Bohr-Sommerfeld tori and relative Poincare series on a complex hyperbolic space

Abstract

Automorphic forms on a bounded symmetric domain D=G/K can be viewed as holomorphic sections of $L^{\otimes k}$, where L is a quantizing line bundle on a compact quotient of D and k is a positive integer. Let $Γ$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of relative Poincaré series associated to loxodromic elements in $Γ$. In complex dimension 2 we describe Bohr-Sommerfeld tori in $Γ\backslash SU(n,1)/U(n)$ associated to hyperbolic elements of $Γ$ and prove that the relative Poincaré series associated to the hyperbolic elements of $Γ$ are not identically zero for large k.