We survey a number of results about rational Cherednik algebra representation theory and its connection to symplectic singularities and their resolutions. Mathematics Subject Classification (2000). Primary 16G, 17B; Secondary 20C, 53D.

We present an explicit product formula for the spherical functions of the compact Gelfand pairs (G,K1) = (SU(p+ q), SU(p)× SU(q)) with p ≥ 2q, which can be considered as the elementary spherical functions of one-dimensional K-type for the Hermitian symmetric spaces G/K with K = S(U(p)× U(q)). Due to results of Heckman, they can be expressed in terms of Heckman-Opdam Jacobi polynomials of type B...

We study the rational Cherednik algebra attached to complex reflection group G(r, 1, 2). Each irreducible representation Sλ of 2) corresponds a standard module Δ(λ) for algebra. give necessary and sufficient conditions existence morphism between two these modules explicit formulas them when they exist.

We study unitarity of lowest weight irreducible representations of rational Cherednik algebras. We prove several general results, and use them to determine which lowest weight representations are unitary in a number of cases. In particular, in type A, we give a full description of the unitarity locus (justified in Subsection 5.1 and the appendix written by S. Griffeth), and resolve a question b...

We study unitarity of lowest weight irreducible representations of rational Cherednik algebras. We prove several general results, and use them to determine which lowest weight representations are unitary in a number of cases. In particular, in type A, we give a full description of the unitarity locus (justified in Subsection 5.1 and the appendix written by S. Griffeth), and resolve a question b...

Heckman-Opdam system of differential equations is holonomic , with regular singularities and has locally |W |-dimensional space of solutions ( cf. corollary 3.9 of [12]), where |W | is the cardinality of the Weyl group W . The system is a generalization of radial parts of Laplace-Casimir operators on symmetric Riemannian spaces of nonpositive curvature and is isomorphic to Calogero-Sutherland m...

Hopf-Hecke algebras and Barbasch-Sahi were defined by the first named author (2016) in order to provide a general framework for study of Dirac cohomology. The aim this paper is explore new examples these definitions contribute their classification. are distinguished an orthogonality condition PBW property. property such as ones considered here has been great interest literature we extend discus...

Abstract We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational the imprimitive complex reflection groups $G(m,p,n)$, when m is even. This gives a new construction mystic which have Artin–Schelter regular rings quantum polynomial invariants. As an application this result, we algebra has finite-dimensional representation if only its counterpart...

A survey is given about recent developments in special functions associated with root systems. The paper addresses a general mathematical audience and it does not assume earlier knowledge of either special functions or root systems. The emphasis is on multivariable orthogonal polynomials: the Heckman-Opdam Jacobi polynomials and Maconald’s q-analogues of these.