Singular localization of \(\mathfrak{g}\)-modules and applications to representation theory. (English) Zbl 1380.17006

Backelin, Erik; Kremnitzer, Kobi

J. Eur. Math. Soc. (JEMS) 17, No. 11, 2763-2787 (2015).

This paper develops a singular version of Beilinson-Bernstein localization theory for a complex semisimple Lie algebra. The main result is a singular version of the localization theorem which connects Lie algebra modules having a prescribed (singular) central character with certain twisted differential operators. As an application, various known results are reproved. In particular, a theorem of Bernstein and S. Gelfand which connects category \(\mathcal{O}\) to Harish-Chandra bimodules; and a theorem of Miličić and Soergel which connects category \(\mathcal{O}\) to a certain category of Whittaker modules.

Reviewer: Volodymyr Mazorchuk (Uppsala)

MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14F10	Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14L35	Classical groups (algebro-geometric aspects)

Keywords:

Lie algebra; localization; category \(\mathcal{O}\); translation functor; singular block; Whittaker module

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