THE SAKELLARIDIS-VENKATESH CONJECTURE FOR U(2)\SO2,3

Abstract

My research object is the spherical variety U(2)\SO(5) where U(2) is a quasi-split unitary group and embedded in a natural way in a split special orthogonal group SO(5). In the local case, my main goal is to give the Plancherel decomposition of L2(U(2)\SO(5)) in terms of the tempered unitary dual of PGSO(4)=PGL2 × PGL2 along the theta correspondence. We will also give the relative character identities in the local case. In the global case, I will give the decomposition of the global U(2)-period of certain irreducible cuspidal representations of SO(5) in terms of the local functionals coming from the local Plancherel decomposition of L2(U(2)\SO(5)).