The classification of blocks in BGG category \(\mathcal{O}\) (Q2182439)

A block in the BGG category \(\mathcal{O}\) of a complex semisimple Lie algebra is determined up to equivalence by its integral Weyl group \(W\) and a parabolic subgroup \(W^\prime \leq W\) by results of [\textit{W. Soergel}, J. Am. Math. Soc. 3, No. 2, 421--445 (1990; Zbl 0747.17008)]. Therefore, a block can be denoted unambigously by \(\mathcal{O}(W,W^\prime)\). The main result of this article is that, for any two finite Weyl groups \(W\) and \(U\) with parabolic subgroups \(W^\prime \leq W\) and \(U^\prime \leq U\), the blocks \(\mathcal{O}(W,W^\prime)\) and \(\mathcal{O}(U,U^\prime)\) are equivalent if and only if the Bruhat orders on \(W / W^\prime\) and \(U / U^\prime\) are isomorphic. As part of the proof, it is observed that any finite-dimensional algebra with simple preserving duality admits at most one quasi-hereditary structure. The author further determines all pairs \((W,W^\prime)\) and \((U,U^\prime)\) such that there exists an isomorphisms between the Bruhat orders on \(W / W^\prime\) and \(U / U^\prime\) and thus obtains a complete classification of the blocks in category \(\mathcal{O}\).