Representation theory and geometry of the flag variety (Q2101374)

The goal of this book is to develop the connections between the representation theory of a complex semisimple Lie group \(G\) and the geometry of the flag variety. W. McGovern begins by introducing finite-dimensional representations, and then outlines the Beilinson-Bernstein construction of irreducible representations of \(G\) from closures of orbits of subgroups of \(G\) in \(X=G/B\). He also shows how to go the other way, from representations to orbit closures via the characteristic variety of a representation. This leads to studying which orbit closures lie in which others and the singularities of orbit closures at points of smaller orbits. McGovern discusses Kazhdan-Lusztig and Lusztig-Vogan polynomials, connecting them to singularities and representations. He also provides combinatorial characterizations of nonsingular orbit closures. Furthermore, he investigates Springer fibers in \(X\) and the singularities of their components. This is a nicely written book, strongly recommended for anyone interested in the geometry of the flag variety.