Nagoya Mathematical Journal

Tensor product structure of affine Demazure modules and limit constructions

G. Fourier and P. Littelmann

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Abstract

Let $\mathfrak{g}$ be a simple complex Lie algebra, we denote by $\widehat{\mathfrak{g}}$ the affine Kac-Moody algebra associated to the extended Dynkin diagram of $\mathfrak{g}$. Let $\Lambda_{0}$ be the fundamental weight of $\widehat{\mathfrak{g}}$ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral $\mathfrak{g}$-coweight $\lambda^{\vee}$, the Demazure submodule $V_{-\lambda^{\vee}}(m\Lambda_{0})$ is a $\mathfrak{g}$-module. We provide a description of the $\mathfrak{g}$-module structure as a tensor product of "smaller" Demazure modules. More precisely, for any partition of $\lambda^{\vee} = \sum_{j} \lambda_{j}^{\vee}$ as a sum of dominant integral $\mathfrak{g}$-coweights, the Demazure module is (as $\mathfrak{g}$-module) isomorphic to $\bigotimes_{j} V_{-\lambda^{\vee}_{j}}(m\Lambda_{0})$. For the "smallest" case, $\lambda^{\vee} = \omega^{\vee}$ a fundamental coweight, we provide for $\mathfrak{g}$ of classical type a decomposition of $V_{-\omega^{\vee}}(m\Lambda_{0})$ into irreducible $\mathfrak{g}$-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the $U_{q}(\mathfrak{g})$-characters of certain finite dimensional $U_{q}'(\widehat{\mathfrak{g}})$-modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules $V_{-\lambda^{\vee}, q}(m\Lambda_{0})$ can be naturally endowed with the structure of a $U_{q}'(\widehat{\mathfrak{g}})$-module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the "smallest" Demazure modules are, when viewed as $\mathfrak{g}$-modules, isomorphic to some KR-modules. For an integral dominant $\widehat{\mathfrak{g}}$-weight $\Lambda$ let $V(\Lambda)$ be the corresponding irreducible $\widehat{\mathfrak{g}}$-representation. Using the tensor product decomposition for Demazure modules, we give a description of the $\mathfrak{g}$-module structure of $V(\Lambda)$ as a semi-infinite tensor product of finite dimensional $\mathfrak{g}$-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

Article information

Source
Nagoya Math. J., Volume 182 (2006), 171-198.

Dates
First available in Project Euclid: 20 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1150810006

Mathematical Reviews number (MathSciNet)
MR2235341

Zentralblatt MATH identifier
1143.22010

Subjects
Primary: 22E46: Semisimple Lie groups and their representations 14M15: Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35]

Citation

Fourier, G.; Littelmann, P. Tensor product structure of affine Demazure modules and limit constructions. Nagoya Math. J. 182 (2006), 171--198. https://projecteuclid.org/euclid.nmj/1150810006


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References

  • N. Bourbaki, Algèbre de Lie IV--VI, Hermann, Paris (1968).
  • V. Chari, On the fermionic formula and the Kirillov-Reshetikhin conjecture , Internat. Math. Res. Notices, 12 (2001), 629--654.
  • M. Demazure, Une nouvelle formule de caractère , Bull. Sc. math., 98 , 163--172 (1974).
  • G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Z. Tsuboi, Paths, crystals and fermionic formula , MathPhys Odyssey 2001: Integrable Models and Beyond In Hormor of Barry M. McCoy, Birkhäuser (2002), 205--272.
  • G. Hatayama, A. N. Kirillov, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Character formulae of $\widehat\mathfraksl_n$-modules and inhomogeneous paths , Nucl. Phys., B536 [PM] (1999), 575--616.
  • J. Hong and S.-J. Kang, Introduction to Quantum Groups and Crystal Bases, Graduate Studies in Mathematics, Vol. 42, AMS (2002).
  • V. Kac, Infinite dimensional Lie algebras, third edition, Cambridge University Press, Cambridge (1990).
  • M. Kleber, Combinatorial structure of finite dimensional representations of Yangians, the simply-laced case , Int. Math. Res. Not., 7 (1997, no. 4), 187--2001.
  • M. Kashiwara, On Level Zero Representations of Quantized Affine Algebras , Duke Math. J., 112 n.1 (2002), 117--175.
  • M. Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules , Publ. Res. Inst. Math. Sci., 41 (2005, no. 1), 223--250.
  • S. Kumar, Kac-Moody Groups, their flag varieties and representation theory, Progress in Mathematics, Birkhäuser Verlag, Boston (2002).
  • S. Kumar, Demazure Character Formula in arbitrary Kac-Moody setting , Invent. Math., 89 (1987), 395--423.
  • A. Kuniba, K. Misra, M. Okado, T. Takagi, and J. Uchiyama, Crystals for Demazure Modules of Classical Affine Lie Algebras , J. Algebra, 208 (1998, no. 1), 185--215.
  • A. Kuniba, K. Misra, M. Okado, T. Takagi, and J. Uchiyama, Demazure Modules and Perfect Crystals , J. Comm. Math. Phys., 192 (1998), 555--567.
  • P. Littelmann, On spherical double cones , J. Algebra, 89 (1994), 142--157.
  • P. Magyar, Littelmann Paths for the Basic Representation of an Affine Lie Algebra , math.RT/0308156.
  • O. Mathieu, Formules de caractères pour les algèbres de Kac-Moody générales , Astérisque, 159--160 (1988).
  • S. Naito and D. Sagaki, Crystal of Lakshmibai-Seshadri Paths associated to an Integral Weight of Level Zero for an Affine Lie Algebra , math.QA/051001.
  • S. Naito and D. Sagaki, Construction of perfect crystals conjecturally corresponding to Kirillov-Reshetikhin modules over twisted quantum affine algebras , math.QA/0503287.
  • Y. Sanderson, Real characters for Demazure modules of rank two affine Lie algebras , J. Algebra, 184 (1996).
  • S. Yamane, Perfect crystals of $U_q(\tt G_2^(1))$ , J. Algebra, 210 (1998, no. 2), 440--486.