Restriction to Levi subalgebras and generalization of the category $𝒪$

Tomasini, Guillaume

Annales de l'Institut Fourier, Tome 63 (2013) no. 1, pp. 37-88.

L’étude de la catégorie de tous les modules sur une algèbre de Lie complexe semi-simple est un problème réputé très difficile. Il est donc utile pour approcher ce problème de se restreindre à des sous-catégories pleines. Ainsi, Bernstein, Gelfand et Gelfand ont introduit une catégorie de modules qui fournit un cadre naturel pour étudier les modules de plus haut poids. Dans cet article, nous définissons une famille de catégories qui généralise la catégorie BGG et nous étudions les modules irréductibles pour une certaine sous-famille. Comme corollaire, nous montrons que certaines de ces catégories sont semi-simples.

The category of all modules over a reductive complex Lie algebra is wild, and therefore it is useful to study full subcategories. For instance, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this paper, we define a family of categories which generalizes the BGG category, and we classify the simple modules for a subfamily. As a consequence, we show that some of the obtained categories are semisimple.

Classification : 17B10, 17B20, 17B22, 17B35, 17B55
Keywords: weight modules, cuspidal modules, branching rules
Mot clés : modules de poids, modules cuspidaux, règles de branchement

[1] Benkart, G.; Britten, D.; Lemire, F. Modules with bounded weight multiplicities for simple Lie algebras, Math. Z., Volume 225 (1997) no. 2, pp. 333-353 | DOI | MR | Zbl

[2] Bernšteĭn, I. N.; Gel’fand, I. M.; Gel’fand, S. I. Differential operators on the base affine space and a study of $𝔤$-modules, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 21-64 | MR | Zbl

[3] Bernšteĭn, I. N.; Gel’fand, I. M.; Gel’fand, S. I. A certain category of $𝔤$-modules, Funkcional. Anal. i Priložen, Volume 10 (1976) no. 2, pp. 1-8 | Zbl

[4] Bourbaki, N. Éléments de mathématique, Masson, Paris, 1981 (Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6], 290 page) | MR | Zbl

[5] Britten, D.; Khomenko, O.; Lemire, F.; Mazorchuk, V. Complete reducibility of torsion free $C_n$-modules of finite degree, J. Algebra, Volume 276 (2004) no. 1, pp. 129-142 | DOI | MR | Zbl

[6] Britten, D.; Lemire, F. A classification of simple Lie modules having a $1$-dimensional weight space, Trans. Amer. Math. Soc., Volume 299 (1987) no. 2, pp. 683-697 | MR | Zbl

[7] Coleman, A.; Futorny, V. Stratified $L$-modules, J. Algebra, Volume 163 (1994) no. 1, pp. 219-234 | DOI | MR | Zbl

[8] Drozd, Y.; Futorny, V.; Ovsienko, S. Harish-Chandra subalgebras and Gel’fand-Zetlin modules, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 79-93 | MR | Zbl

[9] Fernando, S. Lie algebra modules with finite-dimensional weight spaces. I, Trans. Amer. Math. Soc., Volume 322 (1990) no. 2, pp. 757-781 | MR | Zbl

[10] Futorny, V. The weight representations of semisimple finite dimensional Lie algebras, Kiev University (1987) (Ph. D. Thesis)

[11] Futorny, V.; Molev, A.; Ovsienko, S. The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite $W$-algebras, Adv. Math., Volume 223 (2010) no. 3, pp. 773-796 | DOI | MR | Zbl

[12] Grantcharov, D.; Serganova, V. Category of $\mathrm{𝔰𝔭}\left(2n\right)$-modules with bounded weight multiplicities, Mosc. Math. J., Volume 6 (2006) no. 1, pp. 119-134 (222) | MR | Zbl

[13] Grantcharov, D.; Serganova, V. Cuspidal representations of $\mathrm{𝔰𝔩}(n+1)$, Adv. Math., Volume 224 (2010) no. 4, pp. 1517-1547 | DOI | MR | Zbl

[14] Howe, R. Remarks on classical invariant theory, Trans. Amer. Math. Soc., Volume 313 (1989) no. 2, pp. 539-570 | DOI | MR | Zbl

[15] Howe, R. Transcending classical invariant theory, J. Amer. Math. Soc., Volume 2 (1989) no. 3, pp. 535-552 | DOI | MR | Zbl

[16] Humphreys, J. Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, Springer-Verlag, New York, 1978 (Second printing, revised, xii+171 pages) | MR | Zbl

[17] Humphreys, J. Representations of semisimple Lie algebras in the BGG category $𝒪$, Graduate Studies in Mathematics, 94, American Mathematical Society, Providence, RI, 2008 | MR | Zbl

[18] Lemire, F. Irreducible representations of a simple Lie algebra admitting a one-dimensional weight space, Proc. Amer. Math. Soc., Volume 19 (1968), pp. 1161-1164 | DOI | MR | Zbl

[19] Li, J. S. The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups, Duke Math. J., Volume 97 (1999) no. 2, pp. 347-377 | DOI | MR | Zbl

[20] Li, J. S. Minimal representations & reductive dual pairs, Representation theory of Lie groups (Park City, UT, 1998) (IAS/Park City Math. Ser.), Volume 8, Amer. Math. Soc., Providence, RI, 2000, pp. 293-340 | MR | Zbl

[21] Mathieu, O. Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 2, pp. 537-592 | DOI | EuDML | Numdam | MR | Zbl

[22] Mazorchuk, V. Generalized Verma modules, Mathematical Studies Monograph Series, 8, VNTL Publishers, L ′ viv, 2000 | MR | Zbl

[23] Mazorchuk, V. Lectures on ${\mathrm{𝔰𝔩}}_2\left(ℂ\right)$-modules, Imperial College Press, 2010 | MR | Zbl

[24] Mazorchuk, V.; Stroppel, C. Blocks of the category of cuspidal ${\mathrm{𝔰𝔭}}_{2n}$-modules, Pac. J. Math., Volume 251 (2011) no. 1, pp. 183-196 | DOI | MR | Zbl

[25] Mazorchuk, V.; Stroppel, C. Cuspidal ${\mathrm{𝔰𝔩}}_n$-modules and deformations of certain Brauer tree algebras, Adv. Math., Volume 228 (2011) no. 2, pp. 1008-1042 | DOI | MR | Zbl

[26] Penkov, I.; Serganova, V. Generalized Harish-Chandra modules, Mosc. Math. J., Volume 2 (2002) no. 4, pp. 753-767 (806, Dedicated to Yuri I. Manin on the occasion of his 65th birthday) | MR | Zbl

[27] Penkov, I.; Zuckerman, G. Generalized Harish-Chandra modules: a new direction in the structure theory of representations, Acta Appl. Math., Volume 81 (2004) no. 1-3, pp. 311-326 | DOI | MR | Zbl

[28] Przebinda, T. The duality correspondence of infinitesimal characters, Colloq. Math., Volume 70 (1996) no. 1, pp. 93-102 | EuDML | MR | Zbl

[29] Rallis, S.; Schiffmann, G. The orbit and $\theta$ correspondence for some dual pairs, J. Math. Kyoto Univ., Volume 35 (1995) no. 3, pp. 423-493 | MR | Zbl

[30] Rocha-Caridi, A. Splitting criteria for $𝔤$-modules induced from a parabolic and the Berňsteĭ n-Gel${}^{\prime }$fand-Gel${}^{\prime }$fand resolution of a finite-dimensional, irreducible $𝔤$-module, Trans. Amer. Math. Soc., Volume 262 (1980) no. 2, pp. 335-366 | MR | Zbl

[31] Stroppel, C. Category $𝒪$: quivers and endomorphism rings of projectives, Represent. Theory, Volume 7 (2003), p. 322-345 (electronic) | DOI | MR | Zbl

[32] Tomasini, G. Étude de certaines catégories de modules de poids et de leurs restrictions à des paires duales, Université de Strasbourg (2010) (Ph. D. Thesis) | MR | Zbl

[33] Tomasini, G. On a generalisation of Bernstein-Gelfand-Gelfand category $𝒪$, Comptes rendus - Mathematique, Volume 348 (2010), pp. 509-512 | DOI | MR | Zbl