D-modules on representations of Capelli type

Let (G, V) be an irreducible multiplicity-free finite-dimensional representation of a connected reductive complex group G, as classified by V.G. Kac [17], and G' its derived subgroup. Denote by g the Lie algebra of G, and U(g) its universal enveloping algebra. Assume that there exists a polynomial f generating the algebra of G'-invariant polynomials on V (C[V](G') similar or equal to C[f]) and such that f is not an element of C[V]G. Such representations are said to be of Capelli type if the algebra of G-invariant differential operators is the image of the center of U(2) under the differential of the G-action. They fall into eight cases given by R. Howe and T. Umeda [14]: five infinite families and three "exceptional" examples. We prove that the category of regular holonomic D-v-modules invariant under the action of G' is equivalent to the category of graded modules of finite type over a suitable algebra A, except for few special cases. Indeed the Levasseur's conjecture [28, Conjecture 5.17, p. 508] fails in these cases because of the disconnectedness of the stabilizers of some "smaller" orbits. (C) 2017 Elsevier Inc. All rights reserved.