Geodesic
Generalized Weyl modules for twisted current algebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 2, pp. 284-306 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and also their connection with nonsymmetric Macdonald polynomials. As an application, we compute the dimension of the classical Weyl modules in the remaining unknown case.
Keywords: Weyl module
Mots-clés : affine algebra. 
@article{TMF_2017_192_2_a7,
     author = {E. A. Makedonskii and E. B. Feigin},
     title = {Generalized {Weyl} modules for twisted current algebras},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {284--306},
     year = {2017},
     volume = {192},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a7/}
}
TY  - JOUR
AU  - E. A. Makedonskii
AU  - E. B. Feigin
TI  - Generalized Weyl modules for twisted current algebras
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2017
SP  - 284
EP  - 306
VL  - 192
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a7/
LA  - ru
ID  - TMF_2017_192_2_a7
ER  - 
%0 Journal Article
%A E. A. Makedonskii
%A E. B. Feigin
%T Generalized Weyl modules for twisted current algebras
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2017
%P 284-306
%V 192
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a7/
%G ru
%F TMF_2017_192_2_a7
E. A. Makedonskii; E. B. Feigin. Generalized Weyl modules for twisted current algebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 192 (2017) no. 2, pp. 284-306. http://geodesic.mathdoc.fr/item/TMF_2017_192_2_a7/

[1] V. Chari, B. Ion, D. Kus, “Weyl modules for the hyperspecial current algebra”, Internat. Math. Res. Notices, 2015:15 (2015), 6470–6515 | DOI | MR

[2] V. Chari, G. Fourier, P. Senesi, “Weyl modules for the twisted loop algebras”, J. Algebra, 319:12 (2008), 5016–5038 | DOI | MR | Zbl

[3] G. Fourier, D. Kus, “Demazure and Weyl modules: the twisted current case”, Trans. Amer. Math. Soc., 365:11 (2013), 6037–6064 | DOI | MR | Zbl

[4] V. Chari, A. Pressley, “Weyl modules for classical and quantum affine algebras”, Represent. Theory, 5 (2001), 191–223 | DOI | MR | Zbl

[5] V. Chari, S. Loktev, “Weyl, Demazure and fusion modules for the current algebra of $\mathfrak{sl}_{r+1}$”, Adv. Math., 207:2 (2006), 928–960 | DOI | MR | Zbl

[6] G. Fourier, P. Littelmann, “Tensor product structure of affine Demazure modules and limit constructions”, Nagoya Math. J., 182 (2006), 171–198 | DOI | MR | Zbl

[7] G. Fourier, P. Littelmann, “Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions”, Adv. Math., 211:2 (2007), 566–593 | DOI | MR | Zbl

[8] Y. Sanderson, “On the connection between Macdonald polynomials and Demazure characters”, J. Algebraic Combin., 11:3 (2000), 269–275 | DOI | MR | Zbl

[9] B. Ion, “Nonsymmetric Macdonald polynomials and Demazure characters”, Duke Math. J., 116:2 (2003), 299–318 | DOI | MR

[10] V. Chari, B. Ion, “BGG reciprocity for current algebras”, Compos. Math., 151:7 (2015), 1265–1287 | DOI | MR | Zbl

[11] I. Makdonald, Simmetricheski funktsii i mnogochleny Kholla, Mir, M., 1985 | MR | MR | Zbl

[12] I. G. Macdonald, “Affine Hecke algebras and orthogonal polynomials”, Séminaire Bourbaki (1994–1995), Astérisque, 37, Exp. No 797, Sociéte Mathématique de France, Paris, 1996, 189–207 | MR

[13] K. Naoi, “Weyl modules, Demazure modules and finite crystals for non-simply laced type”, Adv. Math., 229:2 (2012), 875–934 | DOI | MR | Zbl

[14] I. Cherednik, “Nonsymmetric Macdonald polynomials”, Internat. Math. Res. Notices, 1995:10 (1995), 483–515 | DOI | MR | Zbl

[15] I. Cherednik, Double Affine Hecke Algebras, London Mathematical Society Lecture Note Series, 319, Cambridge Univ. Press, Cambridge, 2006 | MR

[16] D. Orr, M. Shimozono, Specializations of nonsymmetric Macdonald–Koornwinder polynomials, arXiv: 1310.0279

[17] E. Feigin, I. Makedonskyi, Generalized Weyl modules, alcove paths and Macdonald polynomials, arXiv: 1512.03254

[18] E. Feigin, I. Makedonskyi, D. Orr, Generalized Weyl modules and nonsymmetric $q$-Whittaker functions, arXiv: 1605.01560

[19] I. Cherednik, D. Orr, “Nonsymmetric difference Whittaker functions”, Math. Z., 279:3–4 (2015), 879–938 | DOI | MR | Zbl

[20] I. Cherednik, D. Orr, “One-dimensional nil-DAHA and Whittaker functions. II”, Transform. Groups, 18:1 (2013), 23–59 | DOI | MR | Zbl

[21] A. Ram, M. Yip, “A combinatorial formula for Macdonald polynomials”, Adv. Math., 226:1 (2011), 309–331 | DOI | MR | Zbl

[22] S. Gaussent, P. Littelmann, “LS galleries, the path model, and MV cycles”, Duke Math. J., 127:1 (2005), 35–88 | DOI | MR | Zbl

[23] S. Naito, D. Sagaki, “Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials”, Math. Z., 283:3 (2016), 937–978, arXiv: 1404.2436 | DOI | MR

[24] S. Naito, F. Nomoto, D. Sagaki, Specialization of nonsymmetric Macdonald polynomials at $t=\infty$ and Demazure submodules of level-zero extremal weight modules, arXiv: 1511.07005

[25] G. Lusztig, “Hecke algebras and Jantzen's generic decomposition patterns”, Adv. Math., 37:2 (1980), 121–164 | DOI | MR | Zbl

[26] F. Brenti, S. Fomin, A. Postnikov, “Mixed Bruhat operators and Yang–Baxter equations for Weyl groups”, Internat. Math. Res. Notices, 1999:8 (1999), 419–441 | DOI | MR | Zbl

[27] E. Feigin, I. Makedonskyi, Weyl modules for $\mathfrak{osp}(1,2)$ and nonsymmetric Macdonald polynomials, to appear in Math. Res. Lett, arXiv: 1507.01362

[28] C. Lenart, S. Naito, D. Sagaki, A. Schilling, M. Shimozono, “A uniform model for Kirillov–Reshetikhin crystals I: Lifting the parabolic quantum Bruhat graph”, Internat. Math. Res. Notices, 2015:7 (2015), 1848–1901 | DOI | MR | Zbl

[29] C. Lenart, A. Lubovsky, A uniform realization of the combinatorial $R$-matrix, arXiv: 1503.01765