Demazure modules and Weyl modules: The twisted current case

Fourier, Ghislain; Kus, Deniz

doi:10.1090/S0002-9947-2013-05846-1

Demazure modules and Weyl modules: The twisted current case
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by Ghislain Fourier and Deniz Kus PDF

Trans. Amer. Math. Soc. 365 (2013), 6037-6064 Request permission

Abstract:

We study finite–dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure modules for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover, we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight. These results were previously known for untwisted current algebras and are new for all twisted types.

References

Jonathan Beck and Hiraku Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, 335–402. MR 2066942, DOI 10.1215/S0012-7094-04-12325-2X
R. W. Carter, Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, vol. 96, Cambridge University Press, Cambridge, 2005. MR 2188930, DOI 10.1017/CBO9780511614910
Vyjayanthi Chari, On the fermionic formula and the Kirillov-Reshetikhin conjecture, Internat. Math. Res. Notices 12 (2001), 629–654. MR 1836791, DOI 10.1155/S1073792801000332
Vyjayanthi Chari, Ghislain Fourier, and Tanusree Khandai, A categorical approach to Weyl modules, Transform. Groups 15 (2010), no. 3, 517–549. MR 2718936, DOI 10.1007/s00031-010-9090-9
Vyjayanthi Chari, Ghislain Fourier, and Prasad Senesi, Weyl modules for the twisted loop algebras, J. Algebra 319 (2008), no. 12, 5016–5038. MR 2423816, DOI 10.1016/j.jalgebra.2008.02.030
Vyjayanthi Chari and Sergei Loktev, Weyl, Demazure and fusion modules for the current algebra of $\mathfrak {s}\mathfrak {l}_{r+1}$, Adv. Math. 207 (2006), no. 2, 928–960. MR 2271991, DOI 10.1016/j.aim.2006.01.012
Vyjayanthi Chari and Adriano Moura, The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras, Comm. Math. Phys. 266 (2006), no. 2, 431–454. MR 2238884, DOI 10.1007/s00220-006-0032-2
Vyjayanthi Chari and Andrew Pressley, Integrable and Weyl modules for quantum affine $\textrm {sl}_2$, Quantum groups and Lie theory (Durham, 1999) London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 48–62. MR 1903959
Vyjayanthi Chari and Andrew Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191–223. MR 1850556, DOI 10.1090/S1088-4165-01-00115-7
Michel Demazure, Une nouvelle formule des caractères, Bull. Sci. Math. (2) 98 (1974), no. 3, 163–172. MR 430001
B. Feigin and S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 61–79. MR 1729359, DOI 10.1090/trans2/194/04
B. Feigin and S. Loktev, Multi-dimensional Weyl modules and symmetric functions, Comm. Math. Phys. 251 (2004), no. 3, 427–445. MR 2102326, DOI 10.1007/s00220-004-1166-8
Ghislain Fourier, Tanusree Khandai, Deniz Kus, and Alistair Savage, Local Weyl modules for equivariant map algebras with free abelian group actions, J. Algebra 350 (2012), 386–404. MR 2859894, DOI 10.1016/j.jalgebra.2011.10.018
G. Fourier and P. Littelmann, Tensor product structure of affine Demazure modules and limit constructions, Nagoya Math. J. 182 (2006), 171–198. MR 2235341, DOI 10.1017/S0027763000026866
G. Fourier and P. Littelmann, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), no. 2, 566–593. MR 2323538, DOI 10.1016/j.aim.2006.09.002
Ghislain Fourier, Nathan Manning, and Prasad Senesi, Global Weyl modules for the twisted loop algebra, Abh. Math. Semin. Univ. Hambg. 83 (2013), no. 1, 53–82. MR 3055822, DOI 10.1007/s12188-013-0074-2
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Goro Hatayama, Atsuo Kuniba, Masato Okado, Taichiro Takagi, and Zengo Tsuboi, Paths, crystals and fermionic formulae, MathPhys odyssey, 2001, Prog. Math. Phys., vol. 23, Birkhäuser Boston, Boston, MA, 2002, pp. 205–272. MR 1903978
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Michael Steven Kleber, Finite dimensional representations of quantum affine algebras, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–University of California, Berkeley. MR 2697975
Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198, DOI 10.1007/978-1-4612-0105-2
Michael Lau, Representations of multiloop algebras, Pacific J. Math. 245 (2010), no. 1, 167–184. MR 2602688, DOI 10.2140/pjm.2010.245.167
Olivier Mathieu, Construction du groupe de Kac-Moody et applications, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 5, 227–230 (French, with English summary). MR 932325
Hiraku Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. MR 1808477, DOI 10.1090/S0894-0347-00-00353-2
Katsuyuki Naoi, Weyl modules, Demazure modules and finite crystals for non-simply laced type, arXiv:1012.5480.
Erhard Neher, Alistair Savage, and Prasad Senesi, Irreducible finite-dimensional representations of equivariant map algebras, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2619–2646. MR 2888222, DOI 10.1090/S0002-9947-2011-05420-6