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

• Published in: Representation theory 2000-09, Vol.4 (16), p.398-445
• Main Authors: Smithies, Laura, Taylor, Joseph
• Format: Article
• Language: English
• ISSN: 1088-4165; 1088-4165
• DOI: 10.1090/S1088-4165-00-00076-5

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 D \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 D \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.