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 ${⨁}_{N\in \mathbb{Z}}{H}_N$, where each space $H_N(N\ne 0)$ consists of $|N|$ copies of an irreducible representation of $H(\mathbb{R})$ with central character $e^{2\pi iNz}$. 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 $(\mathrm{mod}d)$ 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 $N\ne 0$ 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.

v3. 63 pages, corrections plus additional material in first two sections v4. 56 pages, corrections

References in corpus (1)

• The Lerch zeta function III. Polylogarithms and special values