A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices

摘要: We exhibit a nice Frobenius splitting σ on G× b where is the Lie algebra of Borel group B upper triangular matrices in general linear G = Gln. What about it, that it descends along familiar maps and specializes to subvarieties manner useful for study singularities closures conjugacy classes nilpotent n by matrices. In particular, we show these are simultaneously split, normal, have rational singularities. The result derived from vanishing theorem will be proved our paper [15]. Note normality has already been Donkin [3]. His method uses lot representation theory employs resolutions invented Kraft Procesi. An alternative approach given G. Lusztig. [11] he showed same occur Schubert varieties Kac-Moody groups affine Weyl groups. Now such infinite dimensional mastered Mathieu’s book [12], Mathieu shows they normal contrast with this, work remains finite dimensions. It relies explicit formulas. Indeed formula product principal minors specialization based an inspection what happens determinants. To descend g G, (along natural map G×b → g, cf. Grothendieck’s “simultaneous resolution” [2]), use Galois theoretic argument. find above generic point action trivial. As preparation computation first spell out trivializations canonical bundles t.