2011/01/03 by Alexey Petukhov, Alexey V. Petukhov, Petukhov, Alexey V.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1101.0472
The notion of $(\frak g, \frak k)$-module have been introduced by I. Penkov, V. Serganova, G. Zuckerman. Main ingredients are Hilbert-Mumford Criterion, A. Beilinson- J. Bernstein localization theorem, O. Gabber theorem
arxiv created 2011/01/03 · openalex publication_date 2011/01/03 · arxiv updated 2011/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \frak g be a reductive Lie algebra over an algebraically closed field of characteristic 0 and \frak k be a reductive in \frak g-subalgebra. Let M be a finitely generated (possibly, infinite-dimensional) \frak g-module. We say that M is a (\frak g, \frak k)-module if M is a direct sum of a (possibly, infinite) amount of simple finite-dimensional \frak k-modules. We say that M is of finite type if M is a (\frak g, \frak k)-module and Hom_\frak k(V, M)<∞ for any simple \frak k-module V. Let X be a variety of all Borel subalgebras of \frak g. Let M be a finitely generated (\frak g, \frak k)-module of finite type. In this article we prove that M is holonomic, i.e. M is governed by some subvariety LM⊂ X and some local system SM on it. Furthermore we provide a finite list in which LM necessarily appear.