Type: Seminar

Distribution: World

CalTitle1: Burrull - The Bernstein presentation of the Hecke algebra

Auth: romanova@10.17.27.14 (arom8272) in SMS-SAML

Let G be a simply connected split reductive group (e.g. SL(n,C)). Let T be a complex torus, P the weight lattice and PV the coweight lattice. Let R be a reduced root system and S a fixed choice of simple roots. We define the affine Weyl group Waff associated to (P,PV,R,RV) as the semidirect product of W and P, where W is the Weyl group W of the root system R. This definition is "abstract" in the sense it does not involve the group G. Let HW be the Hecke algebra associated to W. In this talk I introduce the affine Hecke algebra H associated to (R, P), this algebra was introduced by J. Bernstein, and is isomorphic to the Iwahori-Hecke algebra of a split p-adic group with connected center. It contains HW as a subalgebra and a large complementary corresponding to "translation part." I roughly introduce what is the subject known as "equivariant algebraic K- theory". I will state the existence and some properties of an isomorphism between the group algebra Z[Waff] and the convolution algebra arising from the G-equivariant K-group of the Steinberg variety Z. Furthermore, I will state the existence and some properties of an isomorphism between H and the convolution algebra arising from the G \times C*-equivariant action on the Steinberg variety Z. During the talk, I give some examples, and I will show roughly how the above isomorphisms look like in the case of SL(2, C).