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.
Categories: math.GR