# Endomorphisms in Category O and Schubert Classes

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

• Just found a paper of Stroppel using Endomorphismensatz refferring to Soergel; so I will go ahaead and change this. – user9072 Apr 13 '13 at 17:04
• Great, thanks for pointing this out. – George Melvin Apr 13 '13 at 17:06
• Hi George! I fixed the brace problem by using \lbrace and \rbrace instead. – José Figueroa-O'Farrill Apr 13 '13 at 18:07
• Hi Jos\'{e}! Thanks for the help. I may be around Edinburgh in the summer, if so then I'll drop by and say 'hello'. Hope you're well. Cheers, George – George Melvin Apr 13 '13 at 18:13

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
• Mmm, not sure about that; I never really read that book. You should also be able to use the projective $P(w_0w)$ in place of the tilting $T(w)$; for some reason tilting modules felt better when I was writing the answer, but there's a functor that switches projectives and tiltings and induces the isomorphism $P(w_0)=T(e)$ so its essentially the same story. – Ben Webster Apr 13 '13 at 20:12