Appears in collection : Relative Aspects of the Langlands Program, L-Functions and Beyond Endoscopy / Aspects relatifs du programme de Langlands, fonctions L et beyond endoscopy

I will discuss the transfer of Harish-Chandra characters under the local theta correspondence, in particular in the (almost) equal rank case. More precisely, if $G X H$ is a dual pair in the equal rank setting, it is known that discrete series (resp. tempered) representations lifts to discrete series (resp. tempered) representations. If two such representations correspond under theta lifting, one can ask how their Harish-Chandra characters are related. I will define a space of test functions on each group and a correspondence of their orbital integrals induced by the Weil representation, and show that the resulting transfer of invariant distribution carries the character of a tempered representation to that of its theta lift. I will also explain how the transfer of test functions can be understood geometrically, by relating it to the moment map arising in theta correspondence.