Parahoric Spherical Whittaker Functions

Let $F$ be a non-archimedean local field and let $K_μ$ be a standard parahoric subgroup of $GL_n(F)$. We express the diagonal values of normalized parahoric Whittaker eigenfunctions with trivial central character using blockwise Casselman--Shalika formulas with the appropriate modulus factor. Every irreducible generic Iwahori-spherical representation has a one-dimensional fixed space for a suitable parahoric, with no invariants under a strictly larger parahoric. The full Hecke character of this line determines the representation. We compute its adjacent-block Hecke eigenvalues and, under additional irreducibility hypotheses, obtain an explicit finite formula on every Bruhat cell using an opposite induced model, Whittaker-preserving intertwiners, and Demazure--Whittaker operators. We apply these formulas to local Rankin--Selberg integrals for $GL_n\times GL_1$ and for compatible parahorics in $GL_n\times GL_m$. In the latter case, the lowest Laurent coefficient and the equal-rank functional equation determine the integral. Explicit translations and rescalings then give test vectors for the local $L$-factor.