T-equivariant K-theory of generalized flag varieties.
T-equivariant K-theory of generalized flag varieties.
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DOI：
10.4310/jdg/1214445320
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发表时间：
1987
影响因子：
11.1
通讯作者：
B. Kostant;Santhosh K. P. Kumar
中科院分区：
文献类型：
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作者：
B. Kostant;Santhosh K. P. Kumar
设G为具有Borel子群B和紧极大环面t的Kac-Moody群。类似于Kostant和Kumar [Kostant， B. & Kumar， S. (1986) Proc. Natl。学会科学。[USA 83, 1543-1545]，我们纯粹根据Weyl群W（与G相关联）及其对T的作用定义了一个环Y，通过对偶Y，我们得到了另一个环Psi，我们证明了它与G/B的T等变K理论K(T)（G/B）“正则”同构。现在K(T)（G/B）除了是K(T)（pt.）上近似于A(T)的代数外，还具有Weyl群作用，并且K(T)（G/B）允许某些类似于定义在A(T)上的Demazure算子{D(w)}w[unk] w。我们证明了K(T)（G/B）上的这些结构自然地来自于环y。通过对A(T)-模Psi在1处的“求值”，我们恢复了K（G/B）和上述结构。我们相信本文的许多结果在有限情况下（即G是C上的有限维半单群）也是新的。
Let G be a Kac-Moody group with Borel subgroup B and compact maximal torus T. Analogous to Kostant and Kumar [Kostant, B. & Kumar, S. (1986) Proc. Natl. Acad. Sci. USA 83, 1543-1545], we define a certain ring Y, purely in terms of the Weyl group W (associated to G) and its action on T. By dualizing Y we get another ring Psi, which, we prove, is "canonically" isomorphic with the T-equivariant K-theory K(T)(G/B) of G/B. Now K(T)(G/B), apart from being an algebra over K(T)(pt.) approximately A(T), also has a Weyl group action and, moreover, K(T)(G/B) admits certain operators {D(w)}w[unk]W similar to the Demazure operators defined on A(T). We prove that these structures on K(T)(G/B) come naturally from the ring Y. By "evaluating" the A(T)-module Psi at 1, we recover K(G/B) together with the above-mentioned structures. We believe that many of the results of this paper are new in the finite case (i.e., G is a finite-dimensional semisimple group over C) as well.