Generic Hecke algebra and theta correspondence over finite fields

Jia-Jun Ma · Congling Qiu · Jialiang Zou

2022 · arXiv preprint5 Foruma page views

Primary:math.RTRepresentation TheoryCross-list:math.NTNumber Theory

arXiv:2208.00431v2Source snapshot · 25 Sept 2022View source

Abstract

We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at $1$ can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we generalize the results of Aubert-Michel-Rouquier and Pan on theta correspondence between the Harish-Chandra series.

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First submitted 31 Jul 2022. Metadata retrieved 27 Oct 2026. Page created 27 Oct 2026.

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1 published

errata — Theorem 1.1's statement.

mjj·Oct 3130

arXiv v2

Author's erratum for Generic Hecke algebra and theta correspondence over finite fields, Advances in Mathematics 438 (2024), Article 109444 (opens in a new tab).

In Theorem 1.1 (equation (1.8), page 5; restated in Section 5.4, page 35), replace $c(\pi)$ by $2c(\pi)$ .

The corrected conservation relation is

 $n_{\mathcal{V}'_V}(\pi) + n_{\widetilde{\mathcal{V}}'_V}(\pi) + 2c(\pi) = 2\dim V + \delta.$

This is exactly the relation obtained in the proof in Section 5.4.

Source: Author's errata page (opens in a new tab).

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