Unipotent Ideals and Harish-Chandra Bimodules (Annals of Mathematics Studies, 225)

A groundbreaking book that applies new geometric tools to one of the oldest problems in representation theory, expanding the field and paving the way for further progressIn the 1920s, Hermann Weyl gave a complete classification of the irreducible unitary representations of a compact Lie group G.Around the same time, Fritz Peter and Weyl showed that these irreducible unitary representations are fundamental objects for harmonic analysis on G.If G is instead a noncompact Lie group, such as GL_n(R), the classification of the irreducible unitary G-representations is a much more di?cult problem, one that remains open in general.An idea, with its origins in the work of Kostant, Kirillov, and Vogan, is that the set of irreducible unitary G-representations should contain a finite set of “building blocks,” called unipotent representations, related to the set of nilpotent co-adjoint G-orbits.This book proposes a definition and theory of unipotent representations in the case of when G is a complex