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Representation Theory
Representation Theory
ISSN 1088-4165

 

Equivariant coherent sheaves on the exotic nilpotent cone


Author: Vinoth Nandakumar
Journal: Represent. Theory 17 (2013), 663-681
MSC (2010): Primary 17B45, 20G05; Secondary 14F05
Published electronically: December 23, 2013
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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(\textup {Coh}^G(\mathfrak{N}))$, and deduce that the resulting $ t$-structure on $ D^b(\textup {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(\textup {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

DOI: http://dx.doi.org/10.1090/S1088-4165-2013-00444-2
PII: S 1088-4165(2013)00444-2
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
Article copyright: © Copyright 2013 American Mathematical Society