Symmetric Lie superalgebras and quantum integrability

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Symmetric Lie superalgebra is a complex Lie superalgebra with an involutive automorphism. All involutive automorphisms of simple Lie superalgebras were classified by Serganova, so the list of all symmetric simple Lie superalgebras is known.In contrast to the Lie algebra case the theory of spherical functions for symmetric Lie superalgebras is at a very early stage. The proposed approach to this difficult programme is based on the theory of quantum integrable systems. It goes back to an important observation of Sergeev (2001), who discovered a relation of spherical functions of one of the classical series with the theory of deformed quantum Calogero-Moser systems developed earlier by Chalykh, Feigin and Veselov.A particular case of spherical functions are the characters of finite-dimensional irreducible representations, which generate the Grothendieck ring of the corresponding Lie superalgebra. For basic classical Lie superalgebras these rings were recently explicitly described using Serganova's notion of generalised root systems.The theory of the deformed CM systems provides certain deformations of these Grothendieck rings with the action of the deformed CM operators and their quantum integrals.The conjecture is that the algebra of spherical functions for basic classical symmetric Lie superalgebras can be described as a specialisation of the corresponding family and can be studied using the spectral decomposition of the deformed CM operators.The approach was already very successful in the representation theory of orthosymplectic Lie superalgebras:it was shown that a suitable limit of the super Jacobi polynomials (which are the eigenfunctions of the corresponding deformed CM operators) are nothing else but the Euler characters studied by Penkov and Serganova.It is natural to expect that a similar phenomenon happens for the spherical functions as well.