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A Weyl module stratification of integrable representations. With an appendix by Ryosuke Kodera. (English) Zbl 1445.17005
In this work, the authors construct explicitly a filtration on integrable highest weight modules of affine Lie algebras having the property that the adjoint graded quotient decomposes as a direct sum of Weyl modules. It is shown that the graded multiplicity of such constituents is determined by the corresponding level-restricted Kostka polynomial, providing a geometric interpretation of these polynomials. The case of the level one is studied in detail, and global Weyl modules of current algebras of type ADE are realized in terms of Schubert subvarieties of thick affine Grassmannians.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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[1] Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Volume 67 of Mathematical Surveys and Monographs. American Mathematical Society, 2 edn, Providence, RI (2000) · Zbl 0980.22015
[2] Chari, V.; Fourier, G.; Khandai, T., A categorical approach to weyl modules, Transform. Groups, 15, 517-549, (2010) · Zbl 1245.17004
[3] Chari, V.; Greenstein, J., Current algebras, highest weight categories and quivers, Adv. Math., 216, 811-840, (2007) · Zbl 1222.17010
[4] Chari, V.; Ion, B., BGG reciprocity for current algebras, Compos. Math., 151, 1265-1287, (2015) · Zbl 1337.17016
[5] Chari, V.; Loktev, S., Weyl, Demazure and fusion modules for the current algebra of \({\mathfrak{sl}_{r+1}}\), Adv. Math., 207, 928-960, (2006) · Zbl 1161.17318
[6] Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras. Represent. Theory. 5, 191-223 (electronic) (2001) · Zbl 0989.17019
[7] Cherednik, I.; Feigin, B., Rogers-Ramanujan type identities and Nil-DAHA, Adv. Math., 248, 1050-1088, (2013) · Zbl 1298.33029
[8] Feigin, B.; Feigin, E., Homological realization of restricted Kostka polynomials, Int. Math. Res. Not., 33, 1997-2029, (2005) · Zbl 1162.17313
[9] Feigin, B.; Frenkel, E.; Reshetikhin, N., Gaudin model, Bethe ansatz and critical level, Commun. Math. Phys., 166, 27-62, (1994) · Zbl 0812.35103
[10] Fiebig, P., Centers and translation functors for the category \({{\mathscr{O}}}\) over Kac-Moody algebras, Math. Z., 243, 689-717, (2003) · Zbl 1021.17007
[11] Fishel, S.; Grojnowski, I.; Teleman, C., The strong Macdonald conjecture and Hodge theory on the loop Grassmannian, Ann. Math. (2), 168, 175-220, (2008) · Zbl 1186.17010
[12] Fourier, G.; Littelmann, P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math., 211, 566-593, (2007) · Zbl 1114.22010
[13] Frenkel, I.B., Kac V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62(1), 23-66 (1980/81) · Zbl 0493.17010
[14] Hatayama, G.; Kirillov Anatol, N.; Kuniba, A.; Okado, M.; Takagi, T.; Yamada, Y., Character formulae of \({\widehat{\rm sl}_n}\)-modules and inhomogeneous paths, Nucl. Phys. B, 536, 575-616, (1999) · Zbl 0952.17013
[15] Heckenberger, I.; Kolb, S., On the Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and quantized enveloping algebras, Transform. Groups, 12, 647-655, (2007) · Zbl 1138.17010
[16] Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases, Volume 42 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2002) · Zbl 1134.17007
[17] Kac, V. G.; Kazhdan, D. A., Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. Math., 34, 97-108, (1979) · Zbl 0427.17011
[18] Kac Victor G.: Infinite-Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990) · Zbl 0716.17022
[19] Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki A.: Affine crystals and vertex models. In: Infinite analysis, Part A, B (Kyoto, 1991), Volume 16 of Adv. Ser. Math. Phys., pp. 449-484. World Sci. Publ., River Edge, NJ (1992) · Zbl 0925.17005
[20] Kashiwara, M., On crystal bases of the \(Q\)-analogue of universal enveloping algebras, Duke Math J., 63, 465-516, (1991) · Zbl 0739.17005
[21] Kashiwara, M.; Miwa, T.; Stern, E., Decomposition of \(q\)-deformed Fock spaces, Sel. Math. (N.S.), 1, 787-805, (1995) · Zbl 0857.17013
[22] Kashiwara, M.: The flag manifold of Kac-Moody Lie algebra. In: Proceedings of the JAMI Inaugural Conference, supplement to Amer. J. Math. the Johns Hopkins University Press (1989) · Zbl 0764.17019
[23] Kashiwara, M.: Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra. In: The Grothendieck Festschrift, Vol. II, Volume 87 of Progr. Math., pp. 407-433. Birkhäuser Boston, Boston, MA, (1990)
[24] Kashiwara, M., The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., 71, 839-858, (1993) · Zbl 0794.17008
[25] Kashiwara, M., On level-zero representations of quantized affine algebras, Duke Math. J., 112, 117-175, (2002) · Zbl 1033.17017
[26] Kashiwara, M.; Tanisaki, T., Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J., 77, 21-62, (1995) · Zbl 0829.17020
[27] Kato, S., A homological study of Green polynomials, Ann Sci. Éc. Norm. Supér. (4), 48, 1035-1074, (2015) · Zbl 1367.20038
[28] Kato S.: Demazure character formula for semi-infinite flag varieties. Math. Ann. 371(3), 1769-1801 (2018) arXiv:1605.0279 · Zbl 1398.14053
[29] Kato, S.: Frobenius splitting of thick flag manifolds of Kac-Moody algebras. Int. Math. Res. Not. IMRN, to appear (2018). https://doi.org/10.1093/imrn/rny174
[30] Kazhdan, D.; Lusztig, G., Tensor structures arising from affine Lie algebras, I. J Am. Math. Soc., 6, 905-947, (1993) · Zbl 0786.17017
[31] Khoroshkin, A.: Highest weight categories and macdonald polynomials (2013). arXiv:1312.7053
[32] Kleshchev, A. S., Affine highest weight categories and affine quasihereditary algebras, Proc. Lond. Math. Soc. (3), 110, 841-882, (2015) · Zbl 1360.16010
[33] Kumar, S., Demazure character formula in arbitrary Kac-Moody setting, Invent. Math., 89, 395-423, (1987) · Zbl 0635.14023
[34] Kumar, S.: Kac-Moody Groups, Their Flag Varieties and Representation Theory, Volume 204 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, (2002) · Zbl 1026.17030
[35] Lenart, C.; Naito, S.; Sagaki, D.; Schilling, A.; Shimozono, M., A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at \(t\) = 0 and Demazure characters, Transform. Groups, 22, 1041-1079, (2015) · Zbl 1428.05325
[36] Lenart, C., Naito, S., Sagaki, D., Schilling, A., Shimozono, M.: A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X. Int. Math. Res. Not. IMRN (2016). arXiv:1402.2203 · Zbl 1405.05194
[37] Lusztig, G., Hecke algebras and Jantzen’s generic decomposition patterns, Adv. Math., 37, 121-164, (1980) · Zbl 0448.20039
[38] Naito, S.; Sagaki, D., Lakshmibai-Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations, Compos. Math., 144, 1525-1556, (2008) · Zbl 1234.17010
[39] Naoi, K., Weyl modules, Demazure modules and finite crystals for non-simply laced type, Adv. Math., 229, 875-934, (2012) · Zbl 1305.17009
[40] Masato Okado. mimeo.
[41] Polo, P.: Projective versus injective modules over graded Lie algebras and a particular parabolic category \({{\mathscr{O}}}\) for affine Kac-Moody algebras. preprint.
[42] Schilling, A., Shimozono, M.: Bosonic formula for level-restricted paths. In: Combinatorial Methods in Representation Theory (Kyoto, 1998), Volume 28 of Adv. Stud. Pure Math., pp. 305-325. Kinokuniya, Tokyo (2000) · Zbl 1058.17500
[43] Schilling, A.; Ole Warnaar, S., Inhomogeneous lattice paths, generalized Kostka polynomials and \({A_{n-1}}\) supernomials, Commun. Math. Phys., 202, 359-401, (1999) · Zbl 0935.05090
[44] Soergel, W.: Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory 1, 83-114 (electronic) (1997) · Zbl 0886.05123
[45] Teleman, C., Lie algebra cohomology and the fusion rules, Commun. Math. Phys., 173, 265-311, (1995) · Zbl 0842.17038
[46] Zhu, X., Affine Demazure modules and \(T\)-fixed point subschemes in the affine Grassmannian, Adv. Math., 221, 570-600, (2009) · Zbl 1167.14033
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