Integrability and singularities of Harish-Chandra characters

Let $G$ be a reductive group over a local field $F$ of characteristic $0$. By Harish-Chandra's regularity theorem, the character $\Theta_{\pi}$ of an irreducible, admissible representation $\pi$ of $G$ is given by a locally integrable function $\theta_{\pi}$ on $G$. It is a natural question whether $\theta_{\pi}$ has better integrability properties, namely, whether it is locally $L^{1+\epsilon}$-integrable for some $\epsilon>0$. It turns out that the answer is positive, and this gives rise to a new singularity invariant of representations $\epsilon_{\star}(\pi):=\sup\left\{ \epsilon:\theta_{\pi}\in L_{Loc}^{1+\epsilon}(G)\right\} $, which we explore in this paper. We provide a lower bound on $\epsilon_{\star}(\pi)$ which depends only on the absolute root system of $G$, and explicitly determine $\epsilon_{\star}(\pi)$ in the case of a $p$-adic $\mathrm{GL}_{n}$. This is done by studying integrability properties of the Fourier transforms $\widehat{\xi}_{\mathcal{O}}$ of stable Richardson nilpotent orbital integrals $\xi_{\mathcal{O}}$. We express $\epsilon_{\star}(\widehat{\xi}_{\mathcal{O}})$ as the log-canonical threshold of a suitable relative Weyl discriminant, and use a resolution of singularities algorithm coming from the theory of hyperplane arrangements, to compute it in terms of the partition associated with the orbit. We obtain several applications; firstly, we provide bounds on the multiplicities of $K$-types in irreducible representations of $G$ in the $p$-adic case, where $K$ is an open compact subgroup. We further obtain bounds on the multiplicities of the irreducible representations appearing in the space $L^{2}(K/L)$, where $K$ is a compact simple Lie group, and $L\leq K$ is a Levi subgroup. Finally, we discover surprising applications in random matrix theory, namely to the study of the eigenvalue distribution of powers of random unitary matrices.