On a certain family of $U(\mathfrak{b})$-modules

Abstract

We report on results of Kraskiewicz and the author, and Watanabe on KP modules materializing Schubert polynomials, and filtrations having KP modules as their sub quotients. We discuss applications of the bundles $S_w(E)$ for filtered ample bundles $E$ and KP filtrations to positivity due to Fulton and Watanabe respectively.