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Local rigidity of complex hyperbolic lattices in semisimple Lie groups

  • INKANG KIM (a1) and GENKAI ZHANG (a2)

We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, SU(p, q), Sp(2n + 2, $\mathbb{R}$ ), SO*(2n + 2), SO(2n, 2). This reproves or generalises some results in [2, 9, 11, 15].

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[1] Eastwood, M. and Wolf, J. Branching of representations to symmetric subgroups. Münster J. Math. 4 (2011), 127.
[2] Goldman, W. and Millson, J. Local rigidity of discrete groups acting on complex hyperbolic space. Inv. Math. 88 (1987), 495520.
[3] Helgason, S. Differential Geometry, Lie groups and Symmetric spaces (Academic press, New York, 1978).
[4] Howe, R. θ-series and invariant theory. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 275–285, Proc. Sympos. Pure Math., XXXIII (Amer. Math. Soc., Providence, R.I., 1979).
[5] Humphreys, J. Introduction to Lie Algebras and Representation Theory (Springer-Verlag, New York, 1972).
[6] Ihara, S. Holomorphic embeddings of symmetric domains. J. Math. Soc. Japan 19, no.3 (1967).
[7] Johnson, K. D. Composition series and intertwining operators for the spherical principal series. II. Trans. Amer. Math. Soc. 215 (1976), 269283.
[8] Kim, I. and Pansu, P. Local Rigidity in quaternionic hyperbolic space. J. European Math Society 11 (2009), no 6, 11411164.
[9] Kim, I., Klingler, B. and Pansu, P. Local quaternionic rigidity for complex hyperbolic lattices. J. Inst. Math. Jussieu 11 (2012), no 1, 133159.
[10] Kim, I. and Zhang, G. Eichler–Shimura isomorphism for complex hyperbolic lattices, submitted.
[11] Klingler, B. Local rigidity for complex hyperbolic lattices and Hodge theory. Invent. Math. 184 (2011), no.3, 455498.
[12] Koziarz, V. and Maubon, J. Maximal representations of uniform complex hyperbolic lattices, preprint.
[13] Matsushima, Y. and Murakami, S. On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds. Ann. Math. 78 (1963), 329416.
[14] Matsushima, Y. and Murakami, S. On certain cohomology groups attached to Hermitian symmetric spaces. Osaka J. Math. 2 (1965), 135.
[15] Pozzetti, M. B. Maximal representations of complex hyperbolic lattices in SU(m,n). Geom. Funct. Anal. 25 (2015), 12901332.
[16] Raghunathan, M. S. On the first cohomology of discrete subgroups of semisimple Lie groups. Amer. J. Math. 87 (1965), 103139.
[17] Raghunathan, M. S. Discrete subgroups of Lie groups (Springer-Verlag, Berlin, Heidelberg, New York, 1972).
[18] Satake, I. Holomorphic embeddings of symmetric domains into a Siegel domains. Amer. J. Math. 87, no.2 (1965).
[19] Weil, A. Discrete subgroups of Lie groups, II. Ann. of Math 75 (1962), 97123.
[20] Zucker, S. Locally homogeneous variations of Hodge structure. Enseign. Math. (2) 27 (1982), 243276.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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