Representation Theory and Noncommutative Geometry, Paris

Bounding Harish-Chandra characters

3 mars 2025, 15:00

1h

Amphitheater Darboux (IHP)

Orateur

Anna Szumowicz (IMPAN)

Description

Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters ${\theta }_{\pi }$ evaluated at a regular semisimple element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. Kim, Shin and Templier conjectured that $\frac{{\theta }_{\pi }(\gamma )}{\mathrm{deg}(\pi )}$ tends to $0$ when $\pi$ runs over irreducible supercuspidal representations of $G(F)$ whose central character is unitary and the formal degree of $\pi$ tends to infinity. I will sketch the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open CVcompact modulo center subgroup. If time allows I could also discuss progress on optimizing the bound.