Speaker:  Spencer Leslie (Boston College)

Title:  Endoscopic symmetric varieties and rationality 

Abstract: Let G be a quasi-split reductive group over a field k and let X be a spherical variety. Motivated by applications toward relative trace formulae, we review and refine rationality results of Borovoi-Gagliardi, giving a solution (away from characteristic two) when X is symmetric. In good cases, the answer is intimately connected with the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I then discuss the source of these questions in the theory of endoscopy in the context of the relative Langlands program. Finally, I will outline the construction of endoscopic varieties, which are symmetric varieties of an associated endoscopic group. The construction works for most hyperspherical varieties induced from symmetric varieties.