q-Analogue of the degree zero part of a rational Cherednik algebra

Martin Vrabec (University of Glasgow)

Tuesday 28th March, 2023 16:00-17:00 Maths 311B

Abstract

The degree zero subalgebra of a rational Cherednik algebra is interesting from the point of view of algebra, integrable systems, as well as geometry. This subalgebra is a flat deformation of the skew product of a finite Coxeter group with a quotient of the universal enveloping algebra of gl(n), and it is related to generalised Howe duality.

Inside a double affine Hecke algebra, which depends on two parameters q and t, we define a subalgebra A that may be thought of as a q-deformation of the degree zero part of the corresponding rational Cherednik algebra. We prove that the algebra A is a flat t-deformation of the skew product of a symmetric group with the image of the Drinfeld–Jimbo quantum group for gl(n) under the q-oscillator (Jordan-Schwinger) representation. We find all the defining relations and an explicit PBW basis for the algebra A. We describe its centre and establish a double centraliser property. Further, we develop the connection with integrable systems introduced by van Diejen, which we also generalise. This talk is based on joint work with Misha Feigin.

 

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