  Let g be a complex simple Lie algebra, and h be a Cartan subalgebra. In the
end of 1990s, B. Kostant defined two filtrations on h, one using the Clifford
algebras and the odd analogue of the Harish-Chandra projection $hc: Cl(g) \to
Cl(h)$, and the other one using the canonical isomorphism $\check{h} = h^*$
(here $\check{h}$ is the Cartan subalgebra in the simple Lie algebra
corresponding to the dual root system) and the adjoint action of the principal
sl2-triple. Kostant conjectured that the two filtrations coincide. The two
filtrations arise in very different contexts, and comparing them proved to be a
difficult task. Y. Bazlov settled the conjecture for g of type A using explicit
expressions for primitive invariants in the exterior algebra of g. Up to now
this approach did not lead to a proof for all simple Lie algebras. Recently, A.
Joseph proved that the second Kostant filtration coincides with the filtration
on h induced by the generalized Harish-Chandra projection $(Ug \otimes g)^g \to
Sh \otimes h$ and the evaluation at $\rho \in h^*$. In this note, we prove that
Joseph's result is equivalent to the Kostant Conjecture. We also show that the
standard Harish-Chandra projection $Ug \to Sh$ composed with evaluation at
$\rho$ induces the same filtration on h.
