D-modules on representations of Capelli type

JOURNAL OF ALGEBRA

卷:479

Article

Nang, Philibert^1,2 

[1] ENS, Lab Rech & Math, BP 8637, Libreville, Gabon

[2] Max Planck Inst Math, Vivatsgasse 7, D-53111 Bonn, Germany

关键词: D-modules;    Holonomic V-modules;    Invariant differential operators;    Irreducible representations;    Prehomogeneous vector spaces;    Multiplicity-free spaces;    Capelli identity;    Representations of Capelli type;   

DOI  :  10.1016/j.jalgebra.2016.12.029

来源: Elsevier

【 摘 要 】

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.

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