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This book is an expanded version of the Hermann Weyl Lectures given at the Institute for Advanced Study in January 1986. It outlines some of what is now known about irreducible unitary representations of real reductive groups, providing fairly complete definitions and references, and sketches (at least) of most proofs. The first half of the book is devoted to the three more or less understood constructions of such representations: parabolic induction, complementary series, and cohomological parabolic induction. This culminates in the description of all irreducible unitary representation of the general linear groups. For other groups, one expects to need a new construction, giving "unipotent representations." The latter half of the book explains the evidence for that expectation and suggests a partial definition of unipotent representations.

Lie groups --- Representations of Lie groups --- Lie groups. --- Representations of Lie groups. --- 512.81 --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- 512.81 Lie groups --- Abelian group. --- Adjoint representation. --- Annihilator (ring theory). --- Atiyah–Singer index theorem. --- Automorphic form. --- Automorphism. --- Cartan subgroup. --- Circle group. --- Class function (algebra). --- Classification theorem. --- Cohomology. --- Commutator subgroup. --- Complete metric space. --- Complex manifold. --- Conjugacy class. --- Cotangent space. --- Dimension (vector space). --- Discrete series representation. --- Dixmier conjecture. --- Dolbeault cohomology. --- Duality (mathematics). --- Eigenvalues and eigenvectors. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior algebra. --- Function space. --- Group homomorphism. --- Harmonic analysis. --- Hecke algebra. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Homogeneous space. --- Homomorphism. --- Induced representation. --- Infinitesimal character. --- Inner automorphism. --- Invariant subspace. --- Irreducibility (mathematics). --- Irreducible representation. --- Isometry group. --- Isometry. --- K-finite. --- Kazhdan–Lusztig polynomial. --- Langlands decomposition. --- Lie algebra cohomology. --- Lie algebra representation. --- Lie algebra. --- Lie group action. --- Lie group. --- Mathematical induction. --- Maximal compact subgroup. --- Measure (mathematics). --- Minkowski space. --- Nilpotent group. --- Orbit method. --- Orthogonal group. --- Parabolic induction. --- Principal homogeneous space. --- Principal series representation. --- Projective space. --- Pseudo-Riemannian manifold. --- Pullback (category theory). --- Ramanujan–Petersson conjecture. --- Reductive group. --- Regularity theorem. --- Representation of a Lie group. --- Representation theorem. --- Representation theory. --- Riemann sphere. --- Riemannian manifold. --- Schwartz space. --- Semisimple Lie algebra. --- Sheaf (mathematics). --- Sign (mathematics). --- Special case. --- Spectral theory. --- Sub"ient. --- Subgroup. --- Support (mathematics). --- Symplectic geometry. --- Symplectic group. --- Symplectic vector space. --- Tangent space. --- Tautological bundle. --- Theorem. --- Topological group. --- Topological space. --- Trivial representation. --- Unitary group. --- Unitary matrix. --- Unitary representation. --- Universal enveloping algebra. --- Vector bundle. --- Weyl algebra. --- Weyl character formula. --- Weyl group. --- Zariski's main theorem. --- Zonal spherical function. --- Représentations de groupes de Lie --- Groupes de lie --- Representation des groupes de lie