  Remarks on the Kostant Dirac operator In 1999, Kostant [Kos99] indroduces a
Dirac operator D_g/h associated to any triple (g, h,B), where g is a complex
Lie algebra provided with an ad g-invariant non degenerate nsymetric bilinear
form B, and h is a Lie subalgebra of g such that the bilinear form B is non
degenerate on h. Kostant then shows that the square of this operator safisties
a formula that generalizes the so-called Parthasarathy formula [Par72]. We give
here a new proof of this formula. First we use an induction by stage argument
to reduce the proof of the formula to the particular case where h = 0. In this
case we show that the vanishing of the first ordrer term in the Kostant formula
for D2_g/h is a consequence of classic properties related to Lie algebra
cohomology, and the fact that the square of the cubic term is a scalar follows
from such considerations, together with the Jacobi identity.
