On the composition structure of the twisted Verma modules for $\mathfrak{sl}(3,\mathbb{C})$

Libor Křižka and Petr Somberg

Address: Mathematical Institute of Charles University, Sokolovská 83, 180 00 Praha 8, Czech Republic

E-mail:
krizka@karlin.mff.cuni.cz
somberg@karlin.mff.cuni.cz

Abstract: We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $\mathfrak{sl}(3, \mathbb{C})$, including the explicit structure of singular vectors for both $\mathfrak{sl}(3, \mathbb{C})$ and one of its Lie subalgebras $\mathfrak{sl}(2, \mathbb{C})$, and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as ${D}$-modules on the Schubert cells in the full flag manifold for $\mathop {\rm SL} \nolimits (3, \mathbb{C})$.

AMSclassification: primary 53A30; secondary 22E47, 33C45, 58J70.

Keywords: Lie algebra \mathfrak{sl}(3,\mathbb{C}), twisted Verma modules, composition structure, \mathcal{D}-modules.

DOI: 10.5817/AM2015-5-315