Thesis advisor: Mark Reeder / This thesis contributes to the construction of supercuspidal representations in small residual characteristics. Let G be a connected, quasi-split, semisimple reductive algebraic group defined and quasi-split over a non-archimedean local field k and splitting over a tamely, totally ramified extension of k. To each parahoric subgroup of G(k), Moy and Prasad have attached a natural filtration by compact open subgroups, the first of which is called the pro-unipotent radical of the parahoric subgroup. The first main result of this thesis is to characterize shallow characters of a pro-unipotent radical, those being complex characters that vanish on the smallest Moy-Prasad subgroup containing all commutators of linearly-dependent affine k-root groups. Through low-rank examples, we illustrate how this characterization can be used to explicitly construct all shallow characters. Next, we provide a natural sufficient condition under which a shallow character compactly induces as a direct sum of supercuspidal representations of G(k). Through examples, however, we show that this sufficient condition need not be necessary, all while constructing new supercuspidal representations of Sp_4(k) when p = 2 and the split form of G_2 over k when p = 3. This work extends the construction of the simple supercuspidal representations given by Gross and Reeder and the epipelagic supercuspidal representations given by Reeder and Yu. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.
| Identifer | oai:union.ndltd.org:BOSTON/oai:dlib.bc.edu:bc-ir_109353 |
| Date | January 2022 |
| Creators | Gastineau, Stella Sue |
| Publisher | Boston College |
| Source Sets | Boston College |
| Language | English |
| Detected Language | English |
| Type | Text, thesis |
| Format | electronic, application/pdf |
| Rights | Copyright is held by the author, with all rights reserved, unless otherwise noted. |
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