Working Seminar on Representation Theory

It is known that the number of conjugacy classes of a finite group equals the number of irreducible representations (over complex numbers). The conjugacy classes of a finite group give a natural basis of the cocenter of its group algebra. Thus the above equality can be reformulated as a duality between the cocenter of the group algebra and the Grothendieck group of its finite dimensional representations. For affine Hecke algebras, the situation is much more complicated. First, the cocenter of affine Hecke algebras is harder to understand than the cocenter of group algebras. Second, for an affine Hecke algebra, the dimension of its cocenter is countably infinite and the number of irreducible representations is uncountable infinite. However, the "cocenter-representation duality" is still valid. This is what I am going to explain in this talk. I will also explain my conjecture on the affine Hecke algebras at roots of unity and modular representations of affine Hecke algebras.