Decay to zero of matrix coefficients at Adjoint infinity

The main theorems below are Theorem 9 and Theorem 11. As far as I know, Theorem 9 represents a slight improvement over what currently appears in the literature, and gives a fairly easy proof of Theorem 11 which is due to R. Howe and C. Moore [4]. In the semisimple case, the Howe-Moore result follows from [6] or [7]. The proofs appearing here are relatively elementary and some readers will recognize the influence of R. Ellis and M. Nerurkar [2]. What may be less evident is the connection to N. Kowalsky’s work [5] on dynamics on Lorentz manifolds: Lemma 1 below is a unitary analogue of the fact that an expansive Adjoint action can result in much of the Lie algebra being lightlike. In Hilbert space, the situation is even nicer: an isotropic vector must equal zero, so we get that the Lie algebra of the stabilizer contains all “Kowalsky” vectors for the Adjoint sequence, and not just a codimension one subspace. Moreover, much of the rest of proof of Theorem 9 also uses ideas that were originally developed to describe effectively the collection of simply connected Lie groups admitting an orbit nonproper action on a connected Lorentz manifold. Thus the debt to Kowalsky is significant. In this note, we assume some familiarity with Lie theory and basic unitary representation theory. The exposition should otherwise be self-contained. Please send comments, suggestions or questions to [email protected]. Throughout, “Lie group” means “C∞ real Lie group”, “Lie algebra” means “real Lie algebra” and “Hilbert space” means “complex Hilbert space”. Lie groups are denoted by capital Roman letters and, for any Lie group, its Lie algebra is denoted, without comment, by the corresponding small letter in the fraktur font. The sesquilinear form on a Hilbert space is denoted 〈 · , · 〉. Convergence in the weak topology on a Hilbert space is denoted with ⇀, so vi ⇀ v in a Hilbert space H means: for all w ∈ H, 〈vi, w〉 → 〈v, w〉. The group of bounded operators on H is denoted B(H). The unitary group of a Hilbert space H is denoted U(H). The weak-operator and strong-operator topologies agree on U(H), and U(H) is given this topology. Unitary representations are always assumed continuous in the weak-operator topology (or, equivalently, in the strong-operator topology). I do not plan to publish this note.