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 progress

In 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 difficult 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 reductive Lie group, such as GL_n(C).

This definition is based on the theory of quantizations of symplectic singularities, especially the geometry of nilpotent co-adjoint orbits and their equivariant covers. The main theorems include a geometric classification of unipotent representations, a calculation of their infinitesimal characters, and a proof of their unitarity in the case of classical groups. Although further obstacles remain, this work paves the way for a general theory of unipotent representations of reductive Lie groups, which should in turn form the basis of the classification of irreducible unitary representations.