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Abstract

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 (𝜋,V) of GLn, the space Hom(Cc∞(GLn), V ⊗ Ṽ) is one-dimensional. This space is generated by an element denoted by 𝜁, which can be thought of as an integral against matrix coefficients. In this thesis, we are interested in the so-called "extension problem" of 𝜁. 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 𝜁 lies in the image of the induced map of this embedding, then we say that 𝜁 can be extended to rank at least m. For m=0, the extension problem of 𝜁 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 𝜁 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 𝜁 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 𝜋 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.