Informal Friday Seminar: Burrull -- The Bernstein presentation of the Hecke algebra

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).