Theta functions on tube domains

TL;DR

This paper explores generalized theta functions on tube domains, revealing that their transformation properties restrict their existence to certain symmetric domains, with some exceptions like the exceptional symmetric tube domain.

Contribution

It establishes conditions under which generalized theta functions can exist on tube domains, linking their properties to the geometry of the underlying domains.

Findings

Generalized theta functions are defined over quasi-symmetric Siegel domains.

The existence of a theta transformation formula constrains the domain to be quasi-symmetric.

The exceptional symmetric tube domain does not admit a theta function.

Abstract

We discuss generalizations of classical theta series, requiring only some basic properties of the classical setting. As it turns out, the existence of a generalized theta transformation formula implies that the series is defined over a quasi-symmetric Siegel domain. In particular the exceptional symmetric tube domain does not admit a theta function.