Representation theory of the global Cherednik algebra
Name
1015183341-MIT.pdf
Description
Full printable version
Size
2.96 MB
Format
Adobe PDF
Checksum (MD5)
9e43d11084a2d7c3344199470ac9ac8f
Author(s)
Thompson, Daniel (Daniel Craig)
Advisor(s)
Pavel Etingof.
Date Issued
2017
Publisher
Massachusetts Institute of Technology
Abstract
This thesis studies the representation theory of Cherednik algebras associated to a complex algebraic varieties which carries the action of a finite group. First, we prove that the Knizhnik-Zamolodchikov functor from the category of P-coherent modules for the Cherednik algebra to finite dimensional modules over the Hecke algebra is essentially surjective. Then we begin to use this result to study the analog of category 0 for Cherednik algebras on Riemann surfaces and on products of elliptic curves. In particular we give conditions on the parameters under which this category 0 analog is nonzero. Our next goal is to generalize several basic results from the theory of D-modules to the representation theory of rational Cherednik algebras. We relate characterizations of holonomic modules in terms of singular support and Gelfand-Kirillov dimension. We study pullback, pushforward, and dual on the derived category of (holonomic) Cherednik modules for certain classes of maps between varieties. We prove, in the case of generic parameters for the rational Cherednik algebra, that pushforward with respect to an open affine inclusion preserves holonomicity. Finally, we relate the global sections algebra of the sheaf of Cherednik algebras associated to a smooth quadric hypersurface of Pn to the Dunkl angular momentum algebra of Feigin-Hakobyan. In particular, this lets us relate the angular momentum algebra for a rank 3 Coxeter group to the rank 2 symplectic reflection algebra for a corresponding finite subgroup of SL2.
Description
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2017.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 69-71).
Subjects
Mathematics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
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