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Keywords:
Lie algebra $\mathfrak{sl}(3,\mathbb{C})$; twisted Verma modules; composition structure; $\mathcal{D}$-modules
Lie algebra $\mathfrak{sl}(3,\mathbb{C})$; twisted Verma modules; composition structure; $\mathcal{D}$-modules
Summary:
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 $\mathop {\rm SL} \nolimits (3, \mathbb{C})$.
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