The Howe-Moore property for real and $p$-adic groups

Raf Cluckers (1, 2) , Yves de Cornulier (3) , Nicolas Louvet (4) , Romain Tessera (5) , Alain Valette (4)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Department of mathematics
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 Institut de Mathématiques
5 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We consider in this paper a relative version of the Howe-Moore property, about vanishing at infinity of coefficients of unitary representations.We characterize this property in terms of ergodic measure-preserving actions.We also characterize, for linear Lie groups or p-adic Lie groups, the pairs with the relative Howe-Moore property with respect to a closed, normal subgroup. This involves, in one direction, structural results on locally compact groups all of whose proper closed characteristic subgroups are compact, and, in the other direction, some results about the vanishing at infinity of oscillatory integrals.

Raf Cluckers, Yves de Cornulier, Nicolas Louvet, Romain Tessera, Alain Valette. The Howe-Moore property for real and $p$-adic groups. Mathematica Scandinavica, 2011, 109, pp.201-224. ⟨hal-00832262⟩