Explicit multiplicities in the cuspidal spectrum of SU(n,1)

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.