we consider a 2-dimensional representation of the hecke algebra h(g7; u), whereg7 is the complex re ection group and u is the set of indeterminatesu = (x1; x2; y1; y2; y3; z1; z2; z3):after specializing the indetrminates to non zero complex numbers, we then determine a nec-essary and sucient condition that guarantees the irreducibility of the complex specializationof the representation of the ...

We consider a 2-dimensional representation of the Hecke algebra $H(G_7, u)$, where $G_7$ is the complex reflection group and $u$ is the set of indeterminates $u = (x_1,x_2,y_1,y_2,y_3,z_1,z_2,z_3)$. After specializing the indetrminates to non zero complex numbers, we then determine a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of the repre...

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...

We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke alge...

We define a nontrivial homomorphism from the Hecke algebra of type B onto a reduced algebra of the Hecke algebra of type A at roots of unity. We use this homomorphism to describe semisimple quotients of the Hecke algebra of type B at roots of unity. Using these quotients we determine subfactors obtained from the inclusion of Hecke algebra of type A into Hecke algebras of type B. We also study i...

We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke Algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B. Introduc...

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...

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.

Hecke algebras were first studied because of their role in the representation theory of p-adic groups, or algebraic groups over 1-dimensional local fields. There are two important classes of Hecke algebras. One is spherical Hecke algebras attached to maximal compact open subgroups, and the other is Iwahori-Hecke algebras attached to Iwahori subgroups. A spherical Hecke algebra is isomorphic to ...

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...