231:
occurs exactly once in the correspondence, although φ may be a non-trivial representation. The
Springer correspondence has been described explicitly in all cases by Lusztig, Spaltenstein and Shoji. The correspondence, along with its generalizations due to Lusztig, plays a key role in
210:
386:. The Springer correspondence in this case is a bijection, and in the standard parametrizations, it is given by transposition of the partitions (so that the trivial representation of the Weyl group corresponds to the regular unipotent class, and the
295:
over a finite field. This construction was generalized by
Lusztig, who also eliminated some technical assumptions. Springer later gave a different construction, using the ordinary cohomology with rational coefficients and complex algebraic groups.
79:
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in the process. Generalized
Springer correspondence has been studied by Lusztig and Spaltenstein and by Lusztig in his work on
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407:
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letters. Its irreducible representations over a field of characteristic zero are also parametrized by the partitions of
233:
415:
205:{\displaystyle (u,\phi )\mapsto E_{u,\phi }\quad u\in U(G),\phi \in {\widehat {A(u)}},E_{u,\phi }\in {\widehat {W}}}
702:
Springer, T. A. (1976). "Trigonometric sums, Green functions of finite groups and representations of Weyl groups".
237:
32:
332:
241:
751:
704:
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Partial resolutions of nilpotent varieties. Analysis and topology on singular spaces, II, III (Luminy, 1981)
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and generates a correspondence between the irreducible representations of the Weyl group and the pairs (
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Nilpotent orbits, primitive ideals, and characteristic classes. A geometric perspective in ring theory
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73:), one can associate either an irreducible representation of the Weyl group, or 0. The association
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Kazhdan and
Lusztig found a topological construction of Springer representations using the
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35:
414:, both as a general principle and as a technical tool. Many important results are due to
717:
571:. Astérisque. Vol. 101–102. Société Mathématique de France, Paris. pp. 23–74.
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253:
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780:
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311:. Borho and MacPherson gave yet another construction of the Springer correspondence.
790:
Quelques applications de la cohomologie intersection. SĂ©minaire
Bourbaki, exposé 589
605:
Lusztig, George (1981). "Green polynomials and singularities of unipotent classes".
344:
402:
Springer correspondence turned out to be closely related to the classification of
41:. There is another parameter involved, a representation of a certain finite group
343:
is a unipotent element, the corresponding partition is given by the sizes of the
411:
548:
542:. Progress in Mathematics. Vol. 78. Birkhäuser Boston, Inc., Boston, MA.
21:
749:
Springer, T. A. (1978). "A construction of representations of Weyl groups".
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28:
49:) canonically determined by the unipotent conjugacy class. To each pair (
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725:
667:"On the generalized Springer correspondence for exceptional groups"
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Several approaches to
Springer correspondence have been developed.
630:"On the generalized Springer correspondence for classical groups"
538:
Borho, Walter; Brylinski, Jean-Luc; MacPherson, Robert (1989).
256:'s original construction proceeded by defining an action of
521:
227:. It is known that every irreducible representation of
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sfn error: no target: CITEREFLusztigSpaltenstein1980 (
331:, the unipotent conjugacy classes are parametrized by
578:"A topological approach to Springer's representation"
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490:
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792:. Astérisque. Vol. 92–93. pp. 249–273.
628:Lusztig, George; Spaltenstein, Nicolas (1985).
509:
418:. A geometric approach was developed by Borho,
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8:
53:, φ) consisting of a unipotent element
567:Borho, Walter; MacPherson, Robert (1983).
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223:, φ) modulo conjugation, called the
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576:Kazhdan, Davis; Lusztig, George (1980).
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673:. Algebraic Groups and Related Topics.
636:. Algebraic Groups and Related Topics.
454:
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390:corresponds to the identity element of
215:depends only on the conjugacy class of
522:Borho, Brylinski & MacPherson 1989
284:containing a given unipotent element
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671:Advanced Studies in Pure Mathematics
634:Advanced Studies in Pure Mathematics
20:are certain representations of the
61:and an irreducible representation
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807:Representation theory of groups
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665:Spaltenstein, Nicolas (1985).
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305:Kazhdan–Lusztig polynomials
303:and, allegedly, discovered
238:irreducible representations
29:unipotent conjugacy classes
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479:Kazhdan & Lusztig 1980
290:semisimple algebraic group
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242:finite groups of Lie type
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752:Inventiones Mathematicae
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410:of a complex semisimple
234:Lusztig's classification
18:Springer representations
608:Advances in Mathematics
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260:on the top-dimensional
225:Springer correspondence
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321:special linear group
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16:In mathematics, the
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388:sign representation
765:10.1007/BF01403165
726:10.1007/BF01390009
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351:. All groups
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345:Jordan blocks
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398:Applications
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712:: 173–207.
677:: 317–338.
640:: 289–316.
412:Lie algebra
532:References
424:MacPherson
333:partitions
33:semisimple
22:Weyl group
781:121968560
742:121820241
420:Brylinski
197:^
188:∈
183:ϕ
163:^
143:∈
140:ϕ
122:∈
113:ϕ
99:↦
93:ϕ
801:Category
319:For the
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734:0442103
714:Bibcode
406:in the
366:is the
315:Example
276:of the
236:of the
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63:φ
777:S2CID
738:S2CID
430:Notes
339:: if
288:of a
31:of a
689:ISBN
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761:doi
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