nLab
geometric representation theory

Idea

Geometric representation theory studies representations (of various symmetry objects like algebraic groups, Hecke algebras, quantum groups, quivers etc.) realizing them by geometric means, e.g. by geometrically defined actions on sections of various bundles or sheaves as in geometric quantization (see at orbit method), D-modules, perverse sheaves, deformation quantization modules and so on.

Typically the underlying spaces for the sheaves involved are Grassmannians, flag varieties, configuration spaces and the like. Another important tool are cohomological vanishing theorems in appropriate contexts. Some historical landmarks are the Borel-Weil theorem, the Borel-Weil-Bott theorem, Kazhdan-Lusztig conjecture?, the BBDG decomposition theorem, the Beilinson-Bernstein localization theorem and the Lusztig conjectures?.

Quoting from (MSRI 14):

Representation theory is the study of the basic symmetries of mathematics and physics. Symmetry groups come in many different flavors: finite groups, Lie groups, p-adic groups, loop groups, adélic groups,.. A striking feature of representation theory is the persistence of fundamental structures and unifying themes throughout this great diversity of settings. One such theme is the Langlands philosophy, a vast nonabelian generalization of the Fourier transform of classical harmonic analysis, which serves as a visionary roadmap for the subject and places it at the heart of number theory.

The fundamental aims of geometric representation theory are to uncover the deeper ge-metric and categorical structures underlying the familiar objects of representation theory and harmonic analysis, and to apply the resulting insights to the resolution of classical problems. A groundbreaking example of its success is Beilinson-Bernstein’s generalization of the Borel-Weil-Bott theorem, giving a uniform construction of all representations of Lie groups via the geometric study of differential equations on flag varieties.

The geometric study of representations often reveals deeper layers of structure in the form of categorification. Categorification typically replaces numbers (such as character values) by vector spaces (typically cohomology groups), and vector spaces (such as representation rings) by categories (typically of sheaves). It is a primary explanation for miraculous integrality and positivity properties in algebraic combinatorics. A recent triumph of geometric methods is Ngô’s proof of the Fundamental Lemma, a key technical ingredient in the Langlands program. The proof relies on the cohomological interpretation of orbital integrals, which makes available the deep topological tools of algebraic geometry (such as Hodge theory and the Weil conjectures).

References

• A. Beilinson, J. Bernstein, Localisations de $𝔤$–modules, C. R. Acad. Sci. Paris 292 (1981), 15–18.

• N. Chriss, V. Ginzburg, Representation theory and complex geometry, book

• K. Vilonen, Geometric methods of representation theory, math.AG/0410032

• R. Hotta, K. Takeuchi, T. Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics 236, Birkhäuser 2008

• M. Kashiwara, Equivariant derived category and representation of real semisimple Lie groups, in CIME Summer school Representation Theory and Complex Analysis, Cowling, Frenkel et al. eds. LNM 1931 Springer (pdf).

• A. Borel et al., Algebraic D-modules, Perspectives in Mathematics, Academic Press, 1987.

• M.Kashiwara, W.Schmid, Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups, Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathematics, Birkhäuser, 1994, pp. 457–488.

• Dennis Gaitsgory, Geometric representation theory, 61 pp. 2005 course notes cat0.pdf

• David Kazhdan, Algebraic geometry and representation theory, Proc. ICM 1986, vol. I, 849-852, djvu:201K, pdf:399K

• V. Ginzburg (Paper read by D. Vogan.) Geometrical aspects of representation theory, Proc. ICM 1986, vol. I, 840-848, djvu:487K, pdf:940K

• MSRI program Geometric Represntation Theory (2014)

See also the references at orbit method.