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On measure rigidity of unipotent subgroups of semisimple groups. (English) Zbl 0745.28010
[Joint review with the above cited paper.]
These two important papers are the first two parts of a three part series concerned with measure rigidity properties of unipotent subgroups of real Lie groups.
Various notions of rigidity have been investigated in the theory of real Lie groups and their discrete subgroups. In particular, a subgroup \(U\) of a real Lie group \(G\) is called topologically rigid if for all lattices \(\Gamma\subseteq G\) and \(x\to\Gamma\setminus G\) the closure of the \(U\)- orbit \(xU\) in \(\Gamma\setminus G\) is a homogeneous set, i.e., of the shape \(\Gamma yH\) for some \(y\in G\) and some closed subgroup \(H\subseteq G\) such that \(yHy^{-1}\cap\Gamma\) is again a lattice in \(yHy^{-1}\). One of the outstanding open problems in this field is the question of the validity of Raghunathan’s conjecture, which asserts that every unipotent subgroup of a connected real Lie group \(G\) is topologically rigid. A modification of this conjecture, called Raghunathan’s measure conjecture, replaces topological rigidity by measure rigidity (where \(U\subseteq G\) is called measure rigid if every ergodic \(U\)-invariant Borel probability measure on \(\Gamma\setminus G\) is algebraic).
The proof of this conjecture is accomplished in the series of papers under review. Part I (“Strict measure rigidity \(\ldots\)”) proves strict measure rigidity (assuming \(\Gamma\) only to be discrete) for unipotent subgroups of connected solvable real Lie groups. In fact, a more general theorem (Theorem 1) on algebraicity of measures invariant under the action of unipotent elements is proved which implies (among other results) strict measure rigidity for the connected solvable case. The key steps towards the proof consist in the development of an ergodic theory for measure-preserving actions of nilpotent Lie groups and in establishing the so-called \(R\)-property, a dynamical property of a unipotent subgroup \(N\) (with Lie algebra \({\mathfrak N}\)) of a Lie group \(G\) (with Lie algebra \({\mathfrak G}\)). It allows (roughly) to control the projection onto a complement of \({\mathfrak N}\) in \({\mathfrak G}\) of the adjoint action on \({\mathfrak N}\) of elements of \({\mathfrak G}\) obtained as \(\exp(t_ 1b_ 1)\ldots\exp(t_ rb_ r)\) with suitably restricted \(t_ i\) and a “triangular” basis \(\{b_ 1,\ldots,b_ r\}\) of \({\mathfrak N}\).
The main theorem of Part II (“On measure rigidity \(\ldots\)”) proves the algebraicity of a measure \(\mu\) on an arbitrary Lie group \(G\) which admits an ergodic action of a nilpotent horocyclic element of \({\mathfrak G}\) with certain additional properties. As a corollary of the main theorem the authoress obtains measure rigidity of actions of unipotent elements of a connected semisimple \(G\) on \(G/\Gamma\) for a compatible lattice \(\Gamma\) of \(G\). Other important consequences of the main theorem concern ergodic joinings of unipotent elements and give a classification of such joinings generalizing the results of the authoress and Witte for \(SL_ 2(\mathbb{R})\) [the authoress, Ann. Math., II. Ser. 118, 277-313 (1983; Zbl 0556.28020); D. Witte, Am. J. Math. 109, 927-961 (1987; Zbl 0653.22005)].

MSC:
28D05 Measure-preserving transformations
22E40 Discrete subgroups of Lie groups
37A99 Ergodic theory
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[1] [B]Bowen, R., Weak mixing and unique ergodicity on homogeneous spaces.Israel J. Math., 23 (1976), 267–273. · Zbl 0338.43014 · doi:10.1007/BF02761804
[2] [BM]Brezin, J. &Moore, C., Flows on homogeneous spaces: a new look.Amer. J. Math., 103 (1981), 571–613. · Zbl 0506.22008 · doi:10.2307/2374105
[3] [D1]Dani, S. G., Invariant measures of horospherical flows on noncompact homogeneous spaces.Invent. Math., 47 (1978), 101–138. · Zbl 0384.28015 · doi:10.1007/BF01578067
[4] [D2]–, Invariant measures and minimal sets of horospherical flows.Invent. Math., 64 (1981), 357–385. · Zbl 0498.58013 · doi:10.1007/BF01389173
[5] [EP]Ellis, R. &Perrizo, W., Unique ergodicity of flows on homogeneous spaces.Israel J. Math., 29 (1978), 276–284. · Zbl 0383.22004 · doi:10.1007/BF02762015
[6] [F1]Furstenberg, H., Strict ergodicity and transformations of the torus.Amer. J. Math., 83 (1961), 573–601. · Zbl 0178.38404 · doi:10.2307/2372899
[7] [F2]Furstenberg, H., The unique ergodicity of the horocycle flow, inRecent Advances in Topological Dynamics, 95–115. Springer, 1972.
[8] [GE]Greenleaf, P. &Emerson, W. R., Group structure and pointwise ergodic theorem for connected amenable groups.Adv. in Math., 14 (1974), 153–172. · Zbl 0288.22009 · doi:10.1016/0001-8708(74)90027-9
[9] [H]Humphreys, J.,Introduction to Lie Algebras and Representation Theory. Springer-Verlag, 1972. · Zbl 0254.17004
[10] [J]Jacobson, N.,Lie Algebras. John Wiley, 1962. · Zbl 0121.27504
[11] [M]Margulis, G. A., Discrete subgroups and ergodic theory.Symposium in honor of A. Selberg, Number theory, trace formulas and discrete groups. Oslo, 1989.
[12] [P]Parry, W., Ergodic properties of affine transformations and flows on nilmanifolds.Amer. J. Math., 91 (1969), 757–771. · Zbl 0183.51503 · doi:10.2307/2373350
[13] [R1]Ratner, M., Rigidity of horocycle flows.Ann. of Math., 115 (1982), 597–614. · Zbl 0506.58030 · doi:10.2307/2007014
[14] [R2] –, Factors of horocycle flows.Ergodic Theory Dynamical Systems, 2 (1982), 465–489. · Zbl 0536.58029
[15] [R3] –, Horocycle flows: joinings and rigidity of products.Ann. of Math., 118 (1983), 277–313. · Zbl 0556.28020 · doi:10.2307/2007030
[16] [R4] –, Strict measure rigidity for unipotent subgroups of solvable groups.Invent. Math., 101 (1990), 449–482. · Zbl 0745.28009 · doi:10.1007/BF01231511
[17] [R5] –, Invariant measures for unipotent translations on homogeneous spaces.Proc. Nat. Acad. Sci. U.S.A., 87 (1990), 4309–4311. · Zbl 0819.22004 · doi:10.1073/pnas.87.11.4309
[18] [Ve]Veech, W., Unique ergodicity of horospherical flows.Amer. J. Math., 99 (1977), 827–859. · Zbl 0365.28012 · doi:10.2307/2373868
[19] [W1]Witte, D., Rigidity of some translations on homogeneous spaces.Invent. Math., 81 (1985), 1–27. · Zbl 0571.22007 · doi:10.1007/BF01388769
[20] [W2] –, Zero entropy affine maps on homogeneous spaces.Amer. J. Math., 109 (1987), 927–961. · Zbl 0653.22005 · doi:10.2307/2374495
[21] [W3]Witte, D., Measurable quotients of unipotent translations on homogeneous spaces. To appear. · Zbl 0831.28010
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