Kostant principal filtration and paths in weight lattice
There are several interesting filtrations on the Cartan subalgebra of a complex simple Lie algebra coming from very different contexts: one is the principal filtration coming from the Langlands dual, one is coming from the Clifford algebra associated with a non-degenerate invariant bilinear form, one is coming from the symmetric algebra and the Chevalley projection, and two other ones are coming from the enveloping algebra and Harish-Chandra projections. It is known that all these filtrations coincide. This results from a combination of works of several authors (Rohr, Joseph, Alekseev-Moreau). The remarkable connection between the principal filtration and the Clifford filtration was essentially conjectured by Kostant. The purpose of this thesis is to establish a new correspondence between the enveloping filtration and the symmetric filtration for a simple Lie algebra of type A or C. Together with Rohr's result and Alekseev-Moreau theorem, this provides another proof of Kostant's conjecture for these types, that is, a new proof of Joseph's theorem. Our proof is very different from his approach. The starting point is to use an explicit description of invariants via the standard representation which is possible in types A and C. Then we describe the images of their differentials by the generalised Chevalley and Harish-Chandra projections in term of combinatorial objects, called weighted paths, in the crystal graph of the standard representation. The proofs for types A and C are quite similar, but there are new phenomenons in type C which makes the proof much more tricky in this case.