2022 Fall POSTECH-PMI NUMBER THEORY SEMINAR

• Speaker : Julia Gordon(University of British Columbia, Canada)
• Time : 2022.09.23(Fri) 10:00 ~22:00
• Online Streaming(Zoom)

Title : Orbital integrals as local densities

Abstract : This talk will be mainly an exposition of a different point of view on orbital integrals on reductive groups.

Traditionally an orbital integral at a regular semisimple element is written as an integral with respect to the quotient of a Haar measure on G by a Haar measure on the centralizer of the element.

On the other hand, one could instead think of (stable) orbits of such elements as fibres of a map from G to the Steinberg-Hitchin base, and define the orbital integrals using the resulting quotient measure.

This point of view appears in the work of Frenkel-Langlands-Ngo and subsequent literature on beyond endoscopy. It also allows us to think of orbital integrals on p-adic groups as a kind of local densities, which has connections with number theory questions, such as counting isogeny classes in an isomorphism class of principally polarized abelian varieties over a finite field. The conversion between these two ways of defining an invariant measure on an orbit runs into a surprising number of technical difficulties; I will try to give a complete exposition for the symplectic group (which turns out to be the simplest case).