On the Kostant Conjecture for Clifford Algebras

Let g be a complex simple Lie algebra, and h ⊂ g 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 hcodd : Cl(g) → Cl(h), and the other one using the canonical isomorphism ȟ = h∗ (here ȟ is the Cartan subalgebra in the simple Lie algebra ǧ corresponding to the dual root system) and the adjoint action of the principal sl2-triple š ⊂ ǧ. 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 An using explicit expressions for primitive invariants in the exterior algebra ∧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⊗g) → Sh⊗h and the evaluation at ρ ∈ 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 → Sh composed with evaluation at ρ induces the same filtration on h.