On Algebraic Groups and Homogeneous Spaces

In this paper we will be concerned with orbits of a closed subgroup Z of an algebraic group (respectively Lie group) G on a homogeneous space X for G. More precisely, let D be a closed subgroup of G and let X denote the coset space G/D. Let S be a subgroup of G and let Z denote (GS)0 the identity component of GS, the centralizer of S in G. We consider the orbits of Z on XS, the set of fixed points of S on X. We also treat the more general situation in which S is an algebraic group (respectively Lie group) which acts on G by automorphisms and acts on X compatibly with the action of G; again we consider the orbits of (GS)0 on XS.

Universal torsors of Del Pezzo surfaces and homogeneous spaces

Let $X=G/\Gamma $ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X . We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H -orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, $\Gamma $ is a non-cocompact lattice of G , and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H -orbit to a single compact set in X . This complements our results in the case of an arithmetic lattice.

Let H be a compact subgroup of a locally compact group G, and let μ be a strongly quasi-invariant Radon measure on the homogeneous space G/H. In this article, we show that every element of G/Hˆ, the character space of G/H, determines a nonzero multiplicative linear functional on L1(G/H,μ). Using this, we prove that for all ϕ∈G/Hˆ, the right ϕ-amenability of L1(G/H,μ) and the right ϕ-amenability of M(G/H) are both equivalent to the amenability of G. Also, we show that L1(G/H,μ), as well as M(G/H), is right ϕ-contractible if and only if G is compact. In particular, when H is the trivial subgroup, we obtain the known results on group algebras and measure algebras.

In this paper, two theorems about the compactness of almost invariant operators on homogeneous Banach spaces of distributions (in the sense of Feichtinger [11]) defined on a locally compact abelian group are proved. Our theorems generalize the corresponding results of K. de Leeuw [3] and Tewari and Madan [18] for operators on homogeneous Banach spaces on the circle group and Segal algebras on a compact abelian groups, respectively.

Smooth embeddings of SL(2) and PGL(2)

It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces X X under a linear group G G (or under a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of X X are connected. In this work we prove, under similar assumptions, that the étale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilizers. We also deal with some related questions, concerning strong approximation outside a finite set of valuations. Finally, we prove a compatibility result, suggested to be true by work of Cyril Demarche, between the Brauer-Manin obstruction pairing on quotients G / H G/H , where G G and H H are connected algebraic groups and H H is linear, and certain abelianization morphisms associated with these spaces.

Arithmetic purity of strong approximation for homogeneous spaces

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.

We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=GR with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {ut} ofG and anyg∈G, the time spent inC by the {ut}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g−1utg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].

Let G be a p-adic algebraic group of polynomial growth and H be a closed subgroup of G. We prove the growth conjecture for the homogeneous space G/H, that is, G/H supports a recurrent random walk if and only if G/H has polynomial growth of degree atmost two.

For a homogeneous space with connected or abelian stabilizer of a connected linear algebraic group defined over a number field, a cohomological obstruction to the Hasse principle is defined in terms of Galois hypercohomology with coefficients in a complex of two abelian algebraic groups. This obstruction is proved to be the only obstruction to the Hasse principle. It is proved that up to sign this cohomological obstruction coincides with the Brauer-Manin obstruction.

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.

On the cohomology of Coxeter groups and their finite parabolic subgroups. II by T. Akita Linear source modules and trivial source modules by R. Boltje Resolutions de foncteurs de Mackey by S. Bouc Homotopy uniqueness of $BPU(3)$ by C. Broto and A. Viruel Lowering operators for $GL(n)$ and quantum $GL(n)$ by J. Brundan Homomorphisms in higher complexity quotient categories by J. F. Carlson and W. W. Wheeler On the homotopy theory of $p$-completed classifying spaces by F. R. Cohen and R. Levi $H^*V$ is of bounded type over $\mathcal {A} (p)$ by M. D. Crossley Classifying spaces of central group extensions by J. Dietz and T. Ratliff Sharp homology decompositions for classifying spaces of finite groups by W. G. Dwyer On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$ by P. Fong and R. J. Milgram Transfer and Chern classes for extraspecial $p$-groups by D. J. Green and P. A. Minh The infinite order of the symplectic classes in the cohomology of the stable mapping class group by R. Grieder A variant of the proof of the Landweber Stong conjecture by H.-W. Henn Unstable modules over the Steenrod algebra and cohomology of groups by H.-W. Henn The category of Mackey functors for a compact Lie group by L. G. Lewis, Jr. Comparison of extensions of modules for algebraic groups and quantum groups by Z. Lin The defect of the completion of a Green functor by F. Luca Simple modules of Ariki-Koike algebras by A. Mathas On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$: II by R. J. Milgram On the relation between simple groups and homotopy theory by R. J. Milgram Varieties for $G_rT$-modules by D. K. Nakano On defect groups of the Mackey algebras for finite groups by F. Oda Applications of stable classifying space theory to group cohomology by S. Priddy The spectra $k$ and $kO$ are not Thom spectra by Yu. B. Rudyak The code of a regular generalized quadrangle of even order by N. S. N. Sastry and P. Sin Quivers of blocks with normal defect group by M. Schaps, D. Shapira, and O. Shlomo Cohomology of a homogeneous space $E_6/T^1\cdot SU(6)$ by T. Watanabe Problem session, Seattle 1996 by D. Benson.

We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.