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Local Birkhoff decompositions for loop groups and a finiteness result

  • Manish M. Patnaik EMAIL logo
Published/Copyright: February 14, 2025
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 6

Abstract

Let 𝐆 denote an affine Kac–Moody group, and G its points over the local field 𝔽 q ⁢ ( ( s ) ) . We establish a local Birkhoff decomposition for a subset of G in terms of a pair of subgroups roughly of the form 𝐆 ⁢ ( 𝔽 q ⁢ [ [ s ] ] ) and 𝐆 ⁢ ( 𝔽 q ⁢ [ s - 1 ] ) . Our techniques are global-to-local and use the reduction theory for loop groups due to H. Garland. Building on these ideas, we establish the finiteness of a set whose cardinality is related to spherical R-polynomials in D. Muthiah’s conjectural double-affine Kazhdan–Lusztig theory.

Keywords: Loop groups; p-adic groups; Kazhdan–Lusztig theory
MSC 2020: 22E67; 22E50; 20G44

Communicated by Yongchang Zhu

Funding source: Natural Sciences and Engineering Research Council of Canada

Award Identifier / Grant number: RGPIN-2019-06112

Funding statement: The author was supported by the M. V. Subbarao Professorship in Number Theory and NSERC Grant RGPIN-2019-06112 while this paper was in preparation.

Acknowledgements

This project started as a joint one with Dinakar Muthiah who explained to us the constructions in [30] and proposed a number of intriguing questions. We thank him for the many discussions we have had about these topics and for sharing his thoughts on related matters. This note would not exist but for them. We would also like to thank Howard Garland for explaining to us his proof of a local Birkhoff decomposition in finite dimensions a number of years ago–it is essentially just reproduced here with “standard loop group” modifications (that he also explained to us on different occasions). We thank him for generously sharing his ideas with us over the years. We also thank Auguste Hébert for his detailed comments on this note, and for bringing to our attention his interesting works with Paul Philippe. Finally, we are grateful to the anonymous referee for their detailed reading of this paper and many helpful comments.

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Received: 2024-08-08
Revised: 2024-11-27
Published Online: 2025-02-14
Published in Print: 2025-06-01

Š 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
https://doi.org/10.1515/forum-2024-0376
Keywords for this article
Loop groups; p-adic groups; Kazhdan–Lusztig theory

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
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