The Lerch zeta function and the Heisenberg group

Abstract

This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch $L$-functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group $H^{J}$, called here the sub-Jacobi group, which is a semi-direct product of $GL(1, {\mathbb R})$ with the Heisenberg group $H({\mathbb R})$. The Heisenberg group action on L^2-functions on the Heisenberg nilmanifold $H({\mathbb Z}) \backslash H({\mathbb R})$ decomposes as $\bigoplus_{N \in {\mathbb Z}} H_N$, where each space $H_N~ (N \neq 0)$ consists of $|N|$ copies of an irreducible representation of $H({\mathbb R})$ with central character $e^{2 \pi i Nz}$. The paper shows that show one can further decompose $H_N (N \ne 0)$ into irreducible $H({\mathbb R})$-modules $H_{N,d}(\chi)$ indexed by Dirichlet characters $(\bmod~ d)$ for $d \mid N$, each of which carries an irreducible $H^J$-action. On each $H_{N,d}(\chi)$ there is an action of certain two-variable Hecke operators $\{T_m: m \ge 1\}$; these Hecke operators have a natural global definition on all of $L^2(H({\mathbb Z})\backslash H({\mathbb R}))$, including the space of one-dimensional representations $H_0$. For $H_{N,d}(\chi)$ with $N \neq 0$ suitable Lerch $L$-functions on the critical line $\frac{1}{2} + it$ form a complete family of generalized eigenfunctions (pure continuous spectrum) for a certain linear partial differential operator $\Delta_L$. These Lerch $L$-functions are also simultaneous eigenfunctions for all two-variable Hecke operators $T_m$ and their adjoints $T_m^{\ast}$, provided $(m, N/d) = 1$. Lerch $L$-functions are characterized by this Hecke eigenfunction property.