# Holomorphic extensions of representations: (I) automorphic functions

### Abstract

Let $G$ be a connected, real, semisimple Lie group contained in its complexification $G_{\mathbb{C}}$, and let $K$ be a maximal compact subgroup of $G$. We construct a $K_{\mathbb{C}}$-$G$ double coset domain in $G_{\mathbb{C}}$, and we show that the action of $G$ on the $K$-finite vectors of any irreducible unitary representation of $G$ has a holomorphic extension to this domain. For the resultant holomorphic extension of $K$-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain $L^\infty$ bounds on holomorphically extended automorphic functions on $G/K$ in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maaß forms.

## Authors

Bernhard Krötz

Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, United States

Robert J. Stanton

Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, United States