Explicit multiplicities in the cuspidal spectrum of SU(n,1)
Alexander Stadler
TL;DR
This work delivers an explicit multiplicity formula for integrable discrete series in the cuspidal spectrum of $SU(n,1)$ with principal congruence subgroups $\Gamma(m)$, derived via the Selberg trace formula and parametrized by the Harish-Chandra parameter $\tau$. Central to the result is a closed expression for $m(\Gamma(m),\pi_\tau)$ in terms of $d_\tau$, $\mathrm{vol}(\Gamma\backslash G)$, $h_m$, the discriminant $D_\ell$, and special values $L_\ell(k)$, with a clean parity-dependent refinement incorporating an error term for even $n$. The paper further connects spectral data to arithmetic by computing explicit Volumes through Prasad’s formula, and by bounding error terms, enabling concrete growth estimates for cuspidal cohomology dimensions in towers of congruence subgroups. A notable contribution is the explicit rationality result for $\sqrt{|D_\ell|}\,\frac{L_\ell(2n+1)}{(2\pi)^{2n+1}}$, mirroring Siegel-type rationality in the imaginary quadratic setting and tying spectral multiplicities to values of $L$-functions. Overall, the results extend complex hyperbolic (rank-one) spectral analysis and provide arithmetic consequences for cuspidal cohomology and $L$-function values.”
Abstract
This paper investigates the cuspidal spectrum of the quotient of the real Lie group $G= SU(n,1)$ and a principal congruence subgroup $Γ(m)$ for $m\geq 3$, focusing on the multiplicities of integrable discrete series representations. Using the Selberg trace formula, we derive an explicit formula for the multiplicity $m(Γ(m), π_τ)$ of a representation $π_τ$ of integrable discrete series of $G$ within $L^2(Γ(m) \backslash G)$. The formula involves the Harish-Chandra parameter $τ$, the discriminant $D_\ell$ of the imaginary quadratic field $\ell$ over which $G$ is defined and special values of the Dirichlet $L$-function $L_\ell$ associated to $\ell$. We apply these results on the one hand to compute the cuspidal cohomology of locally symmetric spaces $Γ(m) \backslash G / K$, where $K$ is a maximal compact subgroup of $G$. On the other hand we use them to reprove a known rationality result involving the values of $L_\ell$ at odd positive integers and make them more explicit. This work extends previous studies on real and quaternionic hyperbolic spaces to the complex hyperbolic case, contributing to the understanding of the spectrum of $\mathbb{R}$-rank one algebraic groups.