  Let g be a complex simple Lie algebra and h a Cartan subalgebra. The Clifford
algebra C(g) of g admits a Harish-Chandra map. Kostant conjectured (as
communicated to Bazlov in about 1997) that the value of this map on a (suitably
chosen) fundamental invariant of degree 2m+1 is just the zero weight vector of
the simple 2m+1-dimensional module of the principal s-triple obtained from the
Langlands dual. Bazlov settled this conjecture positively in type A.
  The Kostant conjecture was reformulated (Alekseev-Bazlov-Rohr) in terms of
the Harish-Chandra map for the enveloping algebra U(g) composed with evaluation
at the half sum of the positive roots.
  Here an analogue of the Kostant conjecture is settled by replacing the
Harish-Chandra map by a "generalized Harish-Chandra" map which had been studied
notably by Zhelobenko. The proof involves a symmetric algebra version of the
Kostant conjecture (settled in works of Alekseev-Bazlov-Rohr), the Zhelobenko
invariants in the adjoint case and surprisingly the Bernstein-Gelfand-Gelfand
operators introduced in their study of the cohomology of the flag variety.
