Kazhdan’s Orthogonality Conjecture for Real Reductive Groups

We prove a generalization of Harish-Chandra’s character orthogonality relations for discrete series to arbitrary Harish-Chandra modules for real reductive Lie groups. This result is an analogue of a conjecture by Kazhdan for p-adic reductive groups proved by Bezrukavnikov, and Schneider and Stuhler. Introduction Let G0 be a connected compact Lie group. Denote by M(G0) the category of finite-dimensional representations of G0. Then M(G0) is abelian and semisimple. Denote by K(G0) its Grothendieck group. Let U and U ′ be two finitedimensional representations of G0. Denote by HomG0(U,U ′) the complex vector space of intertwining maps between representations U and U ′. Then the map (U,U ′) 7−→ dim HomG0(U,U ′) extends to a biadditive pairing on K(G0), which we call the multiplicity pairing. For a finite-dimensional representation U of G0, we denote by ΘU its character. Let μG0 be the normalized Haar measure on G0. Then the map (U,U ′) 7−→ ∫ G0 ΘU (g)ΘU ′(g) dμG0(g) extends to another pairing on K(G0). The Schur orthogonality relations for characters of irreducible representations imply that these two pairings are equal. Let T0 be a maximal torus in G0. Denote by g and t the complexified Lie algebras of G0 and T0 respectively. Let R be the root system of the pair (g, t). Let W be the Weyl group of R and [W ] its order. For any root α ∈ R define by e the corresponding homomorphism of T0 in the group of complex numbers of absolute value 1. Let D(t) = ∏ α∈R (1− e(t)) for any t ∈ T0. Let μT0 be the normalized Haar measure on T0. Then we have the Weyl integral formula ∫ G0 f(g) dμG0(g) = 1 [W ] ∫