Let k be a non-Archimedean local field, and Cc∞(GLn) the space of locally constant compactly supported complex-valued functions on the general linear group GLn over k. For every irreducible representation (,V) of GLn, the space Hom(Cc∞(GLn), V ⊗ Ṽ) is one-dimensional. This space is generated by an element denoted by , which can be thought of as an integral against matrix coefficients. In this thesis, we are interested in the so-called "extension problem" of .
More explicitly, for 0≤m≤n, GLn can be embedded into the space R≥m of all n×n matrices over k of rank at least m. If lies in the image of the induced map of this embedding, then we say that can be extended to rank at least m. For m=0, the extension problem of to rank at least 0 has been completely answered in Tate's thesis for n=1, and by Moeglin, Vignéras, Waldspurger, and Minguez for general n. Our goal is to determine the least value m for a given representation such that can be extended to rank at least m.
A representation is said to appear in rank m if Hom(Cc∞(R≥m), V ⊗ Ṽ) is non-trivial. It is natural to conjecture that extends to rank at least m+1 but does not extend to rank at least m where m is the highest rank less than n that appears in. In this thesis, this conjecture is proved for spherical representations, by means of extending Satake transform to the space of K-bi-invariant functions on Mn and obtaining a partial description of the image of the rank filtration under this extended Satake transform.
Some explicit computations for spherical representations of GL₃ are included as motivating examples of the general case.There are also some suggestive calculations for non-spherical representations of GL₂.Science, Faculty ofMathematics, Department ofGraduat
