Computational aspects of Calogero–Moser spaces

Abstract

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the $$\mathbb {C}^\times $$ C × -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a $$\mathbb {Q}$$ Q -factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.

On the Calogero–Moser space associated with dihedral groups II. The equal parameter case

“…Proof. By [8], a presentation of 𝑍 𝑐 is obtained by deforming the generators of 𝑍 0 = C[𝑉 × 𝑉 * ] 𝑊 and deforming the relations. Therefore, in order to prove the theorem, it is sufficient to check that the relations given in the statement are satisfied.…”