Differential operators on quantized flag manifolds at roots of unity. III (Q2237384)

Let \(\mathfrak{g}_k\) be the Lie algebra of a connected semisimple algebraic group over an algebraically closed field \(k\) of positive characteristic. Two important results concerning the sheaf \(\mathcal{D}\) of twisted differential operators on the corresponding flag manifold, which are Beilinson-Bernstein type derived equivalence between the category of certain representations of \(\mathfrak{g}_k\) and that of \(\mathcal{D}\)-modules, and the split Azumaya property of \(\mathcal{D}\) over a certain central subalgebra. The author gives an analogue using quantized flag manifolds and quantized enveloping algebras at roots of unity instead of ordinary flag manifolds and ordinary enveloping algebras in positive characteristics. More specifically, the author describes the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. For the De Concini-Kac type quantized enveloping algebra, where the parameter \(q\) is specialized to a root of unity whose order is a prime power, it follows that the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber, giving a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra.