L² spectral decomposition on the Heisenberg group associated to the action of U(p,q)

T. Godoy and L. Saal

Vol. 193 (2000), No. 2, 327–353

DOI: 10.2140/pjm.2000.193.327

Abstract

Here we consider the Heisenberg group H_n = Cⁿ ×ℜ. U, p + q = n, acts by automorphism on H_n by g ⋅ = .

Let be the standard basis of the Lie algebra of H_n and let

Via the Plancherel inversion formula, we obtain the joint spectral decomposition of L² with respect to L and T

where each S_λ.k is a tempered distribution U invariant satisfying iTS_λ,k = λS_λ,k, LS_λ,k = −S_λ,k. We compute explicitly the distributions S_λ,k and the integral μ_k = ∫ _−∞^+∞f ∗ S_λ,kⁿdλ.

Milestones

Received: 11 August 1998

Published: 1 April 2000