$P$-alcoves, parabolic subalgebras and cocenters of affine Hecke algebras

Abstract

This is a continuation of the sequence of papers \cite{HN2}, \cite{H99} in the study of the cocenters and class polynomials of affine Hecke algebras $\mathrm{ch}$ and their relation to affine Deligne-Lusztig varieties. Let $w$ be a $P$-alcove element, as introduced in \cite{GHKR} and \cite{GHN}. In this paper, we study the image of $T_w$ in the cocenter of $\mathrm{ch}$. In the process, we obtain a Bernstein presentation of the cocenter of $\mathrm{ch}$. We also obtain a comparison theorem among the class polynomials of $\mathrm{ch}$ and of its parabolic subalgebras, which is analogous to the Hodge-Newton decomposition theorem for affine Deligne-Lusztig varieties. As a consequence, we present a new proof of \cite{GHKR} and \cite{GHN} on the emptiness pattern of affine Deligne-Lusztig varieties.

22 pages

References in corpus (3)

• Minimal length elements of extended affine Coxeter groups, II
• $P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties
• Geometric and homological properties of affine Deligne-Lusztig varieties

Cited by in corpus (3)

• Cocenters and representations of affine $0$-Hecke algebras
• Fundamental elements of an affine Weyl group
• The cocenter-representation duality