The Borel-Weil theorem for reductive Lie groups (Q498695)

The aim of the article under review is to realize irreducible admissible representations of real reductive Lie groups of Harish-Chandra class as sheaf cohomologies of finite-rank holomorphic vector bundles. For more detail I will quote the abstract of the paper below, as it should serve as a well-organized summary of the paper: ``In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomology of an equivariant holomorpchic line bundle defined on an open invariant submanifold of a complex flag space. Our main result is the following: suppose \(G_0\) is a real reductive group of Harish-Chandra class and let \(X\) be the associated full complex flag space. Suppose \(\mathcal{O}_\lambda\) is the sheaf of sections of a \(G_0\)-equivariant holomorphic line bundle on \(X\) whose parameter \(\lambda\) (in the usual twisted \(\mathcal{D}\)-module context) is antidominant and regular. Let \(S \subset X\) be a \(G_0\)-orbit and suppose \(U \supset S\) is the smallest \(G_0\)-invariant open submanifold of \(X\) that contains \(S\). From the analytic localization theory of Hecht and Taylor one knows that there is a nonnegative integer \(q\) such that the compactly supported sheaf cohomology gourps \(H^p_c(S,\mathcal{O}|_{S})\) vanish except in degree \(q\), in which case \(H^q_c(S,\mathcal{O}|_{S})\) is the minimal globalization of an associated standard Beilinson-Bernstein module. In this study, we show that the \(q\)-th compactly supported cohomology group \(H^q_c(U,\mathcal{O}|_{U})\) defines, in a natural way, a nonzero submodule of \(H^q_c(S,\mathcal{O}|_{S})\), which is irreducible (i.e., realizes the unique irreducible submodule of \(H^q_c(S,\mathcal{O}|_{S})\)) when an associated algebraic variety is nonsingular. By a tensoring argument, we can show that the result holds, more generally (for nonsingular associated variety), when the representaiton \(H^q_c(S,\mathcal{O}|_{S})\) is what we call a classifying module.'' As the proofs for the main results involve some ideas from [\textit{T. Bratten}, Compos. Math. 106, No. 3, 283--319 (1997; Zbl 0928.22014); Beitr. Algebra Geom. 49, No. 1, 33--57 (2008; Zbl 1152.22013)], the argument of the proofs requires the \(\mathcal{D}\)-module theory in depth. Toward the main results the authors summarize preliminary results in a friendly manner. Especially the reader may find the example of the introduction useful to understand the theory behind the main results of this article.