Asymptotics of $p$-adic groups, mostly $SL_2$ [Colloquium]

Abstract: Let $p$ be a prime number and $ Q_p$ the field of $p$-adic numbers. The representations of a cousin of the Galois group of an algebraic closure of $ Q_p$ are related (the {\bf Langlands's bridge}) to the representations of reductive $p$-adic groups, for instance $SL_2(Q_p), GL_n(Q_p) $. The irreducible representations $\pi$ of reductive $p$-adic groups are easier to study than those of the Galois groups but they are rarely finite dimensional. Their classification is very involved but their behaviour around the identity, that we call the ``asymptotics'' of $\pi$, are expected to be more uniform. We shall survey what is known (joint work with Guy Henniart), and what it suggests.

number theory

Audience: researchers in the topic

Comments: Colloquium, no livestream

UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.