Representation Theory of Lie Groups

Jeffrey Adams, University of Maryland, College Park.David Vogan, Massachusetts Institute of Technology, Cambridge, MA.

• A. W. Knapp and P. E. Trapa, Representations of semisimple Lie groups: IntroductionSome representations of $SL(n,\mathbb{R})$Semsimple groups and structure theoryIntroduction to representation theoryCartan subalgebras and highest weightsAction by the Lie algebraCartan subgroups and global charactersDiscrete series and asymptoticsLanglands classificationBibliographyR. Zierau, Representations in Dolbeault cohomology: IntroductionComplex flag varieties and orbits under a real formOpen $G_0$-orbitsExamples, homogeneous bundlesDolbeault cohomology, Bott-Borel-Weil theoremIndefinite harmonic theoryIntertwining operators IIntertwining operators IIThe linear cycle spaceBibliographyL. Barchini, Unitary representations attached to elliptic orbits. A geometric approach: IntroductionGlobalizationsDolbeault cohomology and maximal globalization$L^2$-cohomology and discrete series representationsIndefinite quantizationBibliographyD. A. Vogan, Jr., The method of adjoint orbits for real reductive groups: IntroductionSome ideas from mathematical physicsThe Jordan decomposition and three kinds of quantizationComplex polarizationsThe Kostant-Sekiguchi correspondenceQuantizing the action of $K$Associated graded modulesA good basis for associated graded modulesProving unitarityExercisesBibliographyK. Vilonen, Geometric methods in representation theory: IntroductionOverviewDerived categories of constructible sheavesEquivariant derived categoriesFunctors to representationsMatsuki correspondence for sheavesCharacteristic cylesThe character formulaMicrolocalization of Matsuki = SekiguchiHomological algebra (appendix by M. Hunziker)BibliographyJian-Shu Li, Minimal representations and reductive dual pairs: IntroductionThe oscillator representationModelsDualityClassificationUnitarityMinimal representations of classical groupsDual pairs in simple groupsBibliography

Altogether, the volume brings a coherent description of an important and beautiful part of representation theory, which certainly will be of substantial use for postgraduate students and mathematicians interested in the area"". - European Mathematical Society Newsletter