Integrability and singularities of Harish-Chandra characters
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866910704861184000 |
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| author | Glazer, Itay Gordon, Julia Hendel, Yotam I. |
| author_facet | Glazer, Itay Gordon, Julia Hendel, Yotam I. |
| contents | Let $G$ be a reductive group over a local field $F$ of characteristic $0$. By Harish-Chandra's regularity theorem, the character $Θ_π$ of an irreducible, admissible representation $π$ of $G$ is given by a locally integrable function $θ_π$ on $G$. It is a natural question whether $θ_π$ has better integrability properties, namely, whether it is locally $L^{1+ε}$-integrable for some $ε>0$. It turns out that the answer is positive, and this gives rise to a new singularity invariant of representations $ε_{\star}(π):=\sup\left\{ ε:θ_π\in L_{Loc}^{1+ε}(G)\right\} $, which we explore in this paper.
We provide a lower bound on $ε_{\star}(π)$ which depends only on the absolute root system of $G$, and explicitly determine $ε_{\star}(π)$ in the case of a $p$-adic $\mathrm{GL}_{n}$. This is done by studying integrability properties of the Fourier transforms $\widehatξ_{\mathcal{O}}$ of stable Richardson nilpotent orbital integrals $ξ_{\mathcal{O}}$. We express $ε_{\star}(\widehatξ_{\mathcal{O}})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01591 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integrability and singularities of Harish-Chandra characters Glazer, Itay Gordon, Julia Hendel, Yotam I. Representation Theory Algebraic Geometry 20G05, 14B05, 20G05 (Primary) 14N20, 17B08, 22E30, 22E35, 22E46, 32S22, 43A30 (Secondary) Let $G$ be a reductive group over a local field $F$ of characteristic $0$. By Harish-Chandra's regularity theorem, the character $Θ_π$ of an irreducible, admissible representation $π$ of $G$ is given by a locally integrable function $θ_π$ on $G$. It is a natural question whether $θ_π$ has better integrability properties, namely, whether it is locally $L^{1+ε}$-integrable for some $ε>0$. It turns out that the answer is positive, and this gives rise to a new singularity invariant of representations $ε_{\star}(π):=\sup\left\{ ε:θ_π\in L_{Loc}^{1+ε}(G)\right\} $, which we explore in this paper. We provide a lower bound on $ε_{\star}(π)$ which depends only on the absolute root system of $G$, and explicitly determine $ε_{\star}(π)$ in the case of a $p$-adic $\mathrm{GL}_{n}$. This is done by studying integrability properties of the Fourier transforms $\widehatξ_{\mathcal{O}}$ of stable Richardson nilpotent orbital integrals $ξ_{\mathcal{O}}$. We express $ε_{\star}(\widehatξ_{\mathcal{O}})$ 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. |
| title | Integrability and singularities of Harish-Chandra characters |
| topic | Representation Theory Algebraic Geometry 20G05, 14B05, 20G05 (Primary) 14N20, 17B08, 22E30, 22E35, 22E46, 32S22, 43A30 (Secondary) |
| url | https://arxiv.org/abs/2312.01591 |