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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

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