Square integrable representations and Hecke eigenvalues over a local non-archimedian field

Abstract

Let G be a split connected semi-simple group over a local non-archimedean field F. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack bpo of G-bundles on ℙ¹ endowed with a trivialization at $0$ and ∞. This space is a representation of G(F)× G(F) which contains the regular representation of G(F). We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation π of G(F) defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of G(F) with an Iwahori fixed vector defines a ℂ-point on the Calogero-Moser space associated to G. We also show that for G=PGL(N) the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of π on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when F is a finite field.

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BibTeX RIS

BibTeX

@misc{braverman2026square,
  title = {{Square integrable representations and Hecke eigenvalues over a local non-archimedian field}},
  author = {Alexander Braverman and David Kazhdan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17317},
  eprint = {2610.17317},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17317},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Square integrable representations and Hecke eigenvalues over a local non-archimedian field
AU  - Alexander Braverman
AU  - David Kazhdan
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17317
UR  - https://arxiv.org/abs/2610.17317
AB  - Let G be a split connected semi-simple group over a local non-archimedean field F. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack bpo of G-bundles on ℙ^1 endowed with a trivialization at $0$ and ∞. This space is a representation of G(F)× G(F) which contains the regular representation of G(F). We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation π of G(F) defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of G(F) with an Iwahori fixed vector defines a ℂ-point on the Calogero-Moser space associated to G. We also show that for G=PGL(N) the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of π on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when F is a finite field.
ER  -

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