Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD _w .D ₁ is the only invariant control set, and the subsetW(S)={w:D _w =D ₁} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold. Research partially supported by CNPq grant n^o 50.13.73/91-8     

Transitive actions on Hopf homogeneous spaces

In this paper we investigate transitive actions of compact connected Lie groups on certain spaces X which are not spheres, whose dimension is not too small and whose rational cohomology algebra is an exterior algebra on homogeneous generators of odd degree. In case X is a simply connected classical group, a 3-connected real or a 5-connected complex or a quaternionic Stiefel manifold, we obtain (in principle) the classification of the transitive actions on X up to equivariant homeomorphism.     

Deformation of proper actions on reductive homogeneous spaces

Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of standard compact quotients of G/H, that is, of quotients of G/H by discrete groups Γ that are uniform lattices in some closed reductive subgroup L of G acting properly and cocompactly on G/H. For L of real rank 1, we prove that after a small deformation in G, such a group Γ keeps acting properly discontinuously and cocompactly on G/H. More generally, we prove that the properness of the action of any convex cocompact subgroup of L on G/H is preserved under small deformations, and we extend this result to reductive homogeneous spaces G/H over any local field. As an application, we obtain compact quotients of SO(2n, 2)/U(n, 1) by Zariski-dense discrete subgroups of SO(2n, 2) acting properly discontinuously.     

Reversibility properties of semigroup actions on homogeneous spaces

This paper studies reversibility of subsemigroups acting on homogeneous spaces. The reversor set of a subsemigroup is defined and it is related to the invariant control sets for semigroups acting on certain homogeneous spaces. Let G be a connected noncompact semi-simple Lie group with finite center. Let L be a subgroup of?G. Assume that S is a subsemigroup of G with intS???. The main result characterizes the reversibility of the S-action on G/L in terms of the actions of S and L on the flag manifolds of G.     

On mixing in infinite measure spaces

Summary Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote -algebra (=tail -field). A transformation T is called mixing if (T ^–n A) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures ₁,₂, in total variation norm.Research of this author is supported by the National Science Foundation (U.S.A.) under grant GP 7693.     

Properly discontinuous group actions on affine homogeneous spaces

Let G be a real algebraic group, H ≤ G an algebraic subgroup containing a maximal reductive subgroup of G, and Γ a subgroup of G acting on G/H by left translations. We conjecture that Γ is virtually solvable provided its action on G/H is properly discontinuous and ΓG/H is compact, and we confirm this conjecture when G does not contain simple algebraic subgroups of rank ≥2. If the action of Γ on G/H (which is isomorphic to an affine linear space Aⁿ) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for n ≤ 5.     

Joinings of higher rank torus actions on homogeneous spaces

Publications mathématiques de l'IHÉS - We show that joinings of higher rank torus actions on $S$ -arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.     

Higher cohomology of parabolic actions on certain homogeneous spaces

We show that for a parabolic R ^d -action on PSL(2,R) ^d /Γ, the cohomologies in degrees 1 through d ? 1 trivialize, and we give the obstructions to solving the degree-d coboundary equation, along with bounds on Sobolev norms of primitives. In previous papers, we have established these results for certain Anosov systems. This work extends the methods of those papers to systems that are not Anosov. The main new idea is defining special elements of representation spaces that allow us to modify the arguments from the previous papers. We discuss how to generalize this strategy to R ^d -systems coming from a product of Lie groups, as in the systems analyzed here.     

On infinite products of non-Archimedean measure spaces

We study generalized ‘probabilistic measures’ taking values in non-Archimedean fields (in particular, fields of p-adic numbers). We prove the theorem on the existence of probability on a product of non-Archimedean probabilistic spaces.     

Zonal measure algebras on isotropy irreducible homogeneous spaces

This paper analyzes the convolution algebra $\text{M(K}\text{G}\text{K}\text{)}$ of zonal measures on a Lie group G, with compact subgroup K, primarily for the case when $\text{M(K}\text{G}\text{K}\text{)}$ is commutative and $\text{G}\text{K}$ is isotropy irreducible. A basic result for such (G, K) is that the convolution of dim $\text{G}\text{K}$ continuous (on $\text{G}\text{K}$) zonal measures is absolutely continuous. Using this, the spectrum (maximal ideal space) of $\text{M(K}\text{G}\text{K}\text{)}$ is determined and shown to be in 1-1 correspondence with the bounded Borel spherical functions. Also, certain asymptotic results for the continuous spherical functions are derived. For the special case when G is compact, all the idempotents in $\text{M(K}\text{G}\text{K}\text{)}$ are determined.     

On grand Lebesgue spaces on sets of infinite measure

We consider grand Lebesgue spaces on sets of infinite measure and study the dependence of these spaces on the choice of the so‐called. We also consider Mikhlin and Marcinkiewicz theorems on Fourier multipliers in the setting of grand spaces.     

Measurable linear manifolds in products of linear spaces with measure

We consider some properties of measurable linear manifolds in products of linear spaces with measure which play an important part in the theory of measurable and polylinear functionals in such spaces. An example is adduced of an application of the results obtained in order to investigate the properties of series of independent random variables.Translated from Matematicheskie Zametki, Vol. 5, No. 5, pp. 623–634, May, 1969.In conclusion I should like to thank G. E. Shilov for his attention to my work.     

Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G _ℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G _ℤ; that ifH is epimorphic inG then the action ofH onG/G _ℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G _ℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.     

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,?)×…×SL(2,?)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Liv?ic’s theorem for Anosov flows for a standard partially hyperbolic ? ^d - or ? ^d - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,?) ^d /Γ and are a step in the direction of the “top-degree cohomology” part.     

Bounded Kähler class rigidity of actions on Hermitian symmetric spaces

We prove that the conjugacy class of a Zariski dense representation , q>p?1, of a finitely generated group Γ is completely determined by the pull-back via π of a bounded cohomology class in defined in terms of the Kähler form on the associated symmetric space. Under the assumption that is finite dimensional, we show that, up to equivalence, there is only a finite number of such representations for fixed q>p?1; moreover, under the hypothesis that injects into , we estimate the total number of such representations (for all q>p?1) to be bounded above by .