Shuffling functors and spherical twists on $D^\mathrm b(\mathcal O_0)$

Abstract

For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\mathcal{O}$ is of particular interest in representation theory. It is known that Irving's shuffling functors $\mathrm{Sh}_{w}$, indexed by elements $w\in W$ of the Weyl group, induce an action of the braid group $B_W$ associated to $W$ on the derived categories $D^\mathrm{b}(\mathcal{O}_λ)$ of blocks of $\mathcal{O}$. We show that for maximal parabolic subalgebras $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the parabolic subgroup $W_\mathfrak{p}=S_{n-1}\times S_1$ of $S_n$, the derived shuffling functors $\mathbf{L}\mathrm{Sh}{s_i}$ are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives $P^\mathfrak{p}(w)$ are spherical objects, and the associated twist functors are naturally isomorphic to $\mathbf{L}\mathrm{Sh}{w}[1]$ as auto-equivalences of $D^\mathrm{b}(\mathcal{O}^\mathfrak{p})$. We give an overview of the main properties of the BGG category $\mathcal{O}$, the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of $\mathcal{O}$ and the module categories of certain path algebras.