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arXiv · 1310.3940

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

Xuhua He·Sian Nie
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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 $\ch$ 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 $\ch$. In the process, we obtain a Bernstein presentation of the cocenter of $\ch$. We also obtain a comparison theorem among the class polynomials of $\ch$ 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.

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Xuhua He, Sian Nie. 2013-10-15. $P$-alcoves, parabolic subalgebras and cocenters of affine Hecke algebras. https://arxiv.org/abs/1310.3940

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