Supercuspidal representations of GL(n) over a non-archimedean local field : distinction by a unitary or orthogonal subgroup, base change and automorphic induction

Overview & Synopsis

In this thesis, we consider several concrete examples of the relation among the local Langlands correspondence, its functoriality and the problem of distinction. Let F/F_0 be a finite cyclic extension of non-archimedean locally compact fields of residue characteristic p and let R be an algebraically closed field of characteristic l different from p. In the first part, we assume F/F_0 to be quadratic and p to be odd, and we study the irreducible representations of GL_n(F) over R distinguished by a unitary subgroup. We completely solve the problem for supercuspidal representations and get partial results for generic representations. Meanwhile we also define an l-modular version of the cyclic base change lift. In the second part, we assume F=F_0 and p odd, and we fully characterize the complex supercuspidal representations of GL_n(F) distinguished by an orthogonal subgroup. In the final part for F/F_0 tamely ramified, we study the base change lift and the automorphic induction for complex supercuspidal representations via the simple type theory.