Finiteness Theorems for Kac–Moody Groups over Non-archimedean Local Fields

Abstract We prove the finiteness of formal analogs spherical function (Spherical Finiteness), ${\textbf c}$-function (Gindikin–Karpelevich and obtain a analog Harish-Chandra’s limit (Approximation Theorem) relating in setting $p$-adic Kac–Moody groups. The theorems imply that Gindikin–Karpelevich integral is well defined local settings. These results extend Braverman–Garland–Kazhdan–Patnaik’s affine from [ 4] provide an algebraic combinatorial Gaussent–Rousseau [10] Hébert [14].