A spectral identity between symmetric spaces

Abstract

Let π be a cuspidal automorphic representation ofGL2n. We prove an identity between two spectral distributions onSp2n andGL2n respectively. The first is the spherical distribution with respect toSpn×Spnof the residual Eisenstein series induced from π. The second is the weighted spherical distribution of π with respect toGLn×GLnand a certain degenerate Eisenstein series. A similar identity relates the pair (U2n,Spn) and (GLn/E,GLn/F) whereE/F is the quadratic extension defining the quasi-split unitary groupU2n. We also have a Whittaker version of these trace identities.