A generalization of Springer theory using nearby cycles

• Main Author: Grinberg, Mikhail
• Format: Journal Article
• Language: English
• Published: 08-02-1998
• Subjects: Mathematics - Algebraic Geometry; Mathematics - Representation Theory
• DOI: 10.48550/arxiv.math/9802042

Let g be a complex semisimple Lie algebra, and f : g --> g/G the adjoint
quotient map. Springer theory of Weyl group representations can be seen as the
study of the singularities of f. We give a generalization of Springer theory to
visible, polar representations. It is a class of rational representations of
complex reductive groups, for which the invariant theory works by analogy with
the adjoint representations. Let G|V be such a representation, f : V --> V/G
the quotient map, and P the sheaf of nearby cycles of f. We show that the
Fourier transform of P is an intersection homology sheaf on V*. Associated to
G|V, there is a finite complex reflection group W, called the Weyl group of
G|V. We describe the endomorphism ring of P as a deformation of the group
algebra of W.