Twisted Selberg Zeta Function

• Twisted Selberg zeta function is a complex-analytic spectral invariant that encodes geometric and representation-theoretic data via closed geodesics on compact hyperbolic manifolds.
• It employs methods such as heat trace expansions and transfer operator techniques to establish meromorphic continuation and determinant representations on locally symmetric spaces.
• The function reveals deep connections between analytic torsion, spectral geometry, and prime geodesic theorems, with implications for arithmetic quantum chaos and orbifold extensions.

The twisted Selberg zeta function is a complex-analytic, dynamical spectral invariant attached to a compact hyperbolic manifold or orbifold and a finite-dimensional (not necessarily unitary) representation of its fundamental group. It generalizes Selberg’s original zeta function by incorporating arbitrary (possibly non-unitary) twists, thereby encoding finer geometric, topological, and representation-theoretic data via closed geodesics and the spectrum of naturally associated differential operators. The adelic, functional equation, meromorphic continuation, and determinant formula aspects of the twisted Selberg zeta reveal profound links between representation theory, analytic number theory, and global analysis on locally symmetric spaces.

1. Precise Definition and Twisted Product Structure

Let $X = \Gamma \backslash \widetilde{X}$ be a compact real hyperbolic manifold of odd dimension $d=2n+1$, with $\Gamma$ a discrete, cocompact lattice in $G = SO^0(d,1)$ or its spin cover. Let $\chi : \Gamma \to GL(V_\chi)$ be a finite-dimensional complex representation (not assumed unitary), and $\sigma \in \widehat{M}$ an irreducible unitary representation of $M$, the centralizer of the Cartan subgroup $A$ in the maximal compact subgroup $K$.

Each nontrivial hyperbolic $\gamma \in \Gamma$ is conjugated in $G$ as $g^{-1}\gamma g = m_\gamma a_\gamma$, with $m_\gamma \in M$, $a_\gamma \in A \simeq \mathbb{R}^+$, and $l(\gamma)>0$ the length of the corresponding closed geodesic.

The twisted Selberg zeta function is defined by the convergent infinite product over primitive conjugacy classes: $$Z(s;\sigma,\chi) = \prod_{[\gamma] \text{ prime}} \prod_{k=0}^\infty \det\left[ \mathrm{Id} - \left(\chi(\gamma) \otimes \sigma(m_\gamma) \otimes S^k(\mathrm{Ad}(m_\gamma a_\gamma)|_{\overline{\mathfrak n}})\right) e^{-(s+|\rho|)l(\gamma)} \right]$$ where $|\rho| = \tfrac{1}{2} \dim \mathfrak{n}$ is the half-sum of positive roots, $\overline{\mathfrak n}$ is the opposite nilpotent subalgebra, and $S^k$ refers to the $k$-th symmetric power representation. For surfaces and orbifolds, $\chi$ may be defined on $\pi_1(X_1)$, often incorporating contributions from elliptic and parabolic classes as in the general determinant factorizations (Fedosova, 2015, Jorgenson et al., 18 Dec 2025).

The logarithmic derivative isolates geometric contributions: $$\frac{d}{ds} \log Z(s;\sigma,\chi) = \sum_{[\gamma] \neq e} \frac{l(\gamma)}{n_\Gamma(\gamma)} L_\mathrm{sym}(\gamma;\sigma) e^{-s l(\gamma)}$$ with $$L_\mathrm{sym}(\gamma;\sigma) = \frac{\operatorname{tr}(\chi(\gamma) \otimes \sigma(m_\gamma)) e^{-|\rho| l(\gamma)}}{\det[\mathrm{Id} - \mathrm{Ad}(m_\gamma a_\gamma)|_{\overline{\mathfrak n}}]}$$ (Spilioti, 2015).

2. Analytic Properties and Meromorphic Continuation

For sufficiently large $\Re s$, $Z(s;\sigma,\chi)$ converges absolutely and uniformly on compacta. Meromorphic continuation to the entire complex plane $\mathbb{C}$ is established via two main analytic approaches:

• Heat Trace and Resolvent Methods: Using the asymptotic expansion of $\operatorname{Tr} e^{-t A_\chi^\sharp(\sigma)}$ for the twisted Bochner–Laplace operator $A_\chi^\sharp(\sigma)$ acting on sections of $E(\sigma) \otimes V_\chi$, the spectral theory yields the extension, placing zeros and poles of $Z(s;\sigma,\chi)$ at values determined by the eigenvalues of $A_\chi^\sharp(\sigma)$ (Spilioti, 2015, Fedosova, 2015, Spilioti, 2015).
• Transfer Operator Formalism: For geometrically finite Fuchsian groups $\Gamma$ and twists with non-expanding cusp monodromy, $Z_{\Gamma,\chi}(s)$ can be realized as the Fredholm determinant of a transfer operator $\mathcal{L}_{s,\chi}$, allowing meromorphic continuation with singularities prescribed by the Jordan block structure of the twist at parabolics (Fedosova et al., 2017).

Spectral multiplicities precisely control the order of zeros and poles (Spilioti, 2015, Frahm et al., 2021):

Feature	Description	Governing Spectrum
Zeros/Poles	Location on $s=\pm i \sqrt{t_k}$	$A_\chi^\sharp(\sigma)$
Multiplicity	Algebraic multiplicity $m(t_k)$	Twisted Laplacian

3. Functional Equation and Symmetry Aspects

Twisted Selberg zeta functions typically satisfy deep functional equations relating the values at $s$ and $-s$. For Weyl-invariant twists, the key result is (Spilioti, 2015): $$\frac{Z(s;\sigma,\chi)}{Z(-s;\sigma,\chi)} = \exp\left(-4\pi\, \dim V_\chi\, \operatorname{Vol}(X)\, \int_0^s P_\sigma(r) dr\right)$$ where $P_\sigma$ is the Plancherel polynomial associated to $\sigma$.

For Riemann surfaces twisted by Tate motives, explicit characterization in terms of “absolute automorphy” conditions on the twisting Laurent polynomial determine the functional equation form (Koyama et al., 2020), often up to explicit gamma or sine factors.

Geometrically, the symmetry $s \mapsto -s$ in eigenvalue locations is reflected in spectral parity and trace formula invariance—functional symmetry is a direct analytic consequence of the evenness of $P_\sigma$ (Spilioti, 2015, Frahm et al., 2021).

4. Determinant Representations and Factorization

A foundational principle is the identification of twisted Selberg zeta functions as regularized determinants of naturally associated twisted Laplacians. The general determinant representation is (for Weyl-invariant $\sigma$, all $s$): $$Z(s; \sigma, \chi) = {\det}_{\mathrm{gr}}(A_\chi^\sharp(\sigma) + s^2)\, \exp \left(-2\pi\, \dim V_\chi\, \operatorname{Vol}(X)\, \int_0^s P_\sigma(t) dt\right)$$ where the graded determinant is constructed via zeta regularization of the spectrum of the Laplace operator (Spilioti, 2015).

On hyperbolic surfaces and orbifolds, more refined factorizations split the determinant into hyperbolic, elliptic, identity, and parabolic parts (Fedosova, 2015, Jorgenson et al., 18 Dec 2025, Momeni et al., 2011):

Factor	Contribution	Operator/Formula
Hyperbolic	Primitive closed geodesics	Selberg zeta $Z(s; \rho)$
Identity	Volume/Barnes $G$ contributions	$Z_I(s; \rho)$
Elliptic	Pointwise elliptic data	$Z_\text{ell}(s; \rho)$
Parabolic	Cuspidal singularities	$Z_P(s; \rho)$

Explicit Laplace–Mellin integral transforms relate the logarithmic derivatives of $Z(s; \rho)$ and the regularized determinant of the differential operator, including an explicit “torsion factor” reflecting topological (Euler characteristic) and representation-theoretic data (Jorgenson et al., 18 Dec 2025).

5. Geometric, Spectral, and Topological Interpretations

The twisted Selberg zeta function encodes joint information about the length spectrum of closed geodesics weighted by representation-theoretic data and the spectrum of naturally twisted Laplace operators. Its determinant representation provides spectral invariants such as analytic torsion and links to Ruelle zeta functions and dynamical invariants (Spilioti, 2015, Spilioti, 2015, Frahm et al., 2021).

The asymptotics of the torsion factor in determinant relations (for high-dimensional, non-unitary twists) coincide, up to universal constants, with the behavior of higher-dimensional Reidemeister torsion, illuminating parallels between analytic and combinatorial topology (Jorgenson et al., 18 Dec 2025).

Spectral zeros and poles correspond exactly to Laplace or Dirac eigenvalues, and their orders are given by algebraic multiplicities, supporting applications in spectral geometry, prime geodesic theorems, and analytic number theory (Fedosova, 2015, Frahm et al., 2021).

6. Orbifold, Non-Unitary Extensions, and Transfer Methods

The theory extends to orbifolds and possibly non-unitary twists, incorporating elliptic and parabolic contributions in the trace/determinant formulas. For orbifolds, such terms contribute explicit polynomial corrections, affecting normalization but not spectral locations of zeros (Fedosova, 2015).

For geometrically finite but non-compact groups, convergence and analytic properties depend critically on the “non-expanding cusp monodromy” of the twist. In these cases, transfer operator approaches yield robust analytic continuation, and Venkov–Zograf factorization identities persist (Fedosova et al., 2017).

In all scenarios, the non-unitary nature of the twist enters through the trace and determinant weights, modifying both geometric and spectral contributions and the location/multiplicity of singularities.

7. Applications, Generalizations, and Open Problems

Twisted Selberg zeta functions have applications in the theory of analytic torsion, spectral geometry, dynamical systems (Ruelle zeta), automorphic forms, and arithmetic quantum chaos. They enable explicit calculation of invariants for locally symmetric spaces, including orbifolds and higher rank generalizations.

Current research lines include the extension to infinite-dimensional representations, non-Fuchsian or non-hyperbolic spaces, and achieving full trace-formula spectral interpretation for general non-unitary twists. The correspondence between spectral torsion factors and combinatorial torsion remains an active area, with ongoing work illuminating deep topological and analytic structures (Jorgenson et al., 18 Dec 2025, Spilioti, 2015, Momeni et al., 2011).

Open problems include:

• Complete spectral classification of zeros/poles for general twists.
• Explicit trace formulae for general orbifolds and non-unitary representations.
• Functional equations for non-Weyl-invariant twists and their spectral consequences.