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Symmetric superspaces: slices, radial parts, and invariant functions. (English) Zbl 1371.58011

Krause, Henning (ed.) et al., Representation theory – Current trends and perspectives. In part based on talks given at the last joint meeting of the priority program in Bad Honnef, Germany, in March 2015. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-171-2/hbk; 978-3-03719-671-7/ebook). EMS Series of Congress Reports, 1-11 (2017).
Summary: We study the restriction of invariant polynomials on the tangent space of a Riemannian symmetric supermanifold to a Cartan subspace. We survey known results in the case the symmetric space is a Lie supergroup, and more generally, where the Cartan subspace is even. We then describe an approach to this problem, developed in joint work in progress with K. Coulembier, based on the study of radial parts of differential operators. This leads to a characterisation of the invariant functions for an arbitrary linear isometric action, and as a special case, to a Chevalley restriction theorem valid for the isotropy representation of any contragredient Riemannian symmetric superspace.
For the entire collection see [Zbl 1357.14004].

MSC:

58C50 Analysis on supermanifolds or graded manifolds
58E40 Variational aspects of group actions in infinite-dimensional spaces
17B20 Simple, semisimple, reductive (super)algebras
17B35 Universal enveloping (super)algebras
53C35 Differential geometry of symmetric spaces
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