Rational Cherednik algebras and Schubert cells

TL;DR

This paper explores the connection between rational Cherednik algebras and Schubert cells, revealing how certain algebraic structures relate to geometric objects in the Calogero-Moser space and adelic Grassmannian.

Contribution

It establishes a precise relationship between Lagrangian subvarieties from Cherednik algebra representations and Schubert cells, demonstrating compatibility with space factorization.

Findings

Isomorphism respects factorization properties

Homomorphism spaces correspond to functions on Schubert intersections

Supports of Cherednik algebra representations form smooth Lagrangians

Abstract

The representation theory of rational Cherednik algebras of type A at t=0 gives rise, by considering supports, to a natural family of smooth Lagrangian subvarieties of the Calogero-Moser space. The goal of this article is to make precise the relationship between these Lagrangians and Schubert cells in the adelic Grassmannian. In order to do this we show that the isomorphism, as constructed by Etingof and Ginzburg, from the spectrum of the centre of the rational Cherednik algebra to the Calogero-Moser space is compatible with the factorization property of both of these spaces. As a consequence, the space of homomorphisms between certain representations of the rational Cherednik algebra can be identified with functions on the intersection Schubert cells.