Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Abstract

Let pi _{alpha } be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space A^2_{alpha } (Omega ) on a bounded symmetric domain Omega , of formal dimension d_{pi _{alpha }} > 0. It is shown that if the Bergman kernel k^{(alpha )}_z is a cyclic vector for the restriction pi _{alpha } |_{Gamma } to a lattice Gamma le G (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma } is a frame for A^2_{alpha }(Omega )), then {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} le |Gamma _z|^{-1}. The estimate {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} ge |Gamma _z|^{-1} holds for k^{(alpha )}_z being a p_z-separating vector (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma / Gamma _z} being a Riesz sequence in A^2_{alpha } (Omega )). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for G ={mathrm {PSU}}(1, 1).