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On the Kostant conjecture for Clifford algebras. (English. Abridged French version) Zbl 1264.17006

Summary: Let \(\mathfrak g\) be a complex simple Lie algebra, and \(\mathfrak h \subset \mathfrak g\) be a Cartan subalgebra. In the end of the 1990s, B. Kostant defined two filtrations on \(\mathfrak h\), one using the Clifford algebras and the odd analogue of the Harish-Chandra projection \(\mathrm{hc}_{\text{odd}}: \text{Cl}(\mathfrak g)\to \text{Cl}(\mathfrak h)\), and the other one using the canonical isomorphism \(\check{\mathfrak h}=\mathfrak h^*\) (here \(\check{\mathfrak h}\) is the Cartan subalgebra in the simple Lie algebra corresponding to the dual root system) and the adjoint action of the principal \(\mathfrak{sl}_{2}\)-triple \(\check{\mathfrak s}\subset\check{\mathfrak g}\). 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 \(\mathfrak g\) of type \(A_n\) using explicit expressions for primitive invariants in the exterior algebra of \(\check{\mathfrak g}\). 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 \(\mathfrak h\) induced by the generalized Harish-Chandra projection \((U\mathfrak g \otimes \mathfrak g)^{\mathfrak g} \to S\mathfrak h \otimes \mathfrak h\) and the evaluation at \(\rho \in \mathfrak h^*\). In this note, we prove that Joseph’s result is equivalent to the Kostant Conjecture. We also show that the standard Harish-Chandra projection \(U\mathfrak g \to S\mathfrak h\) composed with evaluation at \(\rho\) induces the same filtration on \(\mathfrak h\).

MSC:

17B20 Simple, semisimple, reductive (super)algebras
17B35 Universal enveloping (super)algebras
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References:

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