The newform K-type and p-adic spherical harmonics

Let \(K: = {\rm{G}}{{\rm{L}}_n}({\cal O})\) denote the maximal compact subgroup of GL_n(F), where F is a nonarchimedean local field with ring of integers \({\cal O}\). We study the decomposition of the space of locally constant functions on the unit sphere in Fⁿ into irreducible K-modules; for F = ℚ_p, these are the p-adic analogues of spherical harmonics. As an application, we characterise the newform and conductor exponent of a generic irreducible admissible smooth representation of GL_n(F) in terms of distinguished K-types. Finally, we compare our results to analogous results in the archimedean setting.