Howe–Moore theorem

Statement

Let $G$ be a connected noncompact real Lie group with simple Lie algebra and finite center. If a strongly continuous unitary representation $\pi$ has no nonzero invariant vectors, then

\langle\pi(g)v,w\rangle\longrightarrow0\qquad(g\to\infty)

for every $v,w$ , where $g\to\infty$ means leaving every compact subset of $G$ . This is the Howe–Moore theorem.

Dynamical application

Apply the theorem to the reduced Koopman representation of an ergodic probability-preserving action, assuming strong continuity. Ergodicity removes invariant vectors, so the action is mixing. The conclusion concerns the whole group tending to infinity, not just a specified sequence of times.

For $G=SL_2(\mathbb R)$ , these hypotheses hold. This is a special rigidity phenomenon of the group; an ergodic action of $\mathbb Z$ , such as an irrational rotation, need not be mixing.

References

Corina Ciobotaru, “A unified proof of the Howe–Moore property”, Theorem 1.1 and the definition of the Howe–Moore property.