On uniqueness of invariant means

The following results on uniqueness of invariant means are shown:

(i) Let be a connected almost simple algebraic group defined over . Assume that , the group of the real points in , is not compact. Let be a prime, and let be the compact -adic Lie group of the -points in . Then the normalized Haar measure on is the unique invariant mean on .

(ii) Let be a semisimple Lie group with finite centre and without compact factors, and let be a lattice in . Then integration against the -invariant probability measure on the homogeneous space is the unique -invariant mean on .

     

Proximality and Equidistribution on the Furstenberg Boundary

Let G be a connected semisimple Lie group with finite center and without compact factors, P a minimal parabolic subgroup of G, and Γ a lattice in G. We prove that every Γ-orbits in the Furstenberg boundary G/P is equidistributed for the averages over Riemannian balls. The proof is based on the proximality of the action of Γ on G/P.*Partially supported by NSF grant 0400631. Mathematics Subject Classification (2000). 37B05, 22E40     

A finiteness theorem of harmonic maps from compact lie groups toO HS(H)

In this article we study the behavior of harmonic maps from compact connected Lie groups with bi-invariant metrics into a Hilbert orthogonal group. In particular, we will demonstrate that any such harmonic map always has image contained within someO(n),n<∞. Since homomorphisms are a special subset of the harmonic maps we get as a corollary an extension of the Peter-Weyl theorem, namely, that every representation of a connected compact Lie group is finite dimensional. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

On a class of stochastic flows driven by quantum Brownian motion

A class of quantum stochastic flows on^*-algebras is introduced which includes both classical flows on Riemannian manifolds and flows induced by Lie group actions onC ^*-algebras. Criteria are established to determine those flows which are unitarily equivalent to ones driven by classical Brownian motion. It is shown that taking complex combinations of the driving coefficients of such flows gives rise to flows which are not of Evans-Hudson type (i.e., all driving coefficients do not preserve the relevant algebra).     

On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups which are diffeomorphic to , for some . After identifying with , the multiplication on can be seen as a map . We show that if is a polynomial map in one of the two (sets of) variables or , then is solvable. Moreover, if one knows that is polynomial in one of the variables, the group is nilpotent if and only if is polynomial in both its variables.

     

On the McCool group M3 and its associated Lie algebra

We prove that the Lie algebra of the McCool group M₃ is torsion free. As a result, we are able to give a presentation for the Lie algebra of M₃. Furthermore, M₃ is a Magnus group.     

Minimal number of singular fibers in a Lefschetz fibration

There exists a (relatively minimal) genus Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus iff and . The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.

     

On a theorem of R. Steinberg on rings of coinvariants

Let be a representation of a finite group over the field . Denote by the algebra of polynomial functions on the vector space . The group acts on and hence also on . The algebra of coinvariants is , where is the ideal generated by all the homogeneous -invariant forms of strictly positive degree. If the field has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that is a Poincaré duality algebra if and only if is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case and give a counterexample in the modular case when .

     

On tails of stationary measures on a class of solvable groups

Let G be a subgroup of GL(R,d) and let (Qn,Mn) be a sequence of i.i.d. random variables with values in Rd?G and law μ. Under some natural conditions there exists a unique stationary measure ν on Rd of the process Xn=MnXn−1+Qn. Its tail properties, i.e. behavior of as t tends to infinity, were described some over thirty years ago by H. Kesten, whose results were recently improved by B. de Saporta, Y. Guivarc'h and E. Le Page. In the present paper we study the tail of ν in the situation when the group G₀ is Abelian and Rd is replaced by a more general nilpotent Lie group N. Thus the tail behavior of ν is described for a class of solvable groups of type NA, i.e. being semi-direct extension of a simply connected nilpotent Lie group N by an Abelian group isomorphic to Rd. Then, due to A. Raugi, (N,ν) can be interpreted as the Poisson boundary of (NA,μ).     

有限Chevalley群作用下的子代数轨道生成的格

讨论了一般情形Chevalley群作用下的子代数轨道生成的格.在同类型格中,研究了不同格之间的包含关系,并对格中子代数的特性及格的几何性进行了刻画.     

On Macdonald's formula for the volume of a compact Lie group

We give a simple proof of Macdonald's formula for the volume of a compact Lie group. Received: May 10, 1997     

On the cellular decomposition of the exceptional Lie group

The present note is to give a cellular decomposition of the compact connected exceptional Lie group .

     

On extensions of sub-Riemannian structures on Lie groups

We define the extension of a left-invariant sub-Riemannian structure in terms of an extension of the underlying Lie group and compatibility of the respective distributions and metrics. We show that geodesics of a structure can be lifted to geodesics of any extension of the structure. In the case of central extensions, we show that the normal geodesics of the minimal extension are the projection (in a sense) of the normal geodesics of any other compatible extension. Several illustrative examples are discussed.     

JSJ-Decompositions of finitely presented groups and complexes of groups

A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips–Sela and Dunwoody–Sageev, and we construct a JSJ-splitting for any finitely presented group with respect to the class of all slender subgroups along which the group splits. Our approach relies on Haefliger’s theory of group actions on CAT(0) spaces. Submitted: October 2003 Revision: February 2005 Accepted: June 2005     

On a theorem about orderability

The purpose of this note is proving a result of Nogura and Shakhmatov [4, Theorem 4]. Furthermore, a characterization of connected spaces with a selection is given. Work supported by the research project “Metodi e problemi in Analisi Reale” of the Italian Ministero dell'Università e della Ricerca Scientifica e Tecnologica.     

Standard tableaux and Klyachko's theorem on Lie representations

We show that for all but two partitions λ of n>6 there exists a standard tableau of shape λ with major index coprime to n. In conjunction with a deep result of Kra?kiewicz and Weyman this provides a new purely combinatorial proof of Klyachko's famous theorem on Lie representations of the general linear group.