The Nil Hecke Ring and Cohomology of
The Nil Hecke Ring and Cohomology of
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尼尔赫克环和上同调
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
Shrawan Kumar
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作者：
B. Kostant;Shrawan Kumar
对于任何（不一定可对称化的）广义1 × 1 Cartan矩阵A，可以将C上的Kac-Moody代数g = g（A）和群G = G（A）联系起来。若A是经典Cartan矩阵，则G是C上的有限维半单代数群。我们称之为有限情形。在一般情况下，一个子代数g; h c B c p，Cartan子代数，Bore 1子代数，抛物子代数，分别。设W是与（g，h）相关联的Weyl群，并设它-I> Iszl~l表示单反射的集合。群W作用在h上（因此作用在它的对偶空间h* 上）。W参数化了广义旗簇G/B = U，（'E，V，.（= Bw ~ 'B/B）。（一个合适的子集W' c W对G/P也是如此。）我们主要关心的是上同调环H（G/B）（更一般地说是H（G/P）），实际上是GJP的任意（左）B-稳定闭子空间的上同调环。H（G/B）除了具有环结构和由Schubert类组成的特殊基（由Schubert胞腔的闭包的对偶给出）外，还是W的模。此外，在有限情形下，环
To any (not necessarily symmetrizable) generalized 1 x 1 Cartan matrix A, one associates a Kac-Moody algebra g = g(A) over C and group G = G(A). If A is a classical Cartan matrix, then G is a finite dimensional semi-simple algebraic group over C. We refer to this as the finite case. In general, one has subalgebras of g; h c b c p, the Cartan subalgebra, Bore1 subalgebra, and a parabolic subalgebra, respectively. One also has the corresponding subgroups H c B c P. Let W be the Weyl group associated to (g, h) and let it-I> Iszl~l denote the set of simple reflections. The group W operates on h (and hence on its dual space h*). W parametrizes the Schubert cell decomposition of the generalized flag variety G/B = U ,(‘E ,,, V,.( = Bw ~ ‘B/B). (A suitable subset W’ c W does the same for G/P.) Our principal concern is the cohomology ring H(G/B) (more generally H(G/P)) and in fact the cohomology ring of arbitrary (left) B-stable closed subspaces of GJP. Now besides having a ring structure and having a distinguished basis consisting of Schubert classes (given by the dual of the closures of Schubert cells, H(G/B) is also a module for W. In addition, in the finite case, a ring