# Workshop on Symmetric Spaces, Their Generalizations, and Special Functions

## Description

Symmetric spaces play a pivotal role in connecting Lie theory to many other areas of mathematics, most notably to harmonic analysis and the theory of special functions. Since the pioneering works of Harish-Chandra and Helgason, an enormous amount of research has been conducted on invariant differential operators of symmetric spaces to establish these connections. The theory of Jack and Macdonald polynomials, which represents an extension of spectral analysis on symmetric spaces, is one of the highlights of this research. The Jack polynomials arise as spherical functions on symmetric spaces. The Macdonald polynomials, on the other hand, are the incarnation of spherical polynomials in the context of a far-reaching generalization of invariant differential operators, called double-affine Hecke algebras. They serve as unifying objects in the theory of special functions, in that their limits yield many families of orthogonal polynomials such as Jack, Hall–Littlewood, and Askey–Wilson polynomials. Macdonald polynomials have also resulted in new links to combinatorics and knot theory.

In the 1990’s, Kostant and Sahi investigated eigenvalues of variants of the celebrated Capelli operator. It turns out that these operators are special cases of a natural basis of invariant differential operators (the Capelli basis) on certain symmetric spaces. To compute the eigenvalues of the operators in the Capelli basis, Sahi introduced a general class of polynomials, depending on several parameters, which are characterized by symmetry and vanishing conditions. Later, Knop and Sahi showed that the top-degree homogeneous part of a one-parameter subfamily of the latter polynomials yields the Jack polynomials. Other important results on these polynomials were obtained by Okounkov and Olshanski.

Quite recently, the aforementioned ideas are being extended to the setting of Lie superalgebras. Real forms and symmetric spaces of simple Lie superalgebras were classified by Serganova, and certain important ingredients of harmonic analysis, including the Chevalley restriction theorem and the Harish-Chandra isomorphism, have been obtained by Sergeev, Kac, Gorelik, and Alldridge. Nevertheless, many aspects of harmonic analysis on superspaces, such as spherical functions and the theory of spherical representations, are still being developed. In the past few years, the organizers together with Alldridge and Serganova have defined an analogous Capelli basis in the super setting, and computed explicit formulas for their eigenvalues. These formulas involve either the Sergeev-Veselov polynomials or the factorial Schur $Q$-polynomials of Ivanov and Okounkov, depending on whether the restricted roots of the symmetric space constitute a graded root system of type $A(m,n)$ or of type $Q(n)$. For symmetric spaces corresponding to root systems of type $BC(m,n)$, the analogous phenomena are less understood and many problems remain open, which we hope to explore during the conference.

The quantized symmetric spaces constitute yet another framework where special functions are expected to arise naturally. Following the introduction of quantized enveloping algebras, Koornwinder and Macdonald raised the question of defining quantizations of symmetric spaces and studying the theory of spherical functions. The coordinate ring of a quantized symmetric space is a suitable $q$-deformation of the ring of regular functions on the corresponding classical space. Significant results on characterization of spherical functions of these spaces have been obtained by Koornwinder, Noumi, Etingof-Kirillov, Dijkhuizen-Stokman, and Letzter. There is strong evidence that one can construct $q$-versions of generalized Capelli operators acting on quantized coordinate rings. It would be extremely interesting to investigate to what extent the properties and applications of these operators in the classical setting are paralleled in the quantum context.