# American Institute of Mathematical Sciences

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

## Boundary unitary representations-irreducibility and rigidity

 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

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.
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.  doi: 10.1007/s00209-002-0442-6.  Google Scholar [2] Marc Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace,, Enseign. Math., 2 (1995), 63.   Google Scholar [3] Marc Burger and Pierre de la Harpe, Constructing irreducible representations of discrete groups,, Proc. Indian Acad. Sci. Math. Sci., 107 (1997), 223.  doi: 10.1007/BF02867253.  Google Scholar [4] M. Burger and S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators,, J. Amer. Math. Soc., 9 (1996), 57.  doi: 10.1090/S0894-0347-96-00196-8.  Google Scholar [5] Chris Connell and Roman Muchnik, Harmonicity of quasiconformal measures and poisson boundaries of hyperbolic spaces,, to appear in GAFA., ().   Google Scholar [6] M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices,, J. Reine Angew. Math., 420 (1991), 85.   Google Scholar [7] Alessandro Figà-Talamanca and Massimo A. Picardello, "Harmonic Analysis on Free Groups,", Lecture Notes in Pure and Applied Mathematics, 87 (1983).   Google Scholar [8] Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees,, Mem. Amer. Math. Soc., 110 (1994).   Google Scholar [9] Alex Furman, Rigidity of group actions on infinite volume homogeneous spaces, II,, preprint., ().   Google Scholar [10] George W. Mackey, "The Theory of Unitary Group Representations,", University of Chicago Press, (1976).   Google Scholar [11] Grigoriy A. Margulis, "On Some Aspects of the Theory of Anosov Systems,", Springer Monographs in Mathematics, (2004).   Google Scholar [12] Chengbo Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature,, Trans. Amer. Math. Soc., 348 (1996), 4965.  doi: 10.1090/S0002-9947-96-01614-5.  Google Scholar

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.  doi: 10.1007/s00209-002-0442-6.  Google Scholar [2] Marc Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace,, Enseign. Math., 2 (1995), 63.   Google Scholar [3] Marc Burger and Pierre de la Harpe, Constructing irreducible representations of discrete groups,, Proc. Indian Acad. Sci. Math. Sci., 107 (1997), 223.  doi: 10.1007/BF02867253.  Google Scholar [4] M. Burger and S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators,, J. Amer. Math. Soc., 9 (1996), 57.  doi: 10.1090/S0894-0347-96-00196-8.  Google Scholar [5] Chris Connell and Roman Muchnik, Harmonicity of quasiconformal measures and poisson boundaries of hyperbolic spaces,, to appear in GAFA., ().   Google Scholar [6] M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices,, J. Reine Angew. Math., 420 (1991), 85.   Google Scholar [7] Alessandro Figà-Talamanca and Massimo A. Picardello, "Harmonic Analysis on Free Groups,", Lecture Notes in Pure and Applied Mathematics, 87 (1983).   Google Scholar [8] Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees,, Mem. Amer. Math. Soc., 110 (1994).   Google Scholar [9] Alex Furman, Rigidity of group actions on infinite volume homogeneous spaces, II,, preprint., ().   Google Scholar [10] George W. Mackey, "The Theory of Unitary Group Representations,", University of Chicago Press, (1976).   Google Scholar [11] Grigoriy A. Margulis, "On Some Aspects of the Theory of Anosov Systems,", Springer Monographs in Mathematics, (2004).   Google Scholar [12] Chengbo Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature,, Trans. Amer. Math. Soc., 348 (1996), 4965.  doi: 10.1090/S0002-9947-96-01614-5.  Google Scholar
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