Dimension formulae of Gelfand-Graev, Jones and their relation to automorphic forms and temperdness of quasiregular representations

Abstract: Vaughan Jones introduced a formula computing the von Neumann dimension for the restriction to a lattice of the left regular representation of a semisimple Lie group.

It is a variant of a formula by Atiah Schmidt computing the formal dimension in the Haris Chandra trace formula for discrete series. It is surprisingly similar (in the case of PSL(2,Z)) to the dimension of the space of automorphic forms and is similar to a formula proved by Gelfand, Graev. We use an extension of this formula to provide a method for computing the formal trace of representations of PSL(2,Q_p) (or more general situations), when analyzing the quasi regular representation on PSL(2,R)/PSL(2,Z). It provides a method to obtain estimates for eigenvalues of Hecke operators.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar