Monday, November 28, 2022 at 2:00 (sharp)

65 St George St, Room 101
Zoom Web Conference

PhD Candidate: Vincent Girard
Supervisor: Fiona Murnaghan
Thesis title: Relatively Supercuspidal Representations of the Symplectic p-adic Groups

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In this thesis, we construct in a concrete manner a family of non-supercuspidal, relatively supercuspidal
representations of symplectic p-adic groups, based on the work of Murnaghan [Mur]. We cover
the symmetric pairs (Sp(4n, F), Sp(2n, E)), (Sp(2n, E), Sp(2n, F)) and (Sp(2n, F), Sp(2k, F) x Sp(2(n-k), F))
where F is a p-adic field of odd residual characteristic, and E is a quadratic field extension of F.

We also look at the symmetric pairs (Sp(2n, F), GL(n, F)) and (Sp(2n, F), U(n, E/F, ε)) for ε an invertible
Hermitian matrix over E/F. For these additional pairs, the above construction doesn’t result
in any non-supercuspidal, relatively supercuspidal representations (despite these pairs admitting
distinguished supercuspidals).

We end with an in-depth look at the case of Sp(2, F) = SL(2, F). We show that in this low-rank
example, for all of the above pairs, all irreducible relatively supercuspidal representations of SL2(F)
are either supercuspidal or obtained from our construction. In particular, all irreducible H-relatively
supercuspidal representations of SL(2, F), for H either GL(1, F) or U(1, E/F, ε), are supercuspidal.

A copy of the thesis can be found here: Girard_Vincent_202212_PhD_thesis

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