A Cauchy - Harish-Chandra integral for a dual pair over a p-adic field

Hung Yean Loke (U of Singapore)

Apr 14. 2022, 11:45 — 12:30

Let (G, G') be an irreducible reductive dual pair in Sp(W) where the rank of G' less or equal to the rank of G. In the case when (G,G') is a real reductive dual pair, Przebinda defines an integral kernel operator Chc. Here Chc stands for the Cauchy Harish-Chandra integral. It is shown that Chc maps the distribution character $\Theta_{V'}$ of an irreducible admissible representation V' of $\widetilde{G}'$ to an invariant eigendistribution $\Theta'_{V'}$ on the group $\widetilde{G}$. Furthermore, if the pair is in the stable range and if the representation V' is unitary, then $\Theta'_{V'} = \Theta_V$, where V is associated to V' via Howe's correspondence. In this talk, we will explain how to transfer the construction of Chc to the p-adic case. We will formulate a conjecture that $\Theta'_{V'}=\Theta_V$. This is a joint work with Tomasz Przebinda.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Minimal Representations and Theta Correspondence (Workshop)
Wee Teck Gan (U of Singapore)
Marcela Hanzer (U Zagreb)
Alberto Minguez (U of Vienna)
Goran Muic (U Zagreb)
Martin Weissman (UC Santa Cruz)