2012/05/09 by Bertram Kostant, Kostant, Bertram
Mathematics · #17B08 #17B22 #17B30 #22Exx #53D05 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B08 #msc:17B22 #msc:17B30 #msc:22Exx #msc:53D05
paper · pdf · doi:10.48550/arxiv.1205.2017
Withdrawn due to critical missing word in abstract, which also affects the first page of the article
arxiv created 2012/05/10 · arxiv updated 2012/05/11
Let G be a complex simply-connected semisimple Lie group and let \frakg= Lie G. Let \frakg = \frakn- +\frakh + \frakn be a triangular decomposition of \frakg. One readily has that Cent U(\frak n) is isomorphic to the ring S(\frak n)\frak\n of symmetric invariants. Using the cascade \cal B of strongly orthogonal roots, some time ago we proved that S(\frak n)^\frak n is a polynomial ring \Bbb C [ξ1,...,ξm] where m is the cardinality of \cal B. Using this result we establish that the maximal coadjoint of N = exp \frak n has codimension m. Let \frak b= \frak h + \frak n so that the corresponding subgroup B is a Borel subgroup of G. Let ℓ = rank \frakg. Then in this paper we prove the theorem that the maximal coadjoint orbit of B has codimension ℓ - m so that the following statements (1) and (2) are equivalent: (1) -1 is in the Weyl group of G (i.e., ℓ = m), and (2), B has a nonempty open coadjoint orbit. We remark that a nilpotent or a semisimple group cannot have a nonempty open coadjoint orbit. Celebrated examples where a solvable Lie group has a nonempty coadjoint orbit are due to Piatetski--Shapiro in his counterexample construction of a bounded complex homogeneous domain which is not of Cartan type.