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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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An analytic Riemann-Hilbert correspondence for semi-simple Lie groups
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by Laura Smithies and Joseph L. Taylor PDF
Represent. Theory 4 (2000), 398-445 Request permission


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:
  • Joseph L. Taylor
  • Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
  • Email:
  • Received by editor(s): July 21, 1999
  • Published electronically: September 12, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Represent. Theory 4 (2000), 398-445
  • MSC (2000): Primary 22E46; Secondary 18D99, 55N91
  • DOI:
  • MathSciNet review: 1780717