Abstract
Abstract We study the endomorphism algebras of Verma modules for rational Cherednik algebras at t = 0. It is shown that, in many cases, these endomorphism algebras are quotients of the centre of the rational Cherednik algebra. Geometrically, they define Lagrangian subvarieties of the generalized Calogero–Moser space. In the introduction, we motivate our results by describing them in the context of derived intersections of Lagrangians.
Highlights
The quest for a direct bridge between the geometric world of Fukaya categories and the world of microlocal sheaves on a symplectic manifold is ongoing; it is currently the subject of intense research
Though considerable progress has recently been made, for instance with Tamarkin’s seminal work on quantizations of Lagrangians [32], it seems that concrete, computable examples of this expected correspondence are still desirable to aid one’s intuition. One abode where such computable, though non-trivial, examples live is that of rational Cherednik algebras, beginning for instance with the results of [29]
Though we have no idea what the appropriate definition of Fukaya category should be in this case, or its possible relation to microlocal sheaves on the generalized Calogero–Moser space X, we study in this article a shadow of such a hoped for relationship
Summary
The quest for a direct bridge between the geometric world of Fukaya categories and the world of (algebraic or analytic) microlocal sheaves on a symplectic manifold is ongoing; it is currently the subject of intense research. One major advantage of working on the generalized Calogero–Moser space, even though it is a singular symplectic variety, is that it is equipped with a canonical quantization by virtue of the fact that it is the centre of the rational Cherednik algebra. A pair of (left or right) quantizable modules M and N are said to have smooth intersection if SuppM ∩ SuppN is contained in the smooth locus of Xc. M is said to be simple if eM is a cyclic Zc-module; the terminology comes from the representation theory of deformation-quantization algebras. Since the definition of Δ(p, λ) makes sense over Ht,c, it is a quantizable module It has recently been shown by Bonnafe and Rouquier [12] that each block Ω of the Calogero–Moser partition of Irr(W ) has a canonical representative λΩ.
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