Transfer of Characters in the Theta Correspondence with One Compact Member

摘要

For an irreducible dual pair (G, G') subset of Sp(W) with one member compact and two representations Pi <-> Pi' appearing in the Howe duality, we give an expression of the character Theta(Pi), of Pi' via the character of Pi. We compute the value of Theta(Pi), on the maximal compact torus T' of G' for the dual pair (G = U(n, C), G' = U(p, q, C)), which are explicit in low dimensions. For (G = U(1, C), G' = U(1, 1, C)), we determine the value of the character on both Cartan subgroups of G'.