Finiteness Theorems for Kac-Moody Groups Over Nonarchimedean Local Fields

TL;DR

This paper establishes finiteness properties of key functions in the representation theory of p-adic Kac-Moody groups, extending classical results and enabling well-defined integrals in this infinite-dimensional setting.

Contribution

It proves finiteness theorems for spherical and c-functions in p-adic Kac-Moody groups, extending known affine results and providing algebraic analogues of geometric theorems.

Findings

Finiteness of spherical functions in p-adic Kac-Moody groups.

Finiteness of the c-function (Gindikin-Karpelevich) in this setting.

Formal analogue of Harish-Chandra's limit relating spherical and c-functions.

Abstract

We prove the finiteness of formal analogues of the spherical function (Spherical Finiteness), the ${\mathbf c}$-function (Gindikin-Karpelevich Finiteness), and obtain a formal analogue of Harish-Chandra's limit (Approximation Theorem) relating spherical and ${\mathbf c}$-function in the setting of $p$-adic Kac-Moody groups. The finiteness theorems imply that the formal analogue of the Gindikin-Karpelevich integral is well defined in local Kac-Moody settings. These results extend the Braverman-Garland-Kazhdan-Patnaik's affine Gindikin-Karpelevich finiteness theorems and provide an algebraic analogue of the geometrical results of Gaussent-Rousseau and H\'{e}bert.