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- W2176529541 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." @default.
- W2176529541 created "2016-06-24" @default.
- W2176529541 creator A5005612207 @default.
- W2176529541 date "2015-11-25" @default.
- W2176529541 modified "2023-09-27" @default.
- W2176529541 title "The Lerch zeta function and the Heisenberg group" @default.
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