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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 di铿僣ult 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 鈥渂uilding blocks,鈥� 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.