Differential operators on quantized flag manifolds at roots of unity
Differential operators on quantized flag manifolds at roots of unity
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DOI：
10.1016/j.aim.2012.04.018
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发表时间：
2010-01
影响因子：
1.7
通讯作者：
T. Tanisaki
中科院分区：
文献类型：
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作者：
T. Tanisaki
量子化的旗流形是普通旗流形的q-模拟，它被实现为一个非交换方案，我们可以利用非交换代数几何的框架定义其上的D-模范畴;然而当参数q是单位根时，Lusztig的Frobenius态射允许我们处理D-量子化旗流形上的模通过普通旗流形上的某个环层上的模。本文证明了普通旗流形上的这层环是其中心上的Azumaya代数。我们还证明了它对某些子集的限制是分裂的Azumaya代数。这些结果类似于Bezrukavnikov-Mirković-Rumynin关于旗流形上正特征的D-模的一些结果。
The quantized flag manifold, which is a q-analogue of the ordinary flag manifold, is realized as a non-commutative scheme, and we can define the category of D-modules on it using the framework of non-commutative algebraic geometry; however, when the parameter q is a root of unity, Lusztig’s Frobenius morphism allows us to handle D-modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold. In this paper, we will show that this sheaf of rings on the ordinary flag manifold is an Azumaya algebra over its center. We also show that its restriction to certain subsets are split Azumaya algebras. These are analogues of some results of Bezrukavnikov–Mirković–Rumynin on D-modules on flag manifolds in positive characteristics.