  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.
