Zeroes of primary summand functions on compact solvmanifolds (Q1196577)

Let \(G\) be a connected and simply connected solvable Lie group with Lie algebra \(\mathfrak G\). Let \(\Gamma\) be a discrete subgroup of \(G\) such that \(M=\Gamma\backslash G\) is compact. We consider the quasi-regular representation \(R\) of \(G\) on \(M\), namely right translation of \(G\) on \(L^ 2(M)\). Then it is well-known that \(L^ 2(M)\) decomposes into the direct sum \(\oplus{\mathcal H}_ \pi\) of mutually orthogonal invariant subspaces \({\mathcal H}_ \pi\), which are finite multiples of the irreducible representation \(\pi\) of \(G\) [cf. \textit{I. M. Gel'fand} et al.: Representation theory and automorphic functions (Moskva 1966; Zbl 0138.072)]. We denote by \((\Gamma\backslash G)^ \land\) the set of irreducible representations of \(G\) appearing in \(R\) and by \((\Gamma\backslash G)^ \land_ \infty\) the set of \(\pi\in(\Gamma\backslash G)^ \land\) having infinite dimension. When \(G\) is the 3-dimensional Heisenberg group \(H_ 3\), \textit{L. Auslander} and \textit{R. Tolimieri} [Abelian harmonic analysis, theta functions and function algebras on a nilmanifold (Lect. Notes Math. 436) (Springer 1975; Zbl 0321.43012)] proved that, for \(\pi\in(\Gamma\backslash G)^ \land_ \infty\), any continuous function in \({\mathcal H}_ \pi\subset L^ 2(\Gamma\backslash H_ 3)\) hast at least one zero on \(\Gamma\backslash H_ 3\). In this paper, the author extends the result of L. Auslander and R. Tolimieri to all compact nilmanifolds, and then studies the 3-dimensional solvable case to show that the result remains valid in most 3-dimensional compact solvmanifolds. The proof consists of Mackey theory or orbit method combined with some topological aspects of such solvmanifolds.