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An analytic Riemann-Hilbert correspondence for semi-simple Lie groups


Authors: Laura Smithies and Joseph L. Taylor
Journal: Represent. Theory 4 (2000), 398-445
MSC (2000): Primary 22E46; Secondary 18D99, 55N91
DOI: https://doi.org/10.1090/S1088-4165-00-00076-5
Published electronically: September 12, 2000
MathSciNet review: 1780717
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Abstract:

Geometric Representation Theory for semi-simple Lie groups has two main sheaf theoretic models. Namely, through Beilinson-Bernstein localization theory, Harish-Chandra modules are related to holonomic sheaves of $\mathcal D$ modules on the flag variety. Then the (algebraic) Riemann-Hilbert correspondence relates these sheaves to constructible sheaves of complex vector spaces. On the other hand, there is a parallel localization theory for globalized Harish-Chandra modules---i.e., modules over the full semi-simple group which are completions of Harish-Chandra modules. In particular, Hecht-Taylor and Smithies have developed a localization theory relating minimal globalizations of Harish-Chandra modules to group equivariant sheaves of $\mathcal D$ modules on the flag variety. The main purpose of this paper is to develop an analytic Riemann-Hilbert correspondence relating these sheaves to constructible sheaves of complex vector spaces and to discuss the relationship between this ``analytic'' study of global modules and the preceding ``algebraic'' study of the underlying Harish-Chandra modules.


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Additional Information

Laura Smithies
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242
Email: smithies@mcs.kent.edu

Joseph L. Taylor
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email: taylor@math.utah.edu

DOI: https://doi.org/10.1090/S1088-4165-00-00076-5
Keywords: Localization, constructible sheaves, equivariant derived category
Received by editor(s): July 21, 1999
Published electronically: September 12, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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