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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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Equivariant coherent sheaves on the exotic nilpotent cone
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by Vinoth Nandakumar PDF
Represent. Theory 17 (2013), 663-681 Request permission

Abstract:

Let $G=Sp_{2n}(\mathbb {C})$, and $\mathfrak {N}$ be Kato’s exotic nilpotent cone. Following techniques used by Bezrukavnikov in 2003 to establish a bijection between $\boldsymbol {\Lambda }^+$, the dominant weights for an arbitrary simple algebraic group $H$, and $\textbf {O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $\widetilde {\mathfrak {N}}$ have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows us to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category $D^b(\mathrm {Coh}^G(\mathfrak {N}))$, and deduce that the resulting $t$-structure on $D^b(\mathrm {Coh}^G(\mathfrak {N}))$ coincides with the perverse coherent $t$-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of the $t$-structure on $D^b(\mathrm {Coh}^G(\mathfrak {N}))$.
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Additional Information
  • Vinoth Nandakumar
  • Affiliation: Department of Mathematics, M.I.T., Cambridge, Massachusetts 02139-4307
  • Email: vinothn@math.mit.edu
  • Received by editor(s): September 13, 2012
  • Received by editor(s) in revised form: April 15, 2013, and July 11, 2013
  • Published electronically: December 23, 2013

  • Dedicated: In loving memory of my grandfather, Nadaraja Rajamanikkam
  • © Copyright 2013 American Mathematical Society
  • Journal: Represent. Theory 17 (2013), 663-681
  • MSC (2010): Primary 17B45, 20G05; Secondary 14F05
  • DOI: https://doi.org/10.1090/S1088-4165-2013-00444-2
  • MathSciNet review: 3145724