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 'Endomorphismensatz' 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* $\lbrace S\_{w}\rbrace\_{w\in W}$, where $S\_{w}$ is the class of the corresponding Schubert cell.


*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?)


2) 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



Endomorphisms in Category O and Schubert Classes
1) I believe there is such a description, though I think its pretty debatable whether it is likely to tell you very much about Schubert calculus.


Category $\mathcal{O}$ has a nice collection of objects called "tilting modules"; these are distinguished by have a Verma and dual Verma filtration (actually all of these are self-dual); the indecomposables are indexed by the elements of the Weyl group (look at the lowest elements whose associated Verma or dual Verma appears in the filtration). 
The self-dual projective $P(w\_0)=T(e)$ is an example of a tilting module. 


Furthermore, these all have graded lifts in the graded version of category $\mathcal{O}$; in particular, there's a way of grading the Hom spaces between these objects so that the endomorphisms of $P(w\_0)$ become $H^\*(G/B)$ with the homological grading. If you choose the gradings correctly, the Hom spaces $Hom(T(e),T(w))$ and $Hom(T(w),T(e))$ have lowest degree $\ell(w)$ and dimension 1 in that degree. I believe the Schubert class for $w$ is (up to scalar) the composition of elements from these Hom spaces in lowest degree.


I won't give a detailed proof, but the point is that from Soergel's work you can identify $Hom(T(e),T(w))\cong Ext^\bullet(\mathbb{C}\_{G/B},\mathbf{IC}\_{S\_w})$ and $Hom(T(w),T(e))\cong Ext^\bullet(\mathbf{IC}\_{S\_w},\mathbb{C}\_{G/B})$ with composition being Yoneda product. This shows that any map $T(e)$ to $T(e)$ that factors through $T(w)$ is a sum of Schubert classes for $S\_{w'}$ with $w'>w$ (it also shows the claim I made about Hom spaces). Thus, $S\_w$ is (up to scalar) the only such element of degree $\ell(w)$.


2) Yes, if you're willing to think about $\mathrm{End}(P(w\_0))$ via its canonical isomorphism with the center of category $\mathcal{O}$. It's the induced action on the center of a categorical braid group action. See Section 3 of this paper of Stroppel: <http://arxiv.org/abs/math.RT/0608234>


