The Harish-Chandra isomorphism for reductive symmetric superpairs

Shaikh

2014

Forum math. Sigma

The superbosonization identity of Littelmann, Sommers and Zirnbauer is a new tool for use in studying universality of random matrix ensembles via supersymmetry, which is applicable to nonGaussian invariant distributions. We give a new conceptual interpretation of this formula, linking it to harmonic superanalysis of Lie supergroups and symmetric superspaces, and in particular, to a supergeneralization of the Riesz distributions. Using the super-Laplace transformation of generalized superfunctions, the theory of which we develop, we reduce the proof to computing the Gindikin gamma function of a Riemannian symmetric superspace, which we determine explicitly.

Section: Proposition 38 the Berezinian Density µ Is H -Invariantmentioning

confidence: 97%

“…On general grounds [2,1], the homogeneous superspace Ω = H.1 admits a nonzero H -invariant Berezinian density, unique up to a constant. Due to the special features of this example, we can give an explicit formula.…”

Section: Preliminariesmentioning

confidence: 99%

Superbosonization via Riesz Superdistributions

Shaikh

2014

Forum math. Sigma

“…In this subsection, we recall the relevant facts from [1], and extend these slightly to include cases such as gl n|n . Definition 1.10.…”

Section: Iwasawa Decompositionmentioning

confidence: 98%

“…∀x ∈ g, g ∈ G 0 : π g Ad(g)x = π 0 (g)π g (x)π 0 (g) −1 Remark 1.7. Some results concerning ordinary representations on locally convex vector spaces apply: (1), (2) and (4) imply together that π g | g0 ,R is continuous [25,Lemma 4.2].…”

Section: Representations Of Supergroup Pairsmentioning

confidence: 99%

Spherical representations of Lie supergroups

Journal of Functional Analysis

Schmittner

2015

The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair $(G,K)$ of even type. Along the way, we compute the Harish-Chandra $c$-function of the symmetric superspace $G/K$. By way of an application, we show that all spherical representations are self-dual in type AIII|AIII.Comment: 37 pages; title changed; substantially revised version; accepted for publication, J. Func. Anal. (2014

“…Besides its relation to representation theory [All10], this subject is of high current interest in mathematical physics, in the study of σ-model approximations of invariant random matrix ensembles, as are applied to disordered metals and topological insulators [Zir91, HHZ05, LSZ08, DSZ10, GLMZ11].…”

Section: Introductionmentioning

confidence: 99%

Berezin integration on non-compact supermanifolds

Journal of Geometry and Physics

Hilgert

Palzer

2012

Abstract. We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free repesentation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes's theorem is derived-here, the boundary integral contains transversal derivatives of arbitrarily high order.