|
In mathematics, the Springer representations are certain representations of the Weyl group ''W'' associated to unipotent conjugacy classes of a semisimple algebraic group ''G''. There is another parameter involved, a representation of a certain finite group ''A''(''u'') canonically determined by the unipotent conjugacy class. To each pair (''u'', φ) consisting of a unipotent element ''u'' of ''G'' and an irreducible representation ''φ'' of ''A''(''u''), one can associate either an irreducible representation of the Weyl group, or 0. The association : depends only on the conjugacy class of ''u'' and generates a correspondence between the irreducible representations of the Weyl group and the pairs (''u'', φ) modulo conjugation, called the Springer correspondence. It is known that every irreducible representation of ''W'' 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 Lusztig's classification of the irreducible representations of finite groups of Lie type. == Construction == Several approaches to Springer correspondence have been developed. T. A. Springer's original construction (1976) proceeded by defining an action of ''W'' on the top-dimensional l-adic cohomology groups of the algebraic variety ''B''''u'' of the Borel subgroups of ''G'' containing a given unipotent element ''u'' of a semisimple algebraic group ''G'' over a finite field. This construction was generalized by Lusztig (1981), who also eliminated some technical assumptions. Springer later gave a different construction (1978), using the ordinary cohomology with rational coefficients and complex algebraic groups. Kazhdan and Lusztig found a topological construction of Springer representations using the Steinberg variety and, allegedly, discovered Kazhdan–Lusztig polynomials in the process. Generalized Springer correspondence has been studied by Lusztig-Spaltenstein (1985) and by Lusztig in his work on character sheaves. Borho and MacPherson (1983) gave yet another construction of the Springer correspondence. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Springer correspondence」の詳細全文を読む スポンサード リンク
|