| Summary: | 碩士 === 國立成功大學 === 數學系應用數學碩博士班 === 106 === The goal of this paper is to determine whether a irreducible representation of GL(4) over a finite field admits a shalika model. We analysis it for two parts:cuspidal representations and noncuspidal representations.
For cuspidal representations, we use a result published from mathematician Dipendra Prasad at 2000. From the result, it is easy to determine whether a cuspidal representation has a Shalika model.
For noncuspidal representations, by the definition of cuspidal, we see a noncuspidal representation as a subrepresentation of an parabolic induction from some parabolic induction subgroup of GL(4). We consider all parabolic inductions of GL(4) and use a well-known theorem Mackey’s theorem to determined whether a parabolic induction has a Shalika model. From Mackey’s theorem, we get some condition for a parabolic induction has a Shalika model and use it to get some result.
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