Twining Character Formula of Kac-Wakimoto Type for Affine Lie Algebras

Naito, Satoshi

doi:10.1090/S1088-4165-02-00120-6

Twining Character Formula of Kac-Wakimoto Type for Affine Lie Algebras
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by Satoshi Naito

Represent. Theory 6 (2002), 70-100

DOI: https://doi.org/10.1090/S1088-4165-02-00120-6

Published electronically: July 16, 2002

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Abstract:

We prove a formula of Kac-Wakimoto type for the twining characters of irreducible highest weight modules of symmetric, noncritical, integrally dominant highest weights over affine Lie algebras. This formula describes the twining character in terms of the subgroup of the integral Weyl group consisting of elements which commute with the Dynkin diagram automorphism. The main tools in our proof are the (Jantzen) translation functor and the existence result of a certain local composition series which is stable under the Dynkin diagram automorphism.

References

Richard Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1988), no. 2, 501–512. MR 943273, DOI 10.1016/0021-8693(88)90275-X
Vinay V. Deodhar, Ofer Gabber, and Victor Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. in Math. 45 (1982), no. 1, 92–116. MR 663417, DOI 10.1016/S0001-8708(82)80014-5
Miloud Benayed, Lie bialgebras real cohomology [Lie bialgebras and cohomology], J. Lie Theory 7 (1997), no. 2, 287–292. MR 1473171, DOI 10.1006/jabr.1996.6907
Jürgen Fuchs, Bert Schellekens, and Christoph Schweigert, From Dynkin diagram symmetries to fixed point structures, Comm. Math. Phys. 180 (1996), no. 1, 39–97. MR 1403859, DOI 10.1007/BF02101182
Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943, DOI 10.1007/BFb0069521
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Victor G. Kac and Minoru Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 14, 4956–4960. MR 949675, DOI 10.1073/pnas.85.14.4956
Seok-Jin Kang and Jae-Hoon Kwon, Graded Lie superalgebras, supertrace formula, and orbit Lie superalgebras, Proc. London Math. Soc. (3) 81 (2000), no. 3, 675–724. MR 1781152, DOI 10.1112/S0024611500012661
Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras. III. Positive rational case, Asian J. Math. 2 (1998), no. 4, 779–832. Mikio Sato: a great Japanese mathematician of the twentieth century. MR 1734129, DOI 10.4310/AJM.1998.v2.n4.a8
Masaki Kashiwara and Toshiyuki Tanisaki, Characters of irreducible modules with non-critical highest weights over affine Lie algebras, Representations and quantizations (Shanghai, 1998) China High. Educ. Press, Beijing, 2000, pp. 275–296. MR 1802178
Shrawan Kumar, Extension of the category ${\scr O}^g$ and a vanishing theorem for the Ext functor for Kac-Moody algebras, J. Algebra 108 (1987), no. 2, 472–491. MR 892913, DOI 10.1016/0021-8693(87)90111-6
Shrawan Kumar, Toward proof of Lusztig’s conjecture concerning negative level representations of affine Lie algebras, J. Algebra 164 (1994), no. 2, 515–527. MR 1271251, DOI 10.1006/jabr.1994.1073
Robert V. Moody and Arturo Pianzola, Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1323858
Satoshi Naito, Embedding into Kac-Moody algebras and construction of folding subalgebras for generalized Kac-Moody algebras, Japan. J. Math. (N.S.) 18 (1992), no. 1, 155–171. MR 1173834, DOI 10.4099/math1924.18.155

Character formula of Kac-Wakimoto type for generalized Kac-Moody algebras

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