Title page for ETD etd-04152010-174523


Type of Document Dissertation
Author Culbertson, Jared Lee
Author's Email Address jared.culbertson@gmail.com
URN etd-04152010-174523
Title Perverse Poisson Sheaves on the Nilpotent Cone
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Achar, Pramod N. Committee Chair
Adkins, William Committee Member
Litherland, Richard A. Committee Member
Morales, Jorge Committee Member
Olafsson, Gestur Committee Member
Ishak, Sherif Dean's Representative
Keywords
  • nilpotent cone
  • Poisson
  • perverse sheaves
Date of Defense 2010-04-12
Availability unrestricted
Abstract
For a reductive complex algebraic group, the associated nilpotent cone is the variety of nilpotent elements in the corresponding Lie algebra. Understanding the nilpotent cone is of central importance in representation theory. For example, the nilpotent cone plays a prominent role in classifying the representations of finite groups of Lie type. More recently, the nilpotent cone has been shown to have a close connection with the affine flag variety and this has been exploited in the Geometric Langlands Program.

We make use of the following important fact. The nilpotent cone is invariant under the coadjoint action of G on the dual Lie algebra and admits a canonical Poisson structure which is compatible in a strong way with the action of G. We exploit this connection to develop a theory of perverse sheaves on the nilpotent cone that is suitable for the G-equivariant Poisson setting. Building on the work of Beilinson--Bernstein--Deligne and Deligne--Bezrukavnikov, we define a new category, the equivariant Poisson derived category and endow it with a new semiorthogonal filtration, the perverse Poisson t-structure. In order to construct the perverse Poisson t-structure, we also prove an axiomatized gluing theorem for semiorthogonal filtrations in the general setting of triangulated categories which generalizes the construction of the perverse coherent sheaves of Deligne--Bezrukavnikov.

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