Parabolic induction and restriction functors for rational Cherednik algebras

Abstract

In this paper we introduce and study a categorical action of the positive part of the Heisenberg Lie algebra on categories of modules over rational Cherednik algebras associated to symmetric groups. We show that the generating functor for this action is exact. We then produce a categorical Heisenberg action on the categories O and show it is the same as one constructed by Shan and Vasserot. Finally, we reduce modulo a large prime p. We show that the functors constituting the action of the positive half of the Heisenberg algebra send simple objects to semisimple ones, and we describe these semisimple objects. Contents 1. Introduction 2. Rational Cherednik algebras 2.1. Definition 2.2. Category O 2.3. Bezrukavnikov-Etingof functors 2.4. Supports 2.5. Shan-Vasserot construction 2.6. Local system of restrictions 3. Quantum Hamiltonian reduction 3.1. Algebra isomorphism 3.2. Etale lifts, I 3.3. Comparison of isomorphisms 3.4. Parametrization of simples 4. Parabolic induction of D-modules 4.1. Categories of equivariant D-modules 4.2. Inverse and direct images 4.3. Induction functors 4.4. Supports 4.5. Application to spherical Cherednik algebras 4.6. Translation and wall-crossing bimodules 5. Heisenberg generator functor 5.1. D-module M ′ λ 5.2. Basics on Harish-Chandra bimodules 5.3. Harish-Chandra bimodules and induction 5.4. Etale lifts, II 5.5. Proof of Proposition 5.7 5.6. Compatibility with wall-crossing 6. Functors between categories O 6.1. Construction and main result 1 2

Rational Cherednik Algebras

“…Using the restriction functors, one can show that Σ(W ) is the union of the Σ(W ) for proper parabolic subgroups W < W and of the set of finite dimensional aspherical representations of H 1,c (W ). For W = S n , this observation allows an inductive determination of the aspherical values, [10]. Remarkably, Bezrukavnikov and Etingof note that the number of aspherical representations matches phenomena in the (C * ) 2 -equivariant small quantum cohomology of Hilb n (C 2 ).…”

“…This an analogue of the BGG category O for semisimple Lie algebras. There are related versions of O c (W ) where h acts by non-zero eigenvalues, but [10] shows that such categories are equivalent to O c (W ) for some subgroup W of W .…”

“…The induction and restriction functors help to determine the set of aspherical values of W , [10]. Using the restriction functors, one can show that Σ(W ) is the union of the Σ(W ) for proper parabolic subgroups W < W and of the set of finite dimensional aspherical representations of H 1,c (W ).…”

Equivariant Slices for Symplectic Cones

“…In fact, the above statement is a corollary of a sheafified statement [Los12, Theorem 2.5.3], which gives a C × -equivariant isomorphism of sheaves of Sym c ⋊ Γ-algebras on the symplectic leaf containinḡ x. We note that in the case where Γ < GL(W ) for W ⊆ V a Lagrangian subspace (so V ∼ = T * W and hence GL(W ) < Sp(V ), one can instead use the argument of [BE09], which is simpler, and again complete along the punctured line instead of the point. We omit further details.…”

On the Smoothness of Centres of Rational Cherednik Algebras in Positive Characteristic

“…The main result of this section, Corollary 5.5, shows that the centre Z c (W, V ) of the rational Cherednik algebra being a regular algebra is equivalent to the block partition of H c (W ) consisting of singletons for all parabolic subgroups W of W . Our proof of this result is based on identifying, as in [5], a certain completion of H c (W ) with an algebra of matricies over a completion of H c (W ) for a suitable parabolic subgroup W of W . Applying Corollary 5.5 to the results of the previous section, we describe precisely for which parameters c the algebra Z c (W, V ) is regular when W = G(m, 1, n).…”