2022, 18: 13-67. doi: 10.3934/jmd.2022002

New time-changes of unipotent flows on quotients of Lorentz groups

Department of Mathematics, Indiana University, Bloomington, IN 47401, USA

Received  October 02, 2020 Revised  June 28, 2021 Published  February 2022

We study the cocompact lattices $ \Gamma\subset SO(n, 1) $ so that the Laplace–Beltrami operator $ \Delta $ on $ SO(n)\backslash SO(n, 1)/\Gamma $ has eigenvalues in $ (0, \frac{1}{4}) $, and then show that there exist time-changes of unipotent flows on $ SO(n, 1)/\Gamma $ that are not measurably conjugate to the unperturbed ones. A main ingredient of the proof is a stronger version of the branching of the complementary series. Combining it with a refinement of the works of Ratner and Flaminio–Forni is adequate for our purpose.

Citation: Siyuan Tang. New time-changes of unipotent flows on quotients of Lorentz groups. Journal of Modern Dynamics, 2022, 18: 13-67. doi: 10.3934/jmd.2022002
References:
[1]

A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, Adv. Math., 385 (2021), 107759, 65 pp. doi: 10.1016/j.aim.2021.107759.

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Fundamental Principles of Mathematical Sciences, 252, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[3]

R. Brooks, Injectivity radius and low eigenvalues of hyperbolic manifolds, J. Reine Angew. Math., 390 (1988), 117-129.  doi: 10.1515/crll.1988.390.117.

[4]

K. Corlette, Hausdorff dimensions of limit sets. Ⅰ, Invent. Math., 102 (1990), 521-541.  doi: 10.1007/BF01233439.

[5]

R. T. De Aldecoa, Spectral analysis of time changes of horocycle flows, J. Mod. Dyn., 6 (2012), 275-285.  doi: 10.3934/jmd.2012.6.275.

[6]

M. Einsiedler, Ratner's theorem on ${\rm{SL}} (2, {\bf{R}})$-invariant measures, Jahresber. Deutsch. Math.-Verein., 108 (2006), 143-164. 

[7]

M. EinsiedlerG. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212.  doi: 10.1007/s00222-009-0177-7.

[8]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.

[9]

G. Forni and C. Ulcigrai, Time-changes of horocycle flows, J. Mod. Dyn., 6 (2012), 251-273.  doi: 10.3934/jmd.2012.6.251.

[10]

I. M. Gel'fand and M. L. Cetlin, Finite-dimensional representations of groups of orthogonal matrices, Doklady Akad. Nauk SSSR (NS), 71 (1950), 1017-1020. 

[11]

N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898717778.

[12]

T. Hirai, On infinitesimal operators of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad., 38 (1962), 83-87. 

[13]

T. Hirai, On irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad., 38 (1962), 258-262. 

[14]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9, Springer-Verlag, New York-Berlin, 1978.

[15]

K. D. Johnson and N. R. Wallach, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc., 229 (1977), 137-173.  doi: 10.1090/S0002-9947-1977-0447483-0.

[16]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[17] A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Landmarks in Mathematics, Princeton University Press, NJ, 2001. 
[18]

B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc., 75 (1969), 627-642.  doi: 10.1090/S0002-9904-1969-12235-4.

[19]

P. D. Lax and R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Functional Analysis, 46 (1982), 280-350.  doi: 10.1016/0022-1236(82)90050-7.

[20]

P. D. Lax and R. S. Phillips, Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces. Ⅰ, Comm. Pure Appl. Math., 37 (1984), 303-328.  doi: 10.1002/cpa.3160370304.

[21]

B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math. (2), 105 (1977), 81-105.  doi: 10.2307/1971026.

[22]

D. J. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, The Pennsylvalia State University, 2006.

[23]

J. J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. (2), 104 (1976), 235-247.  doi: 10.2307/1971046.

[24] D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005. 
[25]

N. Mukunda, Unitary representations of the Lorentz groups: Reduction of the supplementary series under a noncompact subgroup, J. Math. Phys., 9 (1968), 417-431.  doi: 10.1063/1.1664595.

[26]

F. A. Ramírez, Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions, J. Funct. Anal., 265 (2013), 1002-1063.  doi: 10.1016/j.jfa.2013.05.010.

[27]

B. Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Amer. Math. Soc., 80 (1974), 996-1000.  doi: 10.1090/S0002-9904-1974-13609-8.

[28]

M. Ratner, The Cartesian square of the horocycle flow is not loosely Bernoulli, Israel J. Math., 34 (1979), 72-96.  doi: 10.1007/BF02761825.

[29]

M. Ratner, Rigidity of time changes for horocycle flows, Acta Math., 156 (1986), 1-32.  doi: 10.1007/BF02399199.

[30]

D. Ravotti, Parabolic perturbations of unipotent flows on compact quotients of ${\rm{SL}} (3, {\bf{R}})$, Comm. Math. Phys., 371 (2019), 331-351.  doi: 10.1007/s00220-019-03348-0.

[31]

P. Sarnak, Notes on the generalized Ramanujan conjectures, Clay Math. Proc., 4 (2005), 659-685. 

[32]

Y. Shalom, Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math. (2), 152 (2000), 113-182.  doi: 10.2307/2661380.

[33]

L. D. Simonelli, Absolutely continuous spectrum for parabolic flows/maps, Discrete Contin. Dyn. Syst., 38 (2018), 263-292.  doi: 10.3934/dcds.2018013.

[34]

B. Speh and T. N. Venkataramana, On the Restriction of Representations of ${SL} (2, {\bf{C}})$ to ${SL} (2, {\bf{R}})$, Representation Theory, Complex Analysis, and Integral Geometry, Birkhäuser/Springer, New York, 2012,231–249. doi: 10.1007/978-0-8176-4817-6_9.

[35]

R. SchoenS. Wolpert and and S.-T. Yau, Geometric bounds on the low eigenvalues of a compact surface, Proc. Sympos. Pure Math., 36 (1980), 279-285. 

[36]

B. Speh and G. Zhang, Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups, Math. Z., 283 (2016), 629-647.  doi: 10.1007/s00209-016-1614-0.

[37]

E. Thieleker, On the quasi-simple irreducible representations of the Lorentz groups, Trans. Amer. Math. Soc., 179 (1973), 465-505.  doi: 10.1090/S0002-9947-1973-0325856-0.

[38]

E. Thieleker, The unitary representations of the generalized Lorentz groups, Trans. Amer. Math. Soc., 199 (1974), 327-367.  doi: 10.1090/S0002-9947-1974-0379754-8.

[39]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094.  doi: 10.4007/annals.2010.172.989.

[40]

N. Y. Vilenkin, Special Functions and the Theory of Group Representations, 2$^nd$ edition, "Nauka", Moscow, 1991.

[41]

Z. J. Wang, Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020.  doi: 10.1007/s00039-015-0351-6.

[42]

G. Zhang, Discrete components in restriction of unitary representations of rank one semisimple Lie groups, J. Funct. Anal., 269 (2015), 3689-3713.  doi: 10.1016/j.jfa.2015.09.021.

show all references

References:
[1]

A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, Adv. Math., 385 (2021), 107759, 65 pp. doi: 10.1016/j.aim.2021.107759.

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Fundamental Principles of Mathematical Sciences, 252, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[3]

R. Brooks, Injectivity radius and low eigenvalues of hyperbolic manifolds, J. Reine Angew. Math., 390 (1988), 117-129.  doi: 10.1515/crll.1988.390.117.

[4]

K. Corlette, Hausdorff dimensions of limit sets. Ⅰ, Invent. Math., 102 (1990), 521-541.  doi: 10.1007/BF01233439.

[5]

R. T. De Aldecoa, Spectral analysis of time changes of horocycle flows, J. Mod. Dyn., 6 (2012), 275-285.  doi: 10.3934/jmd.2012.6.275.

[6]

M. Einsiedler, Ratner's theorem on ${\rm{SL}} (2, {\bf{R}})$-invariant measures, Jahresber. Deutsch. Math.-Verein., 108 (2006), 143-164. 

[7]

M. EinsiedlerG. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212.  doi: 10.1007/s00222-009-0177-7.

[8]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.

[9]

G. Forni and C. Ulcigrai, Time-changes of horocycle flows, J. Mod. Dyn., 6 (2012), 251-273.  doi: 10.3934/jmd.2012.6.251.

[10]

I. M. Gel'fand and M. L. Cetlin, Finite-dimensional representations of groups of orthogonal matrices, Doklady Akad. Nauk SSSR (NS), 71 (1950), 1017-1020. 

[11]

N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898717778.

[12]

T. Hirai, On infinitesimal operators of irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad., 38 (1962), 83-87. 

[13]

T. Hirai, On irreducible representations of the Lorentz group of n-th order, Proc. Japan Acad., 38 (1962), 258-262. 

[14]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, 9, Springer-Verlag, New York-Berlin, 1978.

[15]

K. D. Johnson and N. R. Wallach, Composition series and intertwining operators for the spherical principal series. I, Trans. Amer. Math. Soc., 229 (1977), 137-173.  doi: 10.1090/S0002-9947-1977-0447483-0.

[16]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[17] A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Landmarks in Mathematics, Princeton University Press, NJ, 2001. 
[18]

B. Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc., 75 (1969), 627-642.  doi: 10.1090/S0002-9904-1969-12235-4.

[19]

P. D. Lax and R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Functional Analysis, 46 (1982), 280-350.  doi: 10.1016/0022-1236(82)90050-7.

[20]

P. D. Lax and R. S. Phillips, Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces. Ⅰ, Comm. Pure Appl. Math., 37 (1984), 303-328.  doi: 10.1002/cpa.3160370304.

[21]

B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math. (2), 105 (1977), 81-105.  doi: 10.2307/1971026.

[22]

D. J. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, The Pennsylvalia State University, 2006.

[23]

J. J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. (2), 104 (1976), 235-247.  doi: 10.2307/1971046.

[24] D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005. 
[25]

N. Mukunda, Unitary representations of the Lorentz groups: Reduction of the supplementary series under a noncompact subgroup, J. Math. Phys., 9 (1968), 417-431.  doi: 10.1063/1.1664595.

[26]

F. A. Ramírez, Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions, J. Funct. Anal., 265 (2013), 1002-1063.  doi: 10.1016/j.jfa.2013.05.010.

[27]

B. Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Amer. Math. Soc., 80 (1974), 996-1000.  doi: 10.1090/S0002-9904-1974-13609-8.

[28]

M. Ratner, The Cartesian square of the horocycle flow is not loosely Bernoulli, Israel J. Math., 34 (1979), 72-96.  doi: 10.1007/BF02761825.

[29]

M. Ratner, Rigidity of time changes for horocycle flows, Acta Math., 156 (1986), 1-32.  doi: 10.1007/BF02399199.

[30]

D. Ravotti, Parabolic perturbations of unipotent flows on compact quotients of ${\rm{SL}} (3, {\bf{R}})$, Comm. Math. Phys., 371 (2019), 331-351.  doi: 10.1007/s00220-019-03348-0.

[31]

P. Sarnak, Notes on the generalized Ramanujan conjectures, Clay Math. Proc., 4 (2005), 659-685. 

[32]

Y. Shalom, Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group, Ann. of Math. (2), 152 (2000), 113-182.  doi: 10.2307/2661380.

[33]

L. D. Simonelli, Absolutely continuous spectrum for parabolic flows/maps, Discrete Contin. Dyn. Syst., 38 (2018), 263-292.  doi: 10.3934/dcds.2018013.

[34]

B. Speh and T. N. Venkataramana, On the Restriction of Representations of ${SL} (2, {\bf{C}})$ to ${SL} (2, {\bf{R}})$, Representation Theory, Complex Analysis, and Integral Geometry, Birkhäuser/Springer, New York, 2012,231–249. doi: 10.1007/978-0-8176-4817-6_9.

[35]

R. SchoenS. Wolpert and and S.-T. Yau, Geometric bounds on the low eigenvalues of a compact surface, Proc. Sympos. Pure Math., 36 (1980), 279-285. 

[36]

B. Speh and G. Zhang, Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups, Math. Z., 283 (2016), 629-647.  doi: 10.1007/s00209-016-1614-0.

[37]

E. Thieleker, On the quasi-simple irreducible representations of the Lorentz groups, Trans. Amer. Math. Soc., 179 (1973), 465-505.  doi: 10.1090/S0002-9947-1973-0325856-0.

[38]

E. Thieleker, The unitary representations of the generalized Lorentz groups, Trans. Amer. Math. Soc., 199 (1974), 327-367.  doi: 10.1090/S0002-9947-1974-0379754-8.

[39]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989-1094.  doi: 10.4007/annals.2010.172.989.

[40]

N. Y. Vilenkin, Special Functions and the Theory of Group Representations, 2$^nd$ edition, "Nauka", Moscow, 1991.

[41]

Z. J. Wang, Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020.  doi: 10.1007/s00039-015-0351-6.

[42]

G. Zhang, Discrete components in restriction of unitary representations of rank one semisimple Lie groups, J. Funct. Anal., 269 (2015), 3689-3713.  doi: 10.1016/j.jfa.2015.09.021.

Figure 1.  A collection of $ \epsilon $-blocks $ \{{\rm{BL}}_{1}, \ldots, {\rm{BL}}_{n}\} $. The solid straight lines are the unipotent orbits in the $ \epsilon $-blocks and the dashed lines are the rest of the unipotent orbits. The bent curves indicate the length defined by the letters
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