These are notes for a talk in the MIT-NEU Graduate seminar on Hecke Algebras and Affine Hecke Algebras (AHAHA) held in Fall 2014. This talk is divided into three parts. In the first, we introduce the affine Hecke algebras and describe a useful basis for the algebra over the ground ring. We then give a complete description of the center of the affine Hecke algebra and prove Kato’s Theorem regard...

We derive an elementary formula for the trace of a Hecke operator acting on a space of algebraic modular forms, as a sum of character values. We describe explicit computations in the case of the unitary group U(4), allowing the determination of the eigenvalues of a certain Hecke operator. This produces numerical evidence for a U(2, 2) analogue of Harder’s conjecture, on congruences between Heck...

The purpose of this article is to determine representation type for all of the Hecke algebras of classical type. To do this, we combine methods from our previous work, which is used to obtain information on their Gabriel quivers, and recent advances in the theory of finite dimensional algebras. Principal computation is for Hecke algebras of type B with two parameters. Then, we show that the rep...

The purpose of this article is to determine representation type for all of the Hecke algebras of classical type. To do this, we combine methods from our previous work, which is used to obtain information on their Gabriel quivers, and recent advances in the theory of finite dimensional algebras. Principal computation is for Hecke algebras of type B with two parameters. Then, we show that the rep...

We consider the double-row (open) transfer matrix constructed from generic representations of various types of Hecke algebras. For different choices of boundary conditions for the relevant integrable lattice model we express the double-row transfer matrix solely in terms of generators of the corresponding Hecke algebra. We then expand the open transfer matrix and extract the associated Murphy e...

Let G be a nite group of Lie type deened over some nite eld GF(q). Let k be a eld of positive characteristic p not dividing q. Hecke functors tie together the representation theory of kG and that of Hecke algebras associated with nite reeection groups. In DDu1] the theory of vertices and sources for such algebras was introduced in the case of Hecke algebras H of type A, (that is Hecke algebras ...

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity. Mathematics Subject Classification (2010). Primary: 11F11, 11F30. Se...

The geometric Langlands program can be described in a natural way by compactifying on a Riemann surface C a twisted version of N = 4 super Yang-Mills theory in four dimensions. The key ingredients are electric-magnetic duality of gauge theory, mirror symmetry of sigma-models, branes, Wilson and ’t Hooft operators, and topological field theory. Seemingly esoteric notions of the geometric Langlan...

We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group Sn over the complex numbers. Interestingly, the unitary representations in the category O for the rational Cherednik algebra of type A studied in [ESG] correspond to the unitary representations of the corresponding Hecke ...