zbMATH — the first resource for mathematics

Naive boundary strata and nilpotent orbits. (Limites de strates naïves et orbites nilpotentes.) (English. French summary) Zbl 1327.14056
Summary: We give a Hodge-theoretic parametrization of certain real Lie group orbits in the compact dual of a Mumford-Tate domain, and characterize the orbits which contain a naive limit Hodge filtration. A series of examples are worked out for the groups \(\mathrm{SU}(2, 1)\), \(\mathrm{Sp}_4\), and \(G_2\).

14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
17B45 Lie algebras of linear algebraic groups
20G99 Linear algebraic groups and related topics
32M10 Homogeneous complex manifolds
Full Text: DOI arXiv
[1] Adams, Jeffrey, Representation theory of real reductive Lie groups (Snowbird, July 2006), 472, Guide to the atlas software: computational representation theory of real reductive groups, 1-37, (2008), Amer. Math. Soc., Providence, RI · Zbl 1175.22001
[2] Adams, Jeffrey; du Cloux, Fokko, Algorithms for representation theory of real reductive groups, J. Inst. Math. Jussieu, 8, 2, 209-259, (2009) · Zbl 1221.22017
[3] Ash, Avner; Mumford, David; Rapoport, Michael; Tai, Yung-Sheng, Smooth compactifications of locally symmetric varieties, x+230 pp., (2010), Cambridge University Press, Cambridge · Zbl 1209.14001
[4] Borel, Armand, Linear algebraic groups, 126, xii+288 pp., (1991), Springer-Verlag, New York · Zbl 0726.20030
[5] Borel, Armand; Tits, Jacques, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math., 27, 55-150, (1965) · Zbl 0145.17402
[6] Carayol, Henri, Cohomologie automorphe et compactifications partielles de certaines variéetés de Griffiths-schmid, Compos. Math., 141, 5, 1081-1102, (2005) · Zbl 1173.11331
[7] Carlson, James A.; Cattani, Eduardo H.; Kaplan, Aroldo G., Algebraic Geometry, Angers, 1979 (A. Beauville, Ed), Mixed Hodge structures and compactifications of siegel’s space (preliminary report), 77-105, (1980), Sijthoff & Noordhoff · Zbl 0471.14002
[8] Cattani, Eduardo; Kaplan, Aroldo, Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure, Invent. Math., 67, 1, 101-115, (1982) · Zbl 0516.14005
[9] Cattani, Eduardo; Kaplan, Aroldo; Schmid, Wilfried, Degeneration of Hodge structures, Ann. of Math. (2), 123, 3, 457-535, (1986) · Zbl 0617.14005
[10] Cattani, Eduardo H., Mixed Hodge structures, compactifications and monodromy weight filtration, 106, 75-100, (1984), Princeton Univ. Press, Princeton, NJ · Zbl 0579.14010
[11] Fels, Gregor; Huckleberry, Alan; Wolf, Joseph A., Cycle spaces of flag domains, 245, xx+339 pp., (2006), Birkhäuser Boston, Inc., Boston, MA · Zbl 1084.22011
[12] Green, Mark; Griffiths, Phillip; Kerr, Matt, Mumford-Tate groups and domains, 183, viii+289 pp., (2012), Princeton University Press, Princeton, NJ · Zbl 1248.14001
[13] Green, Mark; Griffiths, Phillip; Kerr, Matt, Hodge theory, complex geometry, and representation theory, 118, iv+308 pp., (2013), Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI · Zbl 1364.32001
[14] Humphreys, James E., Linear algebraic groups, xiv+247 pp., (1975), Springer-Verlag, New York-Heidelberg · Zbl 0471.20029
[15] Kerr, Matt; Pearlstein, Gregory, Boundary components of Mumford-Tate domains · Zbl 1375.14045
[16] Knapp, Anthony W., Lie groups beyond an introduction, 140, xviii+812 pp., (2002), Birkhäuser Boston, Inc., Boston, MA · Zbl 1075.22501
[17] Matsuki, Toshihiko, Representations of Lie groups, Kyoto, Hiroshima, 1986, (eds. K. Okamoto and T. Oshima), 14, Closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups, 541-559, (1988), Academic Press, Boston, MA · Zbl 0723.22020
[18] Matsuki, Toshihiko, Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. intersections of associated orbits, Hiroshima Math. J., 18, 1, 59-67, (1988) · Zbl 0652.53035
[19] Milne, J. S., Automorphic forms, Shimura varieties, and \(L\)-functions, Vol. I (Ann Arbor, MI, 1988), 10, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, 283-414, (1990), Academic Press, Boston, MA · Zbl 0704.14016
[20] Pearlstein, Gregory J., Variations of mixed Hodge structure, Higgs fields, and quantum cohomology, Manuscripta Math., 102, 3, 269-310, (2000) · Zbl 0973.32008
[21] Pink, R., Arithmetical compactification of mixed Shimura varieties, (1989) · Zbl 0748.14007
[22] Richardson, R. W.; Springer, T. A., The Bruhat order on symmetric varieties, Geom. Dedicata, 35, 1-3, 389-436, (1990) · Zbl 0704.20039
[23] Richardson, R. W.; Springer, T. A., Linear algebraic groups and their representations (Los Angeles, CA, 1992), 153, Combinatorics and geometry of \(K\)-orbits on the flag manifold, 109-142, (1993), Amer. Math. Soc., Providence, RI · Zbl 0840.20039
[24] Robles, C., Schubert varieties as variations of Hodge structure, Selecta Math. (N.S.), 20, 3, 719-768, (2014) · Zbl 1328.32003
[25] Schmid, Wilfried, Variation of Hodge structure: the singularities of the period mapping, Invent. Math., 22, 211-319, (1973) · Zbl 0278.14003
[26] Trapa, P., Computing real Weyl groups
[27] Wolf, Joseph 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, 1121-1237, (1969) · Zbl 0183.50901
[28] Wolf, Joseph A., Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969-1970), Fine structure of Hermitian symmetric spaces, 271-357. Pure and App. Math., Vol. 8, (1972), Dekker, New York · Zbl 0257.32014
[29] Yee, W.-L., Simplifying and unifying Bruhat order for \(B\ G/B, P\ G/B, K\ G/B,\) and \(K\ G/P\)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.