Completeness of Poincaré Series for Automorphic Forms Associated to the Integrable Discrete Series

Abstract

A couple of years ago, Wolf [11] studied the Poincaré series operator ϑ for a homogeneous holomorphic vector bundle E ➛ D over a flag domain D = G/V and an arbitrary discrete subgroup Г⊂ G. He showed that if E ➛ D is nondegenerate (see below), and if G acts on the square integrable cohomology space H 2 s (D; E) by an integrable discrete series representation, where s is the complex dimension of the maximal compact subvariety K/V in D, then every Г-automorphic Lp cohomology class ψ∈ H p s (Г\D; E), 1 ⩽ p ⩽ ∞, is represented by a Poincaré series $$\rm{\Psi\ =\ \vartheta(\phi)\ =\ \sum_{\Upsilon\epsilon\Gamma}\ \Upsilon^\ast\phi\ with\ \phi\ \epsilon\ H^S_P(D;E).}$$ (1.1)