Table of contents for this issue | Previous article
Matt Kerr; Gregory Pearlstein
Naive boundary strata and nilpotent orbits
(Limites de strates naïves et orbites nilpotentes)
Annales de l'institut Fourier, 64 no. 6 (2014), p. 2659-2714, doi: 10.5802/aif.2923
Article PDF | Reviews MR 3331177 | Zbl 06387350
Class. Math.: 14D07, 14M17, 17B45, 20G99, 32M10, 32G20
Keywords: Mumford-Tate groups, Mumford-Tate domains, nilpotent orbits, variation of Hodge structure, Shimura varieties
[1] Jeffrey Adams, Guide to the Atlas software: computational representation theory of real reductive groups, Representation theory of real reductive Lie groups (Snowbird, July 2006), Contemp. Math. 472, Amer. Math. Soc., Providence, RI, 2008, p. 1–37 MR 2454331 | Zbl 1175.22001
[2] Jeffrey Adams & Fokko du Cloux, “Algorithms for representation theory of real reductive groups”, J. Inst. Math. Jussieu 8 (2009) no. 2, p. 209-259 Article | MR 2485793 | Zbl 1221.22017
[3] Avner Ash, David Mumford, Michael Rapoport & Yung-Sheng Tai, Smooth compactifications of locally symmetric varieties, Cambridge University Press, Cambridge, 2010 MR 2590897 | Zbl 1209.14001
[4] Armand Borel, Linear algebraic groups, Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991 MR 1102012 | Zbl 0726.20030
[5] Armand Borel & Jacques Tits, “Groupes réductifs”, Inst. Hautes Études Sci. Publ. Math. (1965) no. 27, p. 55-150 Numdam | MR 207712 | Zbl 0145.17402
[6] Henri Carayol, “Cohomologie automorphe et compactifications partielles de certaines variéetés de Griffiths-Schmid”, Compos. Math. 141 (2005) no. 5, p. 1081-1102 Article | MR 2157130 | Zbl 1173.11331
[7] James A. Carlson, Eduardo H. Cattani & Aroldo G. Kaplan, Mixed Hodge structures and compactifications of Siegel’s space (preliminary report), Algebraic Geometry, Angers, 1979 (A. Beauville, Ed), Sijthoff & Noordhoff, 1980, p. 77–105 MR 605337 | Zbl 0471.14002
[8] Eduardo Cattani & Aroldo Kaplan, “Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure”, Invent. Math. 67 (1982) no. 1, p. 101-115 Article | MR 664326 | Zbl 0516.14005
[9] Eduardo Cattani, Aroldo Kaplan & Wilfried Schmid, “Degeneration of Hodge structures”, Ann. of Math. (2) 123 (1986) no. 3, p. 457-535 Article | MR 840721 | Zbl 0617.14005
[10] Eduardo H. Cattani, Mixed Hodge structures, compactifications and monodromy weight filtration, Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, NJ, 1984 MR 756847 | Zbl 0579.14010
[11] Gregor Fels, Alan Huckleberry & Joseph A. Wolf, Cycle spaces of flag domains, Progress in Mathematics 245, Birkhäuser Boston, Inc., Boston, MA, 2006, A complex geometric viewpoint MR 2188135 | Zbl 1084.22011
[12] Mark Green, Phillip Griffiths & Matt Kerr, Mumford-Tate groups and domains, Annals of Mathematics Studies 183, Princeton University Press, Princeton, NJ, 2012, Their geometry and arithmetic MR 2918237 | Zbl 1248.14001
[13] Mark Green, Phillip Griffiths & Matt Kerr, Hodge theory, complex geometry, and representation theory, CBMS Regional Conference Series in Mathematics 118, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2013 MR 3115136
[14] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, Graduate Texts in Mathematics, No. 21 MR 396773 | Zbl 0471.20029
[15] Matt Kerr & Gregory Pearlstein, “Boundary components of Mumford-Tate domains”, Preprint,
[16] Anthony W. Knapp, Lie groups beyond an introduction, Progress in Mathematics 140, Birkhäuser Boston, Inc., Boston, MA, 2002 MR 1920389 | Zbl 1075.22501
[17] Toshihiko Matsuki, Closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups, Representations of Lie groups, Kyoto, Hiroshima, 1986, (eds. K. Okamoto and T. Oshima), Adv. Stud. Pure Math. 14, Academic Press, Boston, MA, 1988, p. 541–559 MR 1039852 | Zbl 0723.22020
[18] Toshihiko Matsuki, “Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits”, Hiroshima Math. J. 18 (1988) no. 1, p. 59-67 MR 935882 | Zbl 0652.53035
[19] J. S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, Academic Press, Boston, MA, 1990, p. 283–414 MR 1044823 | Zbl 0704.14016
[20] Gregory J. Pearlstein, “Variations of mixed Hodge structure, Higgs fields, and quantum cohomology”, Manuscripta Math. 102 (2000) no. 3, p. 269-310 Article | MR 1777521 | Zbl 0973.32008
[21] R. Pink, Arithmetical compactification of mixed Shimura varieties, Ph. D. Thesis, Univ. Bonn, 1989 MR 1128753 | Zbl 0748.14007
[22] R. W. Richardson & T. A. Springer, “The Bruhat order on symmetric varieties”, Geom. Dedicata 35 (1990) no. 1-3, p. 389-436 Article | MR 1066573 | Zbl 0704.20039
[23] R. W. Richardson & T. A. Springer, Combinatorics and geometry of $K$-orbits on the flag manifold, Linear algebraic groups and their representations (Los Angeles, CA, 1992), Contemp. Math. 153, Amer. Math. Soc., Providence, RI, 1993, p. 109–142 MR 1247501 | Zbl 0840.20039
[24] C. Robles, “Schubert varieties as variations of Hodge structure”, Selecta Math. (N.S.) 20 (2014) no. 3, p. 719-768 Article | MR 3217458
[25] Wilfried Schmid, “Variation of Hodge structure: the singularities of the period mapping”, Invent. Math. 22 (1973), p. 211-319 Article | MR 382272 | Zbl 0278.14003
[26] P. Trapa, “Computing real Weyl groups”, notes from a 2006 lecture by D. Vogan, available at http://www.math.utah.edu/~ptrapa/preprints.html
[27] Joseph A. Wolf, “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), p. 1121-1237 Article | MR 251246 | Zbl 0183.50901
[28] Joseph A. Wolf, Fine structure of Hermitian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Dekker, New York, 1972, p. 271–357. Pure and App. Math., Vol. 8 MR 404716 | Zbl 0257.32014
[29] W.-L. Yee, “Simplifying and unifying Bruhat order for $B\backslash G/B$, $P\backslash G/B$, $K\backslash G/B$, and $K\backslash G/P$”, preprint,
Matt Kerr; Gregory Pearlstein
Naive boundary strata and nilpotent orbits
(Limites de strates naïves et orbites nilpotentes)
Annales de l'institut Fourier, 64 no. 6 (2014), p. 2659-2714, doi: 10.5802/aif.2923
Article PDF | Reviews MR 3331177 | Zbl 06387350
Class. Math.: 14D07, 14M17, 17B45, 20G99, 32M10, 32G20
Keywords: Mumford-Tate groups, Mumford-Tate domains, nilpotent orbits, variation of Hodge structure, Shimura varieties
Résumé - Abstract
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 $SU(2,1)$, $Sp _4$, and $G_2$.
Bibliography
[2] Jeffrey Adams & Fokko du Cloux, “Algorithms for representation theory of real reductive groups”, J. Inst. Math. Jussieu 8 (2009) no. 2, p. 209-259 Article | MR 2485793 | Zbl 1221.22017
[3] Avner Ash, David Mumford, Michael Rapoport & Yung-Sheng Tai, Smooth compactifications of locally symmetric varieties, Cambridge University Press, Cambridge, 2010 MR 2590897 | Zbl 1209.14001
[4] Armand Borel, Linear algebraic groups, Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991 MR 1102012 | Zbl 0726.20030
[5] Armand Borel & Jacques Tits, “Groupes réductifs”, Inst. Hautes Études Sci. Publ. Math. (1965) no. 27, p. 55-150 Numdam | MR 207712 | Zbl 0145.17402
[6] Henri Carayol, “Cohomologie automorphe et compactifications partielles de certaines variéetés de Griffiths-Schmid”, Compos. Math. 141 (2005) no. 5, p. 1081-1102 Article | MR 2157130 | Zbl 1173.11331
[7] James A. Carlson, Eduardo H. Cattani & Aroldo G. Kaplan, Mixed Hodge structures and compactifications of Siegel’s space (preliminary report), Algebraic Geometry, Angers, 1979 (A. Beauville, Ed), Sijthoff & Noordhoff, 1980, p. 77–105 MR 605337 | Zbl 0471.14002
[8] Eduardo Cattani & Aroldo Kaplan, “Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure”, Invent. Math. 67 (1982) no. 1, p. 101-115 Article | MR 664326 | Zbl 0516.14005
[9] Eduardo Cattani, Aroldo Kaplan & Wilfried Schmid, “Degeneration of Hodge structures”, Ann. of Math. (2) 123 (1986) no. 3, p. 457-535 Article | MR 840721 | Zbl 0617.14005
[10] Eduardo H. Cattani, Mixed Hodge structures, compactifications and monodromy weight filtration, Ann. of Math. Stud. 106, Princeton Univ. Press, Princeton, NJ, 1984 MR 756847 | Zbl 0579.14010
[11] Gregor Fels, Alan Huckleberry & Joseph A. Wolf, Cycle spaces of flag domains, Progress in Mathematics 245, Birkhäuser Boston, Inc., Boston, MA, 2006, A complex geometric viewpoint MR 2188135 | Zbl 1084.22011
[12] Mark Green, Phillip Griffiths & Matt Kerr, Mumford-Tate groups and domains, Annals of Mathematics Studies 183, Princeton University Press, Princeton, NJ, 2012, Their geometry and arithmetic MR 2918237 | Zbl 1248.14001
[13] Mark Green, Phillip Griffiths & Matt Kerr, Hodge theory, complex geometry, and representation theory, CBMS Regional Conference Series in Mathematics 118, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2013 MR 3115136
[14] James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, Graduate Texts in Mathematics, No. 21 MR 396773 | Zbl 0471.20029
[15] Matt Kerr & Gregory Pearlstein, “Boundary components of Mumford-Tate domains”, Preprint,
arXiv:1210.5301[16] Anthony W. Knapp, Lie groups beyond an introduction, Progress in Mathematics 140, Birkhäuser Boston, Inc., Boston, MA, 2002 MR 1920389 | Zbl 1075.22501
[17] Toshihiko Matsuki, Closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups, Representations of Lie groups, Kyoto, Hiroshima, 1986, (eds. K. Okamoto and T. Oshima), Adv. Stud. Pure Math. 14, Academic Press, Boston, MA, 1988, p. 541–559 MR 1039852 | Zbl 0723.22020
[18] Toshihiko Matsuki, “Closure relations for orbits on affine symmetric spaces under the action of parabolic subgroups. Intersections of associated orbits”, Hiroshima Math. J. 18 (1988) no. 1, p. 59-67 MR 935882 | Zbl 0652.53035
[19] J. S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, Academic Press, Boston, MA, 1990, p. 283–414 MR 1044823 | Zbl 0704.14016
[20] Gregory J. Pearlstein, “Variations of mixed Hodge structure, Higgs fields, and quantum cohomology”, Manuscripta Math. 102 (2000) no. 3, p. 269-310 Article | MR 1777521 | Zbl 0973.32008
[21] R. Pink, Arithmetical compactification of mixed Shimura varieties, Ph. D. Thesis, Univ. Bonn, 1989 MR 1128753 | Zbl 0748.14007
[22] R. W. Richardson & T. A. Springer, “The Bruhat order on symmetric varieties”, Geom. Dedicata 35 (1990) no. 1-3, p. 389-436 Article | MR 1066573 | Zbl 0704.20039
[23] R. W. Richardson & T. A. Springer, Combinatorics and geometry of $K$-orbits on the flag manifold, Linear algebraic groups and their representations (Los Angeles, CA, 1992), Contemp. Math. 153, Amer. Math. Soc., Providence, RI, 1993, p. 109–142 MR 1247501 | Zbl 0840.20039
[24] C. Robles, “Schubert varieties as variations of Hodge structure”, Selecta Math. (N.S.) 20 (2014) no. 3, p. 719-768 Article | MR 3217458
[25] Wilfried Schmid, “Variation of Hodge structure: the singularities of the period mapping”, Invent. Math. 22 (1973), p. 211-319 Article | MR 382272 | Zbl 0278.14003
[26] P. Trapa, “Computing real Weyl groups”, notes from a 2006 lecture by D. Vogan, available at http://www.math.utah.edu/~ptrapa/preprints.html
[27] Joseph A. Wolf, “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), p. 1121-1237 Article | MR 251246 | Zbl 0183.50901
[28] Joseph A. Wolf, Fine structure of Hermitian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Dekker, New York, 1972, p. 271–357. Pure and App. Math., Vol. 8 MR 404716 | Zbl 0257.32014
[29] W.-L. Yee, “Simplifying and unifying Bruhat order for $B\backslash G/B$, $P\backslash G/B$, $K\backslash G/B$, and $K\backslash G/P$”, preprint,
arXiv:1107.0518v3
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