinfeld’s construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we...