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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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A generalization of Springer theory using nearby cycles
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by Mikhail Grinberg PDF
Represent. Theory 2 (1998), 410-431 Request permission

Abstract:

Let $\mathfrak g$ be a complex semisimple Lie algebra, and $f : {\mathfrak {g}} \to G \backslash \backslash {\mathfrak {g}}$ the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of $f$. In this paper, we give a generalization of Springer theory to visible, polar representations. It is a class of rational representations of reductive groups over $\mathbb C$, for which the invariant theory works by analogy with the adjoint representations. Let $G | V$ be such a representation, $f : V \to G \backslash \backslash V$ 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 ${\mathrm {End}} (P)$ as a deformation of the group algebra ${\mathbb {C}} [W]$.
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Additional Information
  • Mikhail Grinberg
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Ave., Room 2-247, Cambridge, Massachusetts 02139
  • Email: grinberg@math.mit.edu
  • Received by editor(s): May 21, 1998
  • Received by editor(s) in revised form: October 10, 1998
  • Published electronically: December 4, 1998
  • © Copyright 1998 American Mathematical Society
  • Journal: Represent. Theory 2 (1998), 410-431
  • MSC (1991): Primary 14D05, 22E46
  • DOI: https://doi.org/10.1090/S1088-4165-98-00053-3
  • MathSciNet review: 1657203