Boundary unitary representations-irreducibility and rigidity

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January 2011, 5(1): 49-69. doi: 10.3934/jmd.2011.5.49

Boundary unitary representations-irreducibility and rigidity

Uri Bader ^1, and Roman Muchnik ^2,

1.	Mathematics Department, The Technion - Israel Institute of Technology Haifa, 32000, Israel
2.	Department of Mathematics and Computer Science, Lehman College, CUNY, 2500 Johnson Avenue Bronx, NY 10463, United States

Received January 2010 Revised December 2010 Published April 2011

Abstract
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Let $M$ be compact negatively curved manifold, $\Gamma =\pi_1(M)$ and $M$ be its universal cover. Denote by $B =\partial M$ the geodesic boundary of $M$ and by $\nu$ the Patterson-Sullivan measure on $X$. In this note we prove that the associated unitary representation of $\Gamma$ on $L^2(B,\nu)$ is irreducible. We also establish a new rigidity phenomenon: we show that some of the geometry of $M$, namely its marked length spectrum, is reflected in this $L^2$-representations.

Keywords: Marked length spectrum., Boundary representations, Negative curvature, Unitary representations.

Mathematics Subject Classification: Primary: 20C07, 22D40; Secondary: 37A30, 47A3.

Citation: Uri Bader, Roman Muchnik. Boundary unitary representations-irreducibility and rigidity. Journal of Modern Dynamics, 2011, 5 (1) : 49-69. doi: 10.3934/jmd.2011.5.49

References:

[1]	M. E. B. Bekka and M. Cowling, Some irreducible unitary representations of $G(K)$ for a simple algebraic group $G$ over an algebraic number field $K$, Math. Z., 241 (2002), 731-741. doi: 10.1007/s00209-002-0442-6.
[2]	Marc Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace, Enseign. Math., 2 (1995), 63-102.
[3]	Marc Burger and Pierre de la Harpe, Constructing irreducible representations of discrete groups, Proc. Indian Acad. Sci. Math. Sci., 107 (1997), 223-235. doi: 10.1007/BF02867253.
[4]	M. Burger and S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators, J. Amer. Math. Soc., 9 (1996), 57-93. doi: 10.1090/S0894-0347-96-00196-8.
[5]	Chris Connell and Roman Muchnik, Harmonicity of quasiconformal measures and poisson boundaries of hyperbolic spaces, to appear in GAFA.
[6]	M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math., 420 (1991), 85-98.
[7]	Alessandro Figà-Talamanca and Massimo A. Picardello, "Harmonic Analysis on Free Groups," Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker Inc., New York, 1983.
[8]	Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc., 110 (1994), xii+68.
[9]	Alex Furman, Rigidity of group actions on infinite volume homogeneous spaces, II, preprint.
[10]	George W. Mackey, "The Theory of Unitary Group Representations," University of Chicago Press, Chicago, Ill., 1976, Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955, Chicago Lectures in Mathematics.
[11]	Grigoriy A. Margulis, "On Some Aspects of the Theory of Anosov Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska.
[12]	Chengbo Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005. doi: 10.1090/S0002-9947-96-01614-5.

show all references

References:

[1]	M. E. B. Bekka and M. Cowling, Some irreducible unitary representations of $G(K)$ for a simple algebraic group $G$ over an algebraic number field $K$, Math. Z., 241 (2002), 731-741. doi: 10.1007/s00209-002-0442-6.
[2]	Marc Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace, Enseign. Math., 2 (1995), 63-102.
[3]	Marc Burger and Pierre de la Harpe, Constructing irreducible representations of discrete groups, Proc. Indian Acad. Sci. Math. Sci., 107 (1997), 223-235. doi: 10.1007/BF02867253.
[4]	M. Burger and S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators, J. Amer. Math. Soc., 9 (1996), 57-93. doi: 10.1090/S0894-0347-96-00196-8.
[5]	Chris Connell and Roman Muchnik, Harmonicity of quasiconformal measures and poisson boundaries of hyperbolic spaces, to appear in GAFA.
[6]	M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. Reine Angew. Math., 420 (1991), 85-98.
[7]	Alessandro Figà-Talamanca and Massimo A. Picardello, "Harmonic Analysis on Free Groups," Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker Inc., New York, 1983.
[8]	Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Mem. Amer. Math. Soc., 110 (1994), xii+68.
[9]	Alex Furman, Rigidity of group actions on infinite volume homogeneous spaces, II, preprint.
[10]	George W. Mackey, "The Theory of Unitary Group Representations," University of Chicago Press, Chicago, Ill., 1976, Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955, Chicago Lectures in Mathematics.
[11]	Grigoriy A. Margulis, "On Some Aspects of the Theory of Anosov Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska.
[12]	Chengbo Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc., 348 (1996), 4965-5005. doi: 10.1090/S0002-9947-96-01614-5.

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