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On the composition structure of the twisted Verma modules for \(\mathfrak{sl} (3,\mathbb{C})\). (English) Zbl 1363.17010
In this paper the authors describe the composition structure for twisted Verma modules in the BGG category \(\mathcal{O}\) for the Lie algebra \(\mathfrak{sl}(3,\mathbb{C})\). The arguments are based on the use of partial Fourier transforms applied to realizations of twisted Verma modules as \(\mathcal{D}\)-modules on Schubert cells of the full flag manifold of \(\mathrm{SL}(3,\mathbb{C})\).
The same results can also be obtained using projective functors and their properties which can be described in terms of Kazhdan-Lusztig combinatorics. For the particular case of the Lie algebra \(\mathfrak{sl}(3,\mathbb{C})\), the composition structure of twisted (=shuffled) Verma modules in the principal block of \(\mathcal{O}\) can be read off from page 2970 of thereviewer and C. Stroppel [Trans. Am. Math. Soc. 357, No. 7, 2939–2973 (2005; Zbl 1095.17001)].

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20 Simple, semisimple, reductive (super)algebras
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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