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

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