On the composition structure of the twisted Verma modules for 𝔰𝔩 ( 3 , ℂ )

Libor Křižka; Petr Somberg

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 5, page 315-329
  • ISSN: 0044-8753

Abstract

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We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra 𝔰𝔩 ( 3 , ℂ ) , including the explicit structure of singular vectors for both 𝔰𝔩 ( 3 , ℂ ) and one of its Lie subalgebras 𝔰𝔩 ( 2 , 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 SL ( 3 , ℂ ) .

How to cite

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Křižka, Libor, and Somberg, Petr. "On the composition structure of the twisted Verma modules for $\mathfrak {sl}(3,\mathbb {C})$." Archivum Mathematicum 051.5 (2015): 315-329. <http://eudml.org/doc/276135>.

@article{Křižka2015,
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 $\mathop \{\rm SL\} \nolimits (3, \mathbb \{C\})$.},
author = {Křižka, Libor, Somberg, Petr},
journal = {Archivum Mathematicum},
keywords = {Lie algebra $\mathfrak \{sl\}(3,\mathbb \{C\})$; twisted Verma modules; composition structure; $\mathcal \{D\}$-modules; composition structure; twisted Verma module; simple highest weight module; BGG category; Lie algebra $\mathfrak \{sl\}(3,\mathbb \{C\})$},
language = {eng},
number = {5},
pages = {315-329},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the composition structure of the twisted Verma modules for $\mathfrak \{sl\}(3,\mathbb \{C\})$},
url = {http://eudml.org/doc/276135},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Křižka, Libor
AU - Somberg, Petr
TI - On the composition structure of the twisted Verma modules for $\mathfrak {sl}(3,\mathbb {C})$
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 5
SP - 315
EP - 329
AB - 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 $\mathop {\rm SL} \nolimits (3, \mathbb {C})$.
LA - eng
KW - Lie algebra $\mathfrak {sl}(3,\mathbb {C})$; twisted Verma modules; composition structure; $\mathcal {D}$-modules; composition structure; twisted Verma module; simple highest weight module; BGG category; Lie algebra $\mathfrak {sl}(3,\mathbb {C})$
UR - http://eudml.org/doc/276135
ER -

References

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  11. Kobayashi, T., Ørsted, B., Somberg, P., Souček, V., Branching laws for Verma modules and applications in parabolic geometry. I, Adv. Math. 285 (2015), 1–57. (2015) Zbl1327.53044MR3406542
  12. Křižka, L., Somberg, P., Algebraic analysis on scalar generalized Verma modules of Heisenberg parabolic type I.: A n -series, (2015) arXiv:1502.07095. 
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