Principal nilpotent pairs in a semisimple Lie algebra 1

被引:33
作者
Ginzburg, V [1 ]
机构
[1] Univ Chicago, Dept Math, Chicago, IL 60637 USA
关键词
D O I
10.1007/s002220050371
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. Each principal nilpotent pair gives rise to a harmonic polynomial on the Cartesian square of the Cartan subalgebra, that transforms under an irreducible representation of the Weyl group. In the special case of sl(n), the conjugacy classes of principal nilpotent pairs and the irreducible representations of the symmetric group, S-n, are both parametrised (in a compatible way) by Young diagrams. In general, our theory provides a natural generalization to arbitrary Weyl groups of the classical construction of simple S-n-modules in terms of Young's symmetrisers. First results towards a complete classification of all principal nilpotent pairs in a simple Lie algebra are presented at the end of this paper in an Appendix, written by A. Elashvili and D. Panyushev.
引用
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页码:511 / 561
页数:51
相关论文
共 31 条
[1]   ON SPRINGER CORRESPONDENCE FOR SIMPLE-GROUPS OF TYPE-EN (N=6,7,8) [J].
ALVIS, D ;
LUSZTIG, G ;
SPALTENSTEIN, N .
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 1982, 92 (JUL) :65-78
[2]  
[Anonymous], B LONDON MATH SOC
[3]  
[Anonymous], INDAG MATH
[4]   UNIPOTENT REPRESENTATIONS OF COMPLEX SEMISIMPLE GROUPS [J].
BARBASCH, D ;
VOGAN, DA .
ANNALS OF MATHEMATICS, 1985, 121 (01) :41-110
[5]  
BORHO W, 1983, ASTERISQUE, P23
[6]  
Brylinski RK., 1989, J. Amer. Math. Soc, V2, P517, DOI 10.2307/1990941
[7]  
Carter R. W., 1993, Finite groups of lie type. Conjugacy classes and complex characters
[8]  
Chriss N., 1997, Representation Theory and Complex Geometry
[9]  
Collingwood David H., 1993, Mathematics series
[10]  
Dynkin E.B., 1952, AM MATH SOC TRANSL 2, V6, P245