Irreducible decompositions of unitary representations of infinite- dimensional groups

This paper concerns the problem of irreducible decompositions of unitary representations of topological groups G, including the group Diff₀(M) of diffeomorphisms with compact support on smooth manifolds M. It is well known that the problem is affirmative, when G is a locally compact, separable group (cf. [3, 4]). We extend this result to infinite-dimensional groups with appropriate quasi-invariant measures, and, in particular, we show that every continuous unitary representation of Diff₀(M) has an irreducible decomposition under a fairly mild condition. This research was partially supported by a Grant-in-Aid for Scientific Research (No.14540167), Japan Socieity of the Promotion of Science.     

Quasiregular representations of the infinite-dimensional Borel group

The notion of quasiregular (Representation of Lie groups, Nauka, Moscow, 1983) or geometric (Grundlehren der Mathematischen Wissenschaften, Band 220, Springer, Berlin, New York, 1976; Encyclopaedia of Mathematical Science, Vol. 22, Springer, Berlin, 1994, pp. 1-156) representation is well known for locally compact groups. In the present work an analog of the quasiregular representation for the solvable infinite-dimensional Borel group is constructed and a criterion of irreducibility of the constructed representations is presented. This construction uses G-quasi-invariant Gaussian measures on some G-spaces X and extends the method used in Kosyak (Funktsional. Anal. i Prilo?hen 37 (2003) 78-81) for the construction of the quasiregular representations as applied to the nilpotent infinite-dimensional group .     

Hölder continuous homomorphisms between infinite-dimensional Lie groups are smooth

Let f:G→H be a homomorphism between smooth Lie groups modelled on Mackey complete, locally convex real topological vector spaces. We show that if f is Hölder continuous at 1, then f is smooth.     

Scale functions on $p$-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ _p , s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ _p , the scale functions of G and H:=G(K) are related by s _H ∣ _G =s _G ^{[ K :ℚ} _p ^]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998     

Projective limits of Banach-Lie groups

In this paper we characterize commutative Fréchet-Lie groups using the exponential map. In particular we prove that if a commutative Fréchet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.     

The Ricci curvature of finite dimensional approximations to loop and path groups

Let W(G) and L(G) denote the path and loop groups respectively of a connected real unimodular Lie group G endowed with a left-invariant Riemannian metric. We study the Ricci curvature of certain finite dimensional approximations to these groups based on partitions of the interval [0,1]. We find that the Ricci curvatures of the finite dimensional approximations are bounded below independent of partition iff G is of compact type with an Ad-invariant metric.     

Filling in solvable groups and in lattices in semisimple groups

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all -rank one lattices in semisimple groups of -rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion constant.     

Analogues of Cayley graphs for topological groups

We define for a compactly generated totally disconnected locally compact group a graph, called a rough Cayley graph, that is a quasi-isometry invariant of the group. This graph carries information about the group structure in an analogous way to the ordinary Cayley graph for a finitely generated group. With this construction the machinery of geometric group theory can be applied to topological groups. This is illustrated by a study of groups where the rough Cayley graph has more than one end and a study of groups where the rough Cayley graph has polynomial growth. Supported by project J2245 of the Austrian Science Fund (FWF) and be an IEF Marie Curie Fellowship of the Commission of the European Union.     

A remark on convex functions andp-harmonic maps

We prove that the constant maps are the onlyp-harmonic maps for anyp 2 from an arbitrary compact Riemannian manifold into a complete Riemannian manifold which admits a strictly convex function.     

Theory and numerical evaluation of oddoids and evenoids: Oscillatory cuspoid integrals with odd and even polynomial phase functions

The properties of oscillating cuspoid integrals whose phase functions are odd and even polynomials are investigated. These integrals are called oddoids and evenoids, respectively (and collectively, oddenoids). We have studied in detail oddenoids whose phase functions contain up to three real parameters. For each oddenoid, we have obtained its Maclaurin series representation and investigated its relation to Airy–Hardy integrals and Bessel functions of fractional orders. We have used techniques from singularity theory to characterise the caustic (or bifurcation set) associated with each oddenoid, including the occurrence of complex whiskers. Plots and short tables of numerical values for the oddenoids are presented. The numerical calculations used the software package CUSPINT [N.P. Kirk, J.N.L. Connor, C.A. Hobbs, An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives, Comput. Phys. Commun. 132 (2000) 142–165].     

A geometric construction of local representations of local Lie groups

Local transformation groups acting on a manifold X define a natural action of on a space D(X), of functions on X. The natural action induces a local representation of on a Hilbert subspace of the space of distributions on D(X).     

Liouville theorems for exponentially harmonic functions on Riemannian manifolds

Suppose thatM is a complete Riemannian manifolds with nonnegative sectional curvature. We prove that any bounded exponentially harmonic function onM is a constant function.     

Euler equations and approximations for the minimizers of Heisenberg target

In this paper, we study the Euler equations and derive approximations of the minimizers for a Heisenberg group target. There are some techniques in the arguments for proving the results. This is in order to overcome the obstacles which are due to the nonlinear structure of the group laws.     

Lowest weight representations of some infinite dimensional groups on fock spaces

The results of Kashiwara and Vergne on the decomposition of the tensor products of the Segal-Shale-Weil representation are extended to the infinite dimensional case and give all unitary lowest weight representations. Our methods are basically algebraic. When restricted to the finite dimensional case, they yield a new proof.     

Elliptic operators on Lie groups

We review the theory of strongly elliptic operators on Lie groups and describe some new simplifications. Let U be a continuous representation of a Lie group G on a Banach space and a ₁,...,a _d a basis of the Lie algebra g of G. Let A _i=dU(a _i) denote the infinitesimal generator of the continuous one-parameter group t U(exp(-ta _i)) and set % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaale% qajeaObaGaeyySdegaaOGaeyypa0JaamyqamaaBaaajeaWbaGaaeyA% aaWcbeaajaaOdaWgaaqcbaAaamaaBaaajiaObaGaaiiBaaqabaaaje% aObeaakiaacElacaGG3cGaai4TaiaadgeadaWgaaqcbaCaaiaabMga% aSqabaGcdaWgaaWcbaWaaSbaaKGaahaacaGGUbaameqaaaWcbeaaaa% a!4897!\[A^\alpha = A_{\rm{i}} _{_l } \cdot\cdot\cdotA_{\rm{i}} _{_n } \], where =(i ₁,...,i _n) with _j and set ||=n. We analyze properties of mth order differential operators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2da9i% aabccadaaeqaqaaiaadogadaWgaaqcbaCaaiabgg7aHbWcbeaaaKqa% GgaacqGHXoqycaqG7aGaaeiiaiaabYhacqGHXoqycaqG8bGaeyizIm% QaaeyBaaWcbeqdcqGHris5aOGaamyqamaaCaaaleqajeaObaGaeyyS% degaaaaa!4A6C!\[H = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} \le {\rm{m}}} {c_\alpha } A^\alpha \] with coefficients c . If H is strongly elliptic, i.e., % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOuaiaacwgacq% GH9aqpcaqGGaWaaabeaeaacaGGOaaajeaObaGaeyySdeMaae4oaiaa% bccacaqG8bGaeyySdeMaaeiFaiabg2da9iaab2gaaSqab0GaeyyeIu% oakiaabMgacqaH+oaEcaGGPaWaaWbaaSqabKqaGgaacqGHXoqyaaGc% cqGH+aGpcaaIWaaaaa!4C40!\[{\mathop{\rm Re}\nolimits} = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} = {\rm{m}}} ( {\rm{i}}\xi )^\alpha > 0\] for all % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaeyicI4% SaeSyhHe6aaWbaaSqabeaacaWGKbaaaOGaaiixaiaacUhacaaIWaGa% aiyFaaaa!3EAA!\[\xi \in ^d \backslash \{ 0\} \], then we give a simple proof of the theorem that the closure of H generates a continuous (and holomorphic) semigroup on and the action of the semigroup is determined by a smooth, representation independent, kernel which, together with all its derivatives, satisfies mth order Gaussian bounds.     

Infinite dimensional groups and quantum field theory

This paper surveys recent work on representations of infinite dimensional groups and the connection with quantum field theory.     

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF^∞, i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF^∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes-Moscovici and Dru?u-Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF^∞ and isocohomological.