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 $\Gamma$ be a cocompact discrete subgroup of SU(n,1) which acts freely on SU(n,1)/U(n). We suggest a construction of relative Poincar\'e series associated to loxodromic elements in $\Gamma$. In complex dimension 2 we describe Bohr-Sommerfeld tori in $\Gamma\backslash SU(n,1)/U(n)$ associated to hyperbolic elements of $\Gamma$ and prove that the relative Poincar\'e series associated to the hyperbolic elements of $\Gamma$ are not identically zero for large k.