Some homological constructions in representation theory (Q2718476)

This paper is a short informal exposition of the main results of five papers by the author, along with background and motivation. All except one of the papers [\textit{D. Miličić} and \textit{P. Pandžić}, Equivariant derived categories, Zuckerman functors and localization, Progress Math. 158, Birkhäuser, Boston, 209-242 (1998; Zbl 0907.22019)] are unpublished. A main topic is a generalization of the Zuckerman cohomological construction of Harish-Chandra modules for the pair \((\mathfrak{g},K)\), where \(\mathfrak{g}\) is a complex semisimple Lie algebra, and \(K\) a complex algebraic group which acts algebraically on \(\mathfrak{g}\). NEWLINENEWLINENEWLINEThe author starts by carefully defining the category \(\mathcal{M}(\mathfrak{g},K)\) of \((\mathfrak{g},K)\)-modules and describing two methods for studying them: the Zuckerman construction of derived functor modules, and the geometric method due to Beilinson and Bernstein, who obtained \((\mathfrak{g},K)\)-modules as global sections or cohomology of certain sheaves on the flag variety of \(\mathfrak{g}\) (\(\mathcal{D}\)-modules). Then he explains the notion of equivariant derived categories which were introduced by Beilinson-Ginzburg and Bernstein-Lunts in order to overcome the failure of certain derived categories of modules to be equivalent. Finally, the author describes his generalization of derived Zuckerman functors in the equivariant setting and points to a number of applications; these are mainly results about classical Zuckerman functors which may be obtained from analogous results for their equivariant versions. NEWLINENEWLINENEWLINEAs the author mentions in the introduction but not quite brings out in the paper, the new constructions described aim at explaining the relation between the two methods (Zuckerman and Beilinson/Bernstein), by displaying both of them as extensions of the Borel-Bott-Weil theorem in the appropriate context, namely that of equivariant derived categories.