Ind-Varieties of Generalized Flags: A Survey of Results

M. V. Ignatyev ; I. Penkov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Seminar on algebra and geometry of the Samara University, Tome 147 (2018), pp. 3-50.

Résumé

This paper is a review of results on the structure of the homogeneous ind-varieties $G/P$ of the ind-groups $G=GL_{\infty}(\mathbb{C})$, $SL_{\infty}(\mathbb{C})$, $SO_{\infty}(\mathbb{C})$, and $Sp_{\infty}(\mathbb{C})$, subject to the condition that $G/P$ is the inductive limit of compact homogeneous spaces $G_n/P_n$. In this case, the subgroup $P\subset G$ is a splitting parabolic subgroup of $G$ and the ind-variety $G/P$ admits a “flag realization.” Instead of ordinary flags, one considers generalized flags that are, in general, infinite chains $\mathcal{C}$ of subspaces in the natural representation $V$ of $G$ satisfying a certain condition; roughly speaking, for each nonzero vector $v$ of $V$, there exist the largest space in $\mathcal{C}$, which does not contain $v$, and the smallest space in $\mathcal{C}$ which contains $v$. We start with a review of the construction of ind-varieties of generalized flags and then show that these ind-varieties are homogeneous ind-spaces of the form $G/P$ for splitting parabolic ind-subgroups $P\subset G$. Also, we briefly review the characterization of more general, i.e., nonsplitting, parabolic ind-subgroups in terms of generalized flags. In the special case of the ind-grassmannian $X$, we give a purely algebraic-geometric construction of $X$. Further topics discussed are the Bott–Borel–Weil theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of $G/P$ for arbitrary splitting parabolic ind-subgroups $P\subset G$, as well as the orbits of real forms on $G/P$ for $G=SL_{\infty}(\mathbb{C})$.

Détail

Keywords: ind-variety, ind-group, generalized flag, real form.
Mots-clés : Schubert decomposition

Bibliographie

[1] Burbaki N., Gruppy i algebry Li. Gruppy Kokstera i sistemy Titsa. Gruppy, porozhdennye otrazheniyami. Sistemy kornei, Mir, M., 1972

[2] Penkov I. B., Tikhomirov A. S., “O teoreme Barta—Van de Vena—Tyurina—Sato”, Mat. sb., 206:6 (2015), 49–84 | DOI | MR | Zbl

[3] Penkov I. B., Tikhomirov A. S., “Trivialnost vektornykh rassloenii na skruchennykh ind-grassmanianakh”, Mat. sb., 202:1 (2011), 65–104 | DOI | MR | Zbl

[4] Pressli E., Sigal G., Gruppy petel, Mir, M., 1990 | MR

[5] Tyurin A. N., “Konechnomernye rassloeniya na beskonechnykh mnogoobraziyakh”, Izv. AN SSSR. Ser. mat., 40:6 (1976), 1248–1268 | MR | Zbl

[6] Baranov A. A., “Finitary simple Lie algebras”, J. Algebra, 219 (1999), 299–329 | DOI | MR | Zbl

[7] Barth W., Van de Ven A., “A decomposability criterion for algebraic 2-bundles on projective spaces”, Invent. Math., 25 (1974), 91–106 | DOI | MR | Zbl

[8] Billey S., Lakshmibai V., Singular loci of Schubert varieties, Birkhäuser, Boston, MA, 2000 | MR | Zbl

[9] Bjorner A., Brenti F., Combinatorics of Coxeter groups, Springer-Verlag, Berlin, 2005 | MR | Zbl

[10] Bott R., “Homogeneous vector bundles”, Ann. Math., 66 (1957), 203–248 | DOI | MR | Zbl

[11] Dan-Cohen E., “Borel subalgebras of root-reductive Lie algebras”, J. Lie Theory, 18 (2008), 215–241 | MR | Zbl

[12] Dan-Cohen E., Penkov I., “Parabolic and Levi subalgebras of finitary Lie algebras”, Int. Math. Res. Not., 2010, no. 6, 1062–1101 | MR | Zbl

[13] Dan-Cohen E., Penkov I., Snyder N., “Cartan subalgebras of root-reductive Lie algebras”, J. Algebra, 308:2 (2007), 583–611 | DOI | MR | Zbl

[14] Demazure M., “Une démonstration algébrique d'un théorème de Bott”, Invent. Math., 5 (1968), 349–356 | DOI | MR | Zbl

[15] Demazure M., “A very simple proof of Bott’s theorem”, Invent. Math., 33 (1976), 271–272 | DOI | MR | Zbl

[16] Dimitrov I., Penkov I., “Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups”, Int. Math. Res. Not., 2004, no. 55, 2935–2953 | DOI | MR | Zbl

[17] Dimitrov I., Penkov I., “Borel subalgebras of $\mathfrak{gl}(\infty)$”, Resenhas IME-USP, 6:2–3 (2004), 153–163 | MR | Zbl

[18] Dimitrov I., Penkov I., “Weight modules of direct limit Lie algebras”, Int. Math. Res. Not., 1999, no. 5, 223–249 | DOI | MR | Zbl

[19] Dimitrov I., Penkov I., Wolf J. A., “A Bott–Borel–Weil theory for direct limits of algebraic groups”, Am. J. Math., 124 (2002), 955–998 | DOI | MR | Zbl

[20] Fels G., Huckleberry A. T., Wolf J. A., Cycle spaces of flag domains: a complex geometric viewpoint, Birkhäuser/Springer, Boston, 2005 | MR

[21] Fresse L., Penkov I., Schubert decomposition for ind-varieties of generalized flags, arXiv: 1506.08263 [math.RT] | MR

[22] Fresse L., Penkov I., Orbit duality in ind-varieties of maximal flags, arXiv: 1704.03671 [math.AG] | MR

[23] Hartshorne R., “On the de Rham cohomology of algebraic varieties”, Inst. Hautes Études Sci. Publ. Math., 45 (1976), 5–99 | MR

[24] Hristova E., Penkov I., Decomposition of cohomology of vector bundles on homogeneous ind-spaces, arXiv: 1703.05086 [math.RT] | MR

[25] Huang H., “Some extensions of Witt's theorem”, Linear Multilin. Algebra, 57 (2009), 321–344 | DOI | MR | Zbl

[26] Huckleberry A. T., Wolf J. A., “Cycle spaces of real forms of $\operatorname{SL}_n(\mathbb{C})$”, Complex Geometry, Springer-Verlag, Berlin, 2002, 111–133 | DOI | MR | Zbl

[27] Huckleberry A. T., Wolf J. A., “Schubert varieties and cycle spaces”, Duke Math. J., 120 (2003), 229–249 | DOI | MR | Zbl

[28] Huckleberry A. T., Wolf J. A., “Injectivity of the double fibration transform for cycle spaces of flag domains”, J. Lie Theory, 14 (2004), 509–522 | MR | Zbl

[29] Ignatyev M. V., Penkov I., Wolf J. A., “Real group orbits on flag ind-varieties of $SL(\infty,\mathbb{C})$”, Lie Theory and Its Applications in Physics, v. 191, ed. Dobrev V., Springer Proc. Math. Stat., 2016, 111–135 | DOI | MR | Zbl

[30] Kumar S., Kac–Moody groups, their flag varieties and representation theory, Birkhäuser, Boston, MA, 2002 | MR | Zbl

[31] Mathieu O., “Formules de Demazure–Weyl, et généralisation du théorème de Borel–Weil–Bott”, C. R. Acad. Sci. Paris, 303 (1986), 391–394 | MR | Zbl

[32] Mathieu O., “Formules de caractéres pour les algébres de Kac–Moody gé'erales”, Astérisque, 1988, no. 159–160 | MR

[33] Neeb K.-H., Penkov I., “Cartan subalgebras of $\mathfrak{gl}_{\infty}$”, Can. Math. Bull, 46:4 (2003), 597–616 | DOI | MR | Zbl

[34] Penkov I., Tikhomirov A., “Linear ind-grassmannians”, Pure Appl. Math. Q., 10 (2014), 289–323 | DOI | MR | Zbl

[35] Sato E., “On the decomposability of infinitely extendable vector bundles on projective spaces and Grassmann varieties”, J. Math. Kyoto Univ., 17 (1977), 127–150 | DOI | MR | Zbl

[36] Wolf J. A., “The action of a real semisimple Lie group on a complex flag manifold. I. Orbit structure and holomorphic arc components”, Bull. Am. Math. Soc., 75 (1969), 1121–1237 | DOI | MR | Zbl

[37] Wolf J. A., “The action of a real semisimple group on a complex flag manifold. II. Unitary representations on partially holomorphic cohomology spaces”, Mem. Am. Math. Soc., 138 (1974) | MR

[38] Wolf J. A., Cycle spaces of infinite dimensional flag domains, arXiv: 1509.0329 [math.DG] | MR