Symmetry properties of the zero sets of nil-theta functions

Abstract:

Let $N$ denote the three dimensional Heisenberg group, and let $\Gamma$ be a discrete two-generator subgroup of $N$ such that $N/\Gamma$ is compact. Then we may decompose ${L^2}(N/\Gamma )$ into primary summands with respect to the right regular representation $R$ of $N$ on ${L^2}(N/\Gamma )$ as follows: ${L^2}(N/\Gamma ) = \oplus \sum \nolimits _{m \in {\mathbf {Z}}} {{H_m}(\Gamma )}$. It can be shown that for $m \ne 0,{H_m}(\Gamma )$ is a multiplicity space for the representation $R$ of multiplicity $\left | m \right |$. The distinguished subspace theory of ${\text {L}}$. Auslander and ${\text {J}}$. Brezin singles out a finite number of the decompositions of ${H_m}(\Gamma ),m \ne 0$, which are in some ways nicer than the others. They define algebraically an integer valued function, called the index, on the set ${\Omega _m}$ of irreducible closed $R$-invariant subspaces of ${H_m}(\Gamma )$ such that the distinguished subspaces have index one. In this paper, we give an analytic-geometric interpretation of the index. Every space in ${\Omega _m}$ contains a unique (up to constant multiple) special function, called a nil-theta function, that arises as a solution of a certain differential operator on $N/\Gamma$. These nil-theta functions have been shown to be closely related to the classical theta functions. Since the classical theta functions are determined (up to constant multiple) by their zero sets, it is natural to attempt to classify the spaces in ${\Omega _m}$ using various properties of the zero sets of the nil-theta functions lying in these spaces. We define the index of a nil-theta function in ${H_m}(\Gamma )$ using the symmetry properties of its zero set. Our main theorem asserts that the algebraic index of a space in ${\Omega _m}$ equals the index of the unique nil-theta function lying in that space. We have thus an analytic-geometric characterization of the index. We then use these results to give a complete description of the zero sets of those nil-theta functions of a fixed index. We also investigate the behavior of the index under the multiplication of nil-theta functions; i.e. we discuss how the index of the nil-theta function $FG$ relates to the indices of the nil-theta functions $F$ and $G$.