An affine Gindikin-Karpelevich formula
Alexander Braverman, Howard Garland, David Kazhdan, Manish Patnaik
TL;DR
This paper provides an elementary proof of finiteness results for affine Kac-Moody groups over local fields, leading to an affine Gindikin-Karpelevich formula consistent with previous work in positive characteristic.
Contribution
It offers a new elementary proof of key finiteness results, enabling the formulation of an affine Gindikin-Karpelevich formula without relying on algebraic geometry or Bruhat-Tits theory.
Findings
Finiteness results for affine Kac-Moody groups over local fields
Formulation of an affine Gindikin-Karpelevich formula
Connection to affine Macdonald formula for spherical functions
Abstract
In this paper we give an elementary proof of certain finiteness results about affine Kac-Moody groups over a local non-archimedian field K. Our results imply those proven earlier by Braverman-Kazhdan, Braverman-Finkelberg-Kazhdan and Gaussent-Rousseau using either algebraic geometry or a Kac-Moody version of the Bruhat-Tits building. The above finiteness results allow one to formulate an affine version of the Gindikin-Karpelevich formula, which coincides with the one discussed by Braverman-Finkelberg-Kazhdan in the case when K has positive characteristic. We deduce this formula from an affine version of the Macdonald formula for the spherical function, which will be proved in a subsequent publication.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology