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Classification of horizontal $\text{SL}(2)$s

Part of: Families, fibrations Lie algebras and Lie superalgebras Deformations of analytic structures General theory of differentiable manifolds

Published online by Cambridge University Press:  07 December 2015

Colleen Robles
Colleen Robles*
Affiliation:
Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320, USA email robles@math.duke.edu
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Abstract

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We classify the horizontal $\text{SL}(2)$s and $\mathbb{R}$-split polarized mixed Hodge structures on a Mumford–Tate domain.

Keywords

horizontal SL(2)nilpotent orbitpolarized mixed Hodge structureHodge–Tate degeneration

MSC classification

Primary: 14D07: Variation of Hodge structures 32G20: Period matrices, variation of Hodge structure; degenerations 17B08: Coadjoint orbits; nilpotent varieties 58A14: Hodge theory
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 5 , May 2016 , pp. 918 - 954
Copyright
© The Author 2015 

References

Altomani, A., Medori, C. and Nacinovich, M., Orbits of real forms in complex flag manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), 69–109.Google Scholar
Bala, P. and Carter, R. W., Classes of unipotent elements in simple algebraic groups. I, Math. Proc. Cambridge Philos. Soc. 79 (1976), 401–425.Google Scholar
Bala, P. and Carter, R. W., Classes of unipotent elements in simple algebraic groups. II, Math. Proc. Cambridge Philos. Soc. 80 (1976), 1–17.Google Scholar
Brosnan, P. and Pearlstein, G., On the algebraicity of the zero locus of an admissible normal function, Compositio Math. 149 (2013), 1913–1962.Google Scholar
Brosnan, P., Pearlstein, G. and Robles, C., Nilpotent cones and their representation theory, in preparation; an expository chapter in a collection honoring Steve Zucker’s work, 2015.Google Scholar
Čap, A. and Slovák, J., Background and general theory, inParabolic geometries. I, Mathematical Surveys and Monographs, vol. 154 (American Mathematical Society, Providence, RI, 2009).Google Scholar
Cattani, E., Deligne, P. and Kaplan, A., On the locus of Hodge classes, J. Amer. Math. Soc. 8 (1995), 483–506.Google Scholar
Cattani, E. H. and Kaplan, A. G., Extension of period mappings for Hodge structures of weight two, Duke Math. J. 44 (1977), 1–43.Google Scholar
Cattani, E. H. and Kaplan, A. G., Horizontal SL2 -orbits in flag domains, Math. Ann. 235 (1978), 17–35.CrossRefGoogle Scholar
Cattani, E. and Kaplan, A., Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure, Invent. Math. 67 (1982), 101–115.Google Scholar
Cattani, E. and Kaplan, A., On the sl(2)-orbits in Hodge theory, Publ. Math. Inst. Hautes Études Sci. Preprint M/82/58, October 1982.Google Scholar
Cattani, E. and Kaplan, A., Sur la cohomologie L 2 et la cohomologie d’intersection à coefficients dans une variation de structure de Hodge, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), 351–353.Google Scholar
Cattani, E. and Kaplan, A., Degenerating variations of Hodge structure, Astérisque 9 (1989), 67–96; Actes du Colloque de Théorie de Hodge (Luminy, 1987).Google Scholar
Cattani, E. and Kaplan, A., L 2 and intersection cohomologies for a polarizable variation of Hodge structure, Invent. Math. 87 (1987), 217–252.CrossRefGoogle Scholar
Cattani, E., Kaplan, A. and Schmid, W., Degeneration of Hodge structures, Ann. of Math. (2) 123 (1986), 457–535.Google Scholar
Collingwood, D. H. and McGovern, W. M., Nilpotent orbits in semisimple Lie algebras (Van Nostrand Reinhold, New York, 1993).Google Scholar
Deligne, P., Théorie Hodge. II, Publ. Math. Inst. Hautes Études Sci. 40 (1971), 5–57.Google Scholar
Deligne, P., Structures de Hodge mixtes réelles, appendix to [CK82b].Google Scholar
Djoković, D. Ž., Proof of a conjecture of Kostant, Trans. Amer. Math. Soc. 302 (1987), 577–585.Google Scholar
Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Mat. Sb. (N.S.) 30 (1952), 349–462 (3 plates).Google Scholar
Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl. 6 (1957), 111–244; translation of [Dyn52].Google Scholar
Fels, G., Huckleberry, A. and Wolf, J. A., A complex geometric viewpoint, inCycle spaces of flag domains, Progress in Mathematics, vol. 245 (Birkhäuser, Boston, 2006).Google Scholar
Green, M., Griffiths, P. and Kerr, M., Mumford–Tate groups and domains: their geometry and arithmetic, Annals of Mathematics Studies, vol. 183 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Green, M., Griffiths, P. and Kerr, M., Hodge theory, complex geometry, and representation theory, CBMS Regional Conference Series in Mathematics, vol. 118 (American Mathematical Society for the Conference Board of the Mathematical Sciences, Providence, RI, 2013).CrossRefGoogle Scholar
Green, M., Griffiths, P. and Robles, C., Extremal degenerations of polarized Hodge structures, Preprint (2014), arXiv:1403.0646.Google Scholar
Hayama, T. and Pearlstein, G., Asymptotics of degenerations of mixed Hodge structures, Adv. Math. 273 (2015), 380–420.Google Scholar
Kashiwara, M. and Kawai, T., The Poincaré lemma for a variation of polarized Hodge structure, Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), 164–167.Google Scholar
Knapp, A. W., Lie groups beyond an introduction, Progress in Mathematics, vol. 140, second edition (Birkhäuser, Boston, 2002).Google Scholar
Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032.CrossRefGoogle Scholar
Kerr, M. and Pearlstein, G., Naive boundary strata and nilpotent orbits, Ann. Inst. Fourier (Grenoble) 64 (2014), 2659–2714.Google Scholar
Kerr, M. and Robles, C., Hodge theory and real orbits in flag varieties, Preprint (2014),arXiv:1407.4507.Google Scholar
Kato, K. and Usui, S., Classifying spaces of degenerating polarized Hodge structures, Annals of Mathematics Studies, vol. 169 (Princeton University Press, Princeton, NJ, 2009).Google Scholar
Noël, A. G., Nilpotent orbits and theta-stable parabolic subalgebras, Represent. Theory 2 (1998), 1–32 (electronic).Google Scholar
Robles, C., Schubert varieties as variations of Hodge structure, Selecta Math. (N.S.) 20 (2014), 719–768.Google Scholar
Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.Google Scholar
Sekiguchi, J., Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), 127–138.Google Scholar
Usui, S., A numerical criterion for admissibility of semi-simple elements, Tohoku Math. J. (2) 45 (1993), 471–484.CrossRefGoogle Scholar
Vinberg, È. B., The classification of nilpotent elements of graded Lie algebras, Dokl. Akad. Nauk SSSR 225 (1975), 745–748.Google Scholar
Wolf, J. A., The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237.Google Scholar
Zucker, S., Hodge theory with degenerating coefficients. L 2 cohomology in the Poincaré metric, Ann. of Math. (2) 109 (1979), 415–476.Google Scholar