Invariant holonomic systems on symmetric spaces and other polar representations

Abstract

Let g be a complex reductive Lie algebra, with adjoint group G, acting on a symmetric space V , with associated little Weyl group W and discriminant δ. Then G also acts on the ring of differential operators D(V ) and we write τ : g → D(V ) for the differential of this action. Consider the invariant holonomic systemIn the diagonal case, when V = g, this module has been intensively studied. For example, the fact that G has no δ-torsion factor module lies at the heart of Harish-Chandra's regularity theorem, while Hotta and Kashiwara have shown that G is semisimple, which has important consequences for the geometric representation theory of g.We study analogous problems for a symmetric space and, more generally, for a visible stable polar G-representation V . By work of Levasseur and the present authors, there exists a radial parts mapwhere Aκ(W ) is the spherical subalgebra of a Cherednik algebra Hκ(W ). When Aκ(W ) is simple, rad is surjective and we generalise work of Sekiguchi and Galina-Laurent by proving that G has no factor nor submodule that is δ-torsion. This answers a conjecture of Sekiguchi. Moreover we show that G is semisimple if and only if the Hecke algebra Hq(W ) associated to Aκ(W ) is semisimple, thereby answering a conjecture of Levasseur-Stafford.By twisting the radial parts map, we study a family of invariant holonomic systems and, more generally, families of admissible modules on V . We introduce shift functors that allow us to pass between different twists. We show that the image of each simple summand of G under these shift functors is described by Opdam's KZ-twist.