Hecke algebras for the 1st congruence subgroup and bundles on P1 I: the case of finite field

Alexander Braverman; David Kazhdan

doi:10.1090/conm/823/16495

Hecke algebras for the 1st congruence subgroup and bundles on P¹ I: the case of finite field

Alexander Braverman
, David Kazhdan

Einstein Institute of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Let G be a split reductive group over a finite field k in this note we study the space V of finitely supported functions on the set Bun_G (P¹)_{0, ∞}(k) of isomorphism classes of G-bundles on the projective line P¹ endowed with a trivialization at 0 and ∞. We show that V is naturally isomorphic to the regular bimodule over the Hecke algebra A of the group G(k((t))) with respect to the first congruence subgroup. As a byproduct we show that Hecke operators at points different from 0 and ∞ generate the” stable center” of A. We provide an expression of the character of the lifting of an irreducible cuspidal representation of GL (N, k) to GL (N, k′) where k′ is a finite extension of k in terms of these generators. In a subsequent publication we plan to develop analogous constructions in the case when k is replaced by a local non-archimedian field.

Original language	English
Title of host publication	The Versatility of Integrability - In Memory of Igor Krichever
Subtitle of host publication	Isomonodromic Deformations, Painlevé Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022
Editors	Mikhail Bershtein, Anton Dzhamay, Andrei Okounkov
Publisher	American Mathematical Society
Pages	155-168
Number of pages	14
ISBN (Print)	9781470475260
DOIs	https://doi.org/10.1090/conm/823/16495
State	Published - 2025
Event	Isomonodromic Deformations, Painleve Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022 - New York, United States Duration: 7 Oct 2022 → 9 Oct 2022

Publication series

Name	Contemporary Mathematics
Volume	823
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	Isomonodromic Deformations, Painleve Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022
Country/Territory	United States
City	New York
Period	7/10/22 → 9/10/22

Bibliographical note

Publisher Copyright:
© 2025 American Mathematical Society.

Access to Document

10.1090/conm/823/16495

Cite this

Braverman, A., & Kazhdan, D. (2025). Hecke algebras for the 1st congruence subgroup and bundles on P¹ I: the case of finite field. In M. Bershtein, A. Dzhamay, & A. Okounkov (Eds.), The Versatility of Integrability - In Memory of Igor Krichever: Isomonodromic Deformations, Painlevé Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022 (pp. 155-168). (Contemporary Mathematics; Vol. 823). American Mathematical Society. https://doi.org/10.1090/conm/823/16495

Braverman, Alexander ; Kazhdan, David. / Hecke algebras for the 1st congruence subgroup and bundles on P¹ I : the case of finite field. The Versatility of Integrability - In Memory of Igor Krichever: Isomonodromic Deformations, Painlevé Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022. editor / Mikhail Bershtein ; Anton Dzhamay ; Andrei Okounkov. American Mathematical Society, 2025. pp. 155-168 (Contemporary Mathematics).

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Braverman, A & Kazhdan, D 2025, Hecke algebras for the 1st congruence subgroup and bundles on P¹ I: the case of finite field. in M Bershtein, A Dzhamay & A Okounkov (eds), The Versatility of Integrability - In Memory of Igor Krichever: Isomonodromic Deformations, Painlevé Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022. Contemporary Mathematics, vol. 823, American Mathematical Society, pp. 155-168, Isomonodromic Deformations, Painleve Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022, New York, United States, 7/10/22. https://doi.org/10.1090/conm/823/16495

Hecke algebras for the 1st congruence subgroup and bundles on P¹ I: the case of finite field. / Braverman, Alexander; Kazhdan, David.
The Versatility of Integrability - In Memory of Igor Krichever: Isomonodromic Deformations, Painlevé Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022. ed. / Mikhail Bershtein; Anton Dzhamay; Andrei Okounkov. American Mathematical Society, 2025. p. 155-168 (Contemporary Mathematics; Vol. 823).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Braverman A, Kazhdan D. Hecke algebras for the 1st congruence subgroup and bundles on P¹ I: the case of finite field. In Bershtein M, Dzhamay A, Okounkov A, editors, The Versatility of Integrability - In Memory of Igor Krichever: Isomonodromic Deformations, Painlevé Equations, and Integrable Systems, IDPEIS 2022 and Algebraic Geometry, Mathematical Physics, and Solitons, AGMPS 2022. American Mathematical Society. 2025. p. 155-168. (Contemporary Mathematics). doi: 10.1090/conm/823/16495