Nilpotent orbits over ground fields of good characteristic
Nilpotent orbits over ground fields of good characteristic
复制标题
具有良好特性的地面场上的幂零轨道
DOI：
--
复制
发表时间：
2002
期刊：
影响因子：
--
通讯作者：
George J. McNinch
中科院分区：
文献类型：
--
作者：
George J. McNinch
设X是定义在基域F上的连通可约群G的李代数中的F-有理幂零元。设李代数具有非退化不变双线性型。证明了X的中心化子的幂幺根是F-分裂的。这个属性有几个后果。当F关于一个离散赋值是完备的，且剩余域是有限的或代数闭的时，我们给出了G（F）在（F）中有1/2个幂零轨道的统一证明。当剩余域有限时，得到了幂零轨道积分收敛的一个证明。在一些进一步的（相当温和的）假设G，我们证明了收敛的任意轨道积分的李代数和组。Deligne和Ranga Rao（1972）给出了F的特征为0时轨道积分的收敛性。
Abstract.Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in (F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).