Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Introduction
- 1 Auslander–Reiten Theory of Finite-Dimensional Algebras
- 2 τ-tilting Theory – an Introduction
- 3 From Frieze Patterns to Cluster Categories
- 4 Infinite-dimensional Representations of Algebras
- 5 The Springer Correspondence
- 6 An Introduction to Diagrammatic Soergel Bimodules
- 7 A Companion to Quantum Groups
- 8 Infinite-dimensional Lie Algebras and Their Multivariable Generalizations
- 9 An Introduction to Crowns in Finite Groups
- 10 An Introduction to Totally Disconnected Locally Compact Groups and Their Finiteness Conditions
- 11 Locally Analytic Representations of p-adic Groups
- References
5 - The Springer Correspondence
Published online by Cambridge University Press: aN Invalid Date NaN
Book contents
- Frontmatter
- Contents
- Contributors
- Preface
- Introduction
- 1 Auslander–Reiten Theory of Finite-Dimensional Algebras
- 2 τ-tilting Theory – an Introduction
- 3 From Frieze Patterns to Cluster Categories
- 4 Infinite-dimensional Representations of Algebras
- 5 The Springer Correspondence
- 6 An Introduction to Diagrammatic Soergel Bimodules
- 7 A Companion to Quantum Groups
- 8 Infinite-dimensional Lie Algebras and Their Multivariable Generalizations
- 9 An Introduction to Crowns in Finite Groups
- 10 An Introduction to Totally Disconnected Locally Compact Groups and Their Finiteness Conditions
- 11 Locally Analytic Representations of p-adic Groups
- References
Summary
We present some key concepts and tools in the field of geometric representation theory. We review the background necessary to state the Springer correspondence for an arbitrary semisimple Lie algebra. We then study the notion of convolution in Borel–Moore homology and see how to apply it to the Springer correspondence. Finally, we reframe these ideas in the language of perverse sheaves and intersection homology.
- Type
- Chapter
- Information
- Modern Trends in Algebra and Representation Theory , pp. 169 - 211Publisher: Cambridge University PressPrint publication year: 2023
References
[2]Baez, John C. 2001. The octonions. Bulletin of the American Mathematical Society, 39(02), 145–206.Google Scholar
[3]Borho, Walter, and MacPherson, Robert. 1983. Partial resolutions of nilpotent varieties. Pages 23–74 of: Analysis and topology on singular spaces, II, III (Luminy, 1981). Aste´risque, vol. 101. Paris: Soc. Math. France.Google Scholar
[4]Chriss, Neil, and Ginzburg, Victor. 1997. Representation theory and complex geometry. Boston, MA: Birkha¨user Boston Inc.Google Scholar
[5]Concini, De, C., Lusztig, G. and Procesi, C. 1988. Homology of the Zero-Set of a Nilpotent Vector Field on a Flag Manifold. Journal of the American Mathematical Society, 1(1), 15.Google Scholar
[7]Ginsburg, V. 1987. “Lagrangian” construction for representations of Hecke algebras. Advances in Mathematics, 63(1), 100–112.Google Scholar
[8]Goresky, Mark, and MacPherson, Robert. 1980. Intersection homology theory. Topology, 19(2), 135–162.Google Scholar
[9]Goresky, Mark, and MacPherson, Robert. 1983. Intersection homology. II. Invent. Math., 72(1), 77–129.Google Scholar
[10]Hotta, Ryoshi, and Kashiwara, Masaki. 1984. The invariant holonomic system on a semisimple Lie algebra. Inventiones Mathematicae, 75(2), 327–358.Google Scholar
[11]Hotta, Ryoshi, Takeuchi, Kiyoshi, and Tanisaki, Toshiyuki. 2008. D-modules, perverse sheaves, and representation theory. Progress in Mathematics, vol. 236. Boston, MA: Birkha¨user Boston Inc. Translated from the 1995 Japanese edition by Takeuchi.CrossRefGoogle Scholar
[12]Kashiwara, Masaki, and Schapira, Pierre. 1990. Sheaves on Manifolds. Berlin, Heidelberg: Springer Berlin Heidelberg Imprint Springer.Google Scholar
[13]Kazhdan, David, and Lusztig, George. 1980. A topological approach to Springer’s representations. Advances in Mathematics, 38(2), 222–228.Google Scholar
[14]Lusztig, G. 1984. Intersection cohomology complexes on a reductive group. Inventiones Mathematicae, 75(2), 205–272.Google Scholar
[15]Shoji, Toshiaki. 1988. Geometry of orbits and Springer correspondence. Pages 61–140 of: Orbites unipotentes et repre´sentations - I. Groupes finis et Alge`bres de Hecke. Aste´risque, no. 168. Socie´te´ mathe´matique de France.Google Scholar
[16]Spaltenstein, N. 1982. Classes Unipotentes et Sous-groupes de Borel. Springer-Verlag GmbH.CrossRefGoogle Scholar
[17]Springer, T. A. 1976. Trigonometric sums, Green functions of finite groups and representations of Weyl groups. Inventiones Mathematicae, 36(1), 173–207.Google Scholar
[19]Springer, Tonny A. 1978. A construction of representations of Weyl groups. Inventiones Mathematicae, 44(3), 279–293.Google Scholar