Integrability and singularities of Harish-Chandra characters

Abstract

Let $G$ be a reductive group over a local field $F$ of characteristic $0$. By Harish-Chandra's regularity theorem, the character $\hat{I}{˜}_{\ddot{I}}€$ of an irreducible, admissible representation $\ddot{I}€$ of $G$ is given by a locally integrable function $\hat{I}{¸}_{\ddot{I}}€$ on $G$. It is a natural question whether $\hat{I}{¸}_{\ddot{I}}€$ has better integrability properties, namely, whether it is locally $L^{1+\hat{I}\mu }$-integrable for some $\hat{I}\mu >0$. It turns out that the answer is positive, and this gives rise to a new singularity invariant of representations $\hat{I}{\mu }_{\star }(\ddot{I}€):=\sup \left\{\hat{I}\mu :\hat{I}{¸}_{\ddot{I}}€\in L_{Loc}^{1+\hat{I}\mu }(G)\right\}$, which we explore in this paper. We provide a lower bound on $\hat{I}{\mu }_{\star }(\ddot{I}€)$ which depends only on the absolute root system of $G$, and explicitly determine $\hat{I}{\mu }_{\star }(\ddot{I}€)$ in the case of a $p$-adic ${\mathrm{GL}}_n$. This is done by studying integrability properties of the Fourier transforms $\hat{\hat{I}}{¾}_{\mathcal{O}}$ of stable Richardson nilpotent orbital integrals $\hat{I}{¾}_{\mathcal{O}}$. We express $\hat{I}{\mu }_{\star }(\hat{\hat{I}}{¾}_{\mathcal{O}})$ 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 $L\le K$ 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.

40 pages. This is a second version of the paper, where two new applications where added- estimates on the multiplicities of irreducible representations in compact homogeneous spaces, as well as some insights about the eigenvalue distribution of powers of random unitary matrices. Comments are welcome!