On vector-valued automorphic forms on bounded symmetric domains

TL;DR

This paper establishes a spanning result for vector-valued automorphic forms on bounded symmetric domains, providing norm estimates and asymptotic behaviors, especially for the unit ball in complex n-space.

Contribution

It introduces a new spanning result for vector-valued Poincaré series and analyzes asymptotic norms for automorphic forms associated with submanifolds.

Findings

Established a spanning result for vector-valued Poincaré series.

Provided norm estimates and asymptotics for automorphic forms on the unit ball.

Identified special cases with different asymptotic behaviors for CR submanifolds.

Abstract

We prove a spanning result for vector-valued Poincar\'e series on a bounded symmetric domain. We associate a sequence of holomorphic automorphic forms to a submanifold of the domain. When the domain is the unit ball in ${\Bbb{C}}^n$, we provide estimates for the norms of these automorphic forms and we find asymptotics of the norms (as the weight goes to infinity) for a class of totally real submanifolds. We give an example of a CR submanifold of the ball, for which the norms of the associated automorphic forms have a different asymptotic behavior.