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Let g be a complex simple Lie algebra, and h be a Cartan subalgebra. In theend of 1990s, B. Kostant defined two filtrations on h, one using the Cliffordalgebras and the odd analogue of the Harish-Chandra projection $hc: Cl(g) \toCl(h)$, and the other one using the canonical isomorphism $\check{h} = h^*$(here $\check{h}$ is the Cartan subalgebra in the simple Lie algebracorresponding to the dual root system) and the adjoint action of the principalsl2-triple. Kostant conjectured that the two filtrations coincide. The twofiltrations arise in very different contexts, and comparing them proved to be adifficult task. Y. Bazlov settled the conjecture for g of type A using explicitexpressions for primitive invariants in the exterior algebra of g. Up to nowthis approach did not lead to a proof for all simple Lie algebras. Recently, A.Joseph proved that the second Kostant filtration coincides with the filtrationon h induced by the generalized Harish-Chandra projection $(Ug \otimes g)^g \toSh \otimes h$ and the evaluation at $\rho \in h^*$. In this note, we prove thatJoseph's result is equivalent to the Kostant Conjecture. We also show that thestandard Harish-Chandra projection $Ug \to Sh$ composed with evaluation at$\rho$ induces the same filtration on h.

On the Kostant conjecture for Clifford algebra