On the principal series of Gl n extrm {Gl}_{n} over p p adic fields
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The entire principal series of G = Gl n (F), for a padic 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 Gl n (F), then a "principal series” of representations of K is defined. (Formula Presented) 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. © 1973 American Mathematical Society.
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