| Wed, Nov 13, 2019
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Lie Groups Seminar
4:00 PM
MSCS 509 | | On representations of GL(2,F), where F is a non-archimedean local field
Mishty Ray, OSU
| | | Abstract: Let $F$ be a non-archimedean local field of characterisitic 0. More explicitly, think of $F$ as the field of $p$ adic numbers, $\mathbb{Q}_p$, or a finite extension. We aim to look at the classification of smooth irreducible representations of GL($2,F$). The finite dimensional case turns out to be rather simple. Every finite dimensional smooth irreducible representation is one dimensional, and can be given by characters of $F^{\times}$.
Any infinite dimensional smooth irreducible representation is either supercuspidal, principal series, or special. This can be proved by the method of parabolic induction.The standard parabolic for GL($2,F$) is the subgroup of all upper triangular matrices, called the Borel. We can induce a representation of GL($2,F)$ from the Borel, and realize it as a subspace of some principal series representation. Depending on the conditions satisfied by this induced representation, we get the classification. |
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