2008/12/11 by Yuri Bazlov, Bazlov, Yuri
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.0812.2059
v2: added references
arxiv created 2009/04/22 · arxiv updated 2009/12/01
We study the analogue of the Harish-Chandra homomorphism where the universal enveloping algebra is replaced by the Clifford algebra, Cl(g), of a semisimple Lie algebra g. Two main goals are achieved. First, we prove that there is a Harish-Chandra type isomorphism between the subalgebra of g-invariants in Cl(g) and the Clifford algebra of the Cartan subalgebra of g. Second, the Cartan subalgebra is identified, via this isomorphism, with a graded space of the so-called primitive skew-symmetric invariants of g. This leads to a distinguished orthogonal basis of the Cartan subalgebra, which turns out to be induced from the Lie algebra Langlands dual to g via the action of its principal three-dimensional subalgebra. This settles a conjecture of Kostant.