| Résume | A fundamental but difficult question in the representation theory is counting the PBW-degree, the minimal length of the products of f-operators for all positive roots (not only simple) needed to reach any vector from the highest vector.
E. Feigin, G. Fourier and P. Littlemann constructed the corresponding abstract PBW-basis for the Lie algebras of types A,C. A surprising recent conjecture due to the speaker, D. Orr and E. Feigin connects the PBW-degrees in Demazure level-one (affine) modules with the nonsymmetric Macdonald polynomials at t=infinity. This somewhat resembles the link of the BK-filtrartion to the Hall-Littlewood polynomials (q=0); both filtrations are related to the Kostant q-partition functions, though in very different ways. The conjecture was justified by the speaker and E. Feigin for extremal vectors in finite-dimensional irreducible representations (the ``top'' part of the corresponding Demazure module) for classical Lie algebras and G2. |
