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).