math.RT — Representation Theory
A law of large numbers for infinitesimal characters of real reductive Lie algebras
We prove a law of large numbers for the infinitesimal characters of real reductive Lie algebras. Let be a unitary representation of a connected Lie group with reductive Lie algebra. For product states on the tensor powers of whose sites have uniformly bounded moments, for instance identical sites, the infinitesimal character divided by the number of sites concentrates at the spectrum of the limiting mean moment, with probabilities of deviation for every . The generators may be unbounded and the group non-compact. For compact groups this recovers the laws of large numbers of Keyl and Werner and of Botero, Christandl and Vrana, which enter the asymptotic analysis of linear programming bounds for binary codes; spherical designs require the non-compact case. We give concrete examples arising from Howe dual pairs on bosonic Fock space.