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

Abstract

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