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arXiv · 2407.15128

A description of the integral depth-$r$ Bernstein center

Sarbartha Bhattacharya·Tsao-Hsien Chen
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Abstract

In this paper we give a description of the depth-$r$ Bernstein center for non-negative integers $r$ of a reductive simply connected group $G$ over a non-archimedean local field as a limit of depth-$r$ standard parahoric Hecke algebras. Using the description, we construct maps from the algebra of stable functions on the $r$-th Moy-Prasad filtration quotient of hyperspecial parahorics to the depth-$r$ Bernstein center and use them to attach to each depth-$r$ irreducible representation $\pi$ an invariant $\theta(\pi)$, called the depth-$r$ Deligne-Lusztig parameter of $\pi$. We show that $\theta(\pi)$ is equal to the semi-simple part of minimal $K$-types of $\pi$.

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Sarbartha Bhattacharya, Tsao-Hsien Chen. 2024-07-21. A description of the integral depth-$r$ Bernstein center. https://arxiv.org/abs/2407.15128

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