Γ-EQUIVARIANT ^-THEORY OF GENERALIZED FLAG VARIETIES
Γ-EQUIVARIANT ^-THEORY OF GENERALIZED FLAG VARIETIES
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发表时间：
2008
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通讯作者：
B. Kostant;Shrawan Kumar
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作者：
B. Kostant;Shrawan Kumar
对于任意(不一定是对称化的)广义/x/Cartan矩阵A，C上有一个Kac-Moody代数g=g(A)，群G=G(A)。G有一个“标准酉型”K，如果A是经典的Cartan矩阵，则G是C上的有限维半单单连通代数群，K是G的极大紧子群，我们称之为有限情形。一般而言，我们分别有g：f)cbCp的子代数、Cartan子代数、Borel子代数和抛物线子代数。一个群也有相应的子群：Hc-B-CP、复极大环面、Borel子群和抛物线子群。我们用T表示K的紧极大环面HΠK。设W是与(g，ί)相联系的Weyl群，设{^}1<；κ/表示单反射集。群W在紧极大环面T(以及H)上运算，从而在T的特征标群X(T)的群代数R{T：=Z[X(T)]上运算，也在R(T)的商域Q{T上运算。对于任一W-域F，我们可以形成群代数Z[W]与F的Smash积FW。在[19]中，我们取F的域q=ζ？(ϊ(*)f上的所有有理函数，定义了一个适当的子环Rc QW，并证明了R及其“适当”的对偶Λ，以及Λ上的某种Λ-模结构，用H*(G/B)上的各种算子代替了G/B的上同调代数的研究。因此，理解H*(G/B)的问题，特别是理解H*(G/B)上的杯积结构和其他算子的问题，归结为一个纯粹的组合(希望更容易处理)理解环R及其对偶Λ的问题，该问题纯粹地和明确地根据Coxeΐer群W及其在()*上的表示来定义。
To any (not necessarily symmetrizable) generalized / x / Cartan matrix A , one associates a Kac-Moody algebra g = g(A) over C and group G = G(A). G has a "standard unitary form" K. If A is a classical Cartan matrix, then G is a finite dimensional semi-simple simply-connected algebraic group over C and K is a maximal compact subgroup of G. We refer to this as the finite case. In general, one has subalgebras of g: f) c b C p, the Cartan subalgebra, the Borel subalgebra, and a parabolic subalgebra, respectively. One also has the corresponding subgroups: H c B C P, the complex maximal torus, the Borel subgroup, and a parabolic subgroup, respectively. We denote by T the compact maximal torus H Π K of K. Let W be the Weyl group associated to (g, ί)) and let {^}1 < κ / denote the set of simple reflections. The group W operates on the compact maximal torus T (as well as on H) and hence on the group algebra R{T) := Z[X(T)] of the character group X(T) of T and also on the quotient field Q{T) of R(T). For any W-field F , we can form the smash product Fw of the group algebra Z[W] with F. In [19] we took, for F, the field Q = ζ?(ϊ)*) of all the rational functions on f) and defined an appropriate subring R c Qw , and showed that R and its "appropriate" dual Λ, along with a certain Λ-module structure on Λ, replace the study of the cohomology algebra of G/B together with the various operators defined on H*(G/B). Hence the problem of understanding H*(G/B), especially the cup product structure and other operators on H*(G/B), reduced to a purely combinatorial (and hopefully more tractable) problem of understanding the ring R and its "dual" Λ, defined purely and explicitly in terms of the Coxeΐer group W and its representation on ()*.