Let $\mathfrak{g}\supset \mathfrak{b}\supset \mathfrak{h}$ be a complex semisimple Lie algebra, with choice of Borel and Cartan subalgebras, $W$ the Weyl group.
W. Soergel's 'Endomorphismsatz' allows for the identification of $End_{\mathcal{O}_{0}}(P(w_{0}))$ with the algebra of coinvariants $\mathbb{C}[\mathfrak{h}^{\ast}]/\mathbb{C}[\mathfrak{h}^{\ast}]^{W}_{+}$, a finite dimensional quotient of a polynomial algebra, equipped with a $W$-action. Here $\mathcal{O}_{0}$ is the block of the BGG category $\mathcal{O}$ corresponding to the trivial central character.
Moreover, it is a classical result (due to Borel?) that we can identify $\mathbb{C}[\mathfrak{h}^{\ast}]/\mathbb{C}[\mathfrak{h}^{\ast}]^{W}_{+}$ with the cohomology algebra of the flag variety of $\mathfrak{g}$, and that there is a basis of this cohomology algebra given by the Schubert basis $\{S_{w}\}_{w\in W}$, where $S_{w}$ is the class of the corresponding Schubert cell. (I'm having formatting troubles here - the set braces around the Schubert classes are not appearing; any help appreciated)
Question(s): 1) does anyone know to which morphisms in category $\mathcal{O}$ the Schubert classes correspond to under the above identifications?
If yes; is there a 'nice' intrinsic (in terms of category $\mathcal{O}$) description of these morphisms that would give a 'canonical' description of the Schubert classes?
(rubbish, vague question) if no; would this be an interesting/valuable thing to know? (ie, are there any immediate applications?)
- Is there a way to see the $W$-action on the endomorphism algebra in category $\mathcal{O}$?
Also, replace 'Schubert class' by 'first Chern class of tautological bundles' in the above questions; is anything known in this case?
If this is standard material then my apologies; any references/directions would be much appreciated. In particular, any references for Soergel's work (in English/French) would be particularly appreciated.
Cheers, George