  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 now known that all these filtrations coincide. This results
from a combination of works of several authors (Rohr, Joseph, Alekseev and the
second named author), and was essentially conjectured by Kostant. In this
paper, we establish a direct correspondence between the enveloping filtration
and the symmetric filtration for a simple Lie algebra of type A or C. Our proof
is very different from Rohr and Joseph approaches. The idea is to use an
explicit description of the symmetric and enveloping invariants in term of
combinatorial objects, called weighted paths, in the crystal graph of the
standard representation.
