On the principal series of $\textrm {Gl}_{n}$ over$p$-adic fields

Abstract:

The entire principal series of $G = G{l_n}(F)$, for a p-adic field F, is analyzed after the manner of the analysis of Bruhat and Satake for the spherical principal series. If K is the group of integral matrices in $G{l_n}(F)$, then a “principal series” of representations of K is defined. It is shown that precisely one of these occurs, and only once, in a given principal series representation of G. Further, the spherical function algebras attached to these representations of K are all shown to be abelian, and their explicit spectral decomposition is accomplished using the principal series of G. Computation of the Plancherel measure is reduced to MacDonald’s computation for the spherical principal series, as is computation of the spherical functions themselves.