The Harish-Chandra isomorphism for Clifford algebras

Details The Harish-Chandra isomorphism for Clifford algebras The Harish-Chandra isomorphism for Clifford algebras

2008-12-11

arxiv.org/abs/0812.2059v2

Mathematics

Representation Theory

v2: added references

Scientific paper

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.

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