Howe-Moore's theorem

In mathematics, the Howe-Moore theorem, a doctrine of representation theory with applications in ergodic theory .

It says that for every unitary representation of a simple Lie group, all matrix coefficients vanish at infinity, so it holds. ${\displaystyle c_{v,w}}$ ${\displaystyle c_{v,w}\in C_{0}(G)}$

A corollary, also referred to as the Howe-Moore theorem, is that the effect of a simple Lie group on a probability space is highly mixing and, in particular, ergodic . A typical application concerns the effect of a non-compact closed subgroup on the orbit space of a lattice , which is thus strongly mixing and ergodic. For one obtains the ergodicity of the geodetic flow on hyperbolic surfaces of finite area. ${\displaystyle H\subset G}$ ${\displaystyle \Gamma \backslash G}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle G=PSL(2,\mathbb {R} ),H=\left\{\left({\begin{array}{cc}e^{t}&0\\0&e^{-t}\end{array}}\right)\right\}}$

literature

• RE Howe, CC Moore: Asymptotic properties of unitary representations . J. of Funct. Anal. 32: 72-96 (1979).
• MB Bekka, M. Mayer: Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces (London Mathematical Society Lecture Note Series Book 269) , Cambridge University Press, 2000, ISBN 978-0521660303