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Register a new userHolonomicity of relative characters and applications to multiplicity bounds for spherical pairs
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In this paper, we prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results from [KO13,KS16] on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. In order to deduce this application we prove relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group $G$ defined over $mathbb{R}$, the space of $G(mathbb{R})$-equivariant distributions on the manifold of real points of any algebraic $G$-manifold $X$ is finite-dimensional if $G$ has finitely many orbits on $X$.
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These notes are an expanded version of a talk given by the second author. Our main interest is focused on the challenging problem of computing Kronecker coefficients. We decided, at the beginning, to take a very general approach to the problem of studying multiplicity functions, and we survey the various aspects of the theory that comes into play, giving a detailed bibliography to orient the reader. Nonetheless the main general theorems involving multiplicities functions (convexity, quasi-polynomial behavior, Jeffrey-Kirwan residues) are stated without proofs. Then, we present in detail our approach to the computational problem, giving explicit formulae, and outlining an algorithm that calculate many interesting examples, some of which appear in the literature also in connection with Hilbert series.
Consider the restriction of an irreducible unitary representation $pi$ of a Lie group $G$ to its subgroup $H$. Kirillovs revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ u$ occurring in the restriction $pi|_H$ could be read from the coadjoint action of $H$ on $O^G cap pr^{-1}(O^H)$ provided $pi$ and $ u$ are geometric quantizations of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: sqrt{-1} g^{ast} to sqrt{-1} h^{ast}$ is the projection dual to the inclusion $h subset g$ of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits $O^G$ of a semisimple Lie group $G$ corresponding to highest weight modules of scalar type. We prove that the Corwin--Greenleaf number $sharp(O^G cap pr^{-1}(O^H))/H$ is either zero or one for any $H$-coadjoint orbit $O^H$, whenever $(G,H)$ is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits $O^H$ with nonzero Corwin-Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as classical limits of the multiplicity-free branching laws of holomorphic discrete series representations (T.Kobayashi [Progr.Math.2007]).
Let $G$ be a real reductive algebraic group, and let $Hsubset G$ be an algebraic subgroup. It is known that the action of $G$ on the space of functions on $G/H$ is tame if this space is spherical. In particular, the multiplicities of the space $mathcal{S}(G/H)$ of Schwartz functions on $G/H$ are finite in this case. In this paper we formulate and analyze a generalization of sphericity that implies finite multiplicities in $mathcal{S}(G/H)$ for small enough irreducible representations of $G$.
Following the ideas of Ginzburg, for a subgroup $K$ of a connected reductive $mathbb{R}$-group $G$ we introduce the notion of $K$-admissible $D$-modules on a homogeneous $G$-variety $Z$. We show that $K$-admissible $D$-modules are regular holonomic when $K$ and $Z$ are absolutely spherical. This framework includes: (i) the relative characters attached to two spherical subgroups $H_1$ and $H_2$, provided that the twisting character $chi_i$ factors through the maximal reductive quotient of $H_i$, for $i = 1, 2$; (ii) localization on $Z$ of Harish-Chandra modules; (iii) the generalized matrix coefficients when $K(mathbb{R})$ is maximal compact. This complements the holonomicity proven by Aizenbud--Gourevitch--Minchenko. The use of regularity is illustrated by a crude estimate on the growth of $K$-admissible distributions which based on tools from subanalytic geometry.
The goal of this paper is to establish relative perturbation bounds, tailored for empirical covariance operators. Our main results are expansions for empirical eigenvalues and spectral projectors, leading to concentration inequalities and limit theorems. Our framework is very general, allowing for a huge variety of stationary, ergodic sequences, requiring only $p > 4$ moments. One of the key ingredients is a specific separation measure for population eigenvalues, which we call the relative rank. Developing a new algebraic approach for relative perturbations, we show that this relative rank gives rise to necessary and sufficient conditions for our concentration inequalities and limit theorems.
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