Cocenters and representations of affine Hecke algebras

Abstract

In this paper, we study the relation between the cocenter and the representation theory of affine Hecke algebras. The approach is based on the interaction between the rigid cocenter, an important subspace of the cocenter, and the dual object in representation theory, the rigid quotient of the Grothendieck group of finite dimensional representations.2010 Mathematics Subject Classification. 20C08, 22E50.

Some Results on Affine Deligne–lusztig Varieties

“…The explicit computation of the class polynomials is very difficult at present. Note that there is a close relation between the cocenter and representations of affine Hecke algebras [7]. One may hope that some progress in the representation theory of affine Hecke algebras would also advance our knowledge on affine Deligne-Lusztig varieties.…”

Cocenters and representations of pro-𝑝 Hecke algebras

“…We say that an elementw ∈W (1) is straight if (w n ) = n (w) for any n ∈ N. By [ [12], σ-conjugacy classes of connected reductive p-adic groups [8] and representations of affine Hecke algebras with non-zero parameters [4].…”

“…Similar maps for affine Hecke algebras with generic non-zero parameters are studied in the joint work of Ciubotaru and the first-named author [4]. It is proved in [4] that the trace map is injective and there is a "perfect pairing" between the rigid-cocenter and rigid-representations ofH(q, c).…”

“…It is proved in [4] that the trace map is injective and there is a "perfect pairing" between the rigid-cocenter and rigid-representations ofH(q, c). (1) If (sws −1 ) = (w) and π(sws …”

“…For affine Hecke algebras, both the dimension of the cocenter and the number of irreducible representations are infinite and the counting-number method does not simply apply. In [4], we introduced the rigid cocenter and rigid quotient of the Grothendieck group of representations, and showed that they form a perfect pairing under the trace map, and the whole cocenter and all the finite dimensional representations can be understood via the rigid part of the parabolic subalgebras. 0.3.…”