G-invariant Functions Research Articles

Fourier extension estimates for symmetric functions and applications to nonlinear Helmholtz equations

We establish weighted L^p-Fourier extension estimates for O(N-k) times O(k)-invariant functions defined on the unit sphere {mathbb {S}}^{N-1}, allowing for exponents p below the Stein–Tomas critical exponent frac{2(N+1)}{N-1}. Moreover, in the more general setting of an arbitrary closed subgroup G subset O(N) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation -Δu-u=Q(x)|u|p-2u,u∈W2,p(RN),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\Delta u - u = Q(x)|u|^{p-2}u, \\quad u \\in W^{2,p}({\\mathbb {R}}^{N}), \\end{aligned}$$\\end{document}where Q is a nonnegative bounded and G-invariant weight function.

On equicontinuous factors of flows on locally path-connected compact spaces

AbstractWe consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \textrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

Partial coherent state transforms, G × T-invariant Kähler structures and geometric quantization of cotangent bundles of compact Lie groups

In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain G×T-invariant functions on the cotangent bundle of a compact connected Lie group G with maximal torus T. Namely, we will take the Hamiltonian flows of one G×G-invariant function, h, and one G×T-invariant function, f. Acting with these complex time Hamiltonian flows on G×G-invariant Kähler structures gives new G×T-invariant, but not G×G-invariant, Kähler structures on T⁎G. We study the Hilbert spaces Hτ,σ corresponding to the quantization of T⁎G with respect to these non-invariant Kähler structures. On the other hand, by taking the vertical Schrödinger polarization as a starting point, the above G×T-invariant Hamiltonian flows also generate families of mixed polarizations P0,σ, σ∈C, Imσ>0. Each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a Kähler structure on the leaves of a foliation of T⁎G. The geometric quantization of T⁎G with respect to these mixed polarizations gives rise to unitary partial coherent state transforms, corresponding to KSH maps as defined in [11,12].

Symmetries on manifolds: generalizations of the radial lemma of Strauss

For a compact subgroup G of the group of isometries acting on a Riemannian manifold M we investigate subspaces of Besov and Triebel–Lizorkin type which are invariant with respect to the group action. Our main aim is to extend the classical Strauss lemma under suitable assumptions on the Riemannian manifold by proving that G-invariance of functions implies certain decay properties and better local smoothness. As an application we obtain inequalities of Caffarelli–Kohn–Nirenberg type for G-invariant functions. Our results generalize those obtained in Skrzypczak (Rev Mat Iberoam 18:267–299, 2002). The main tool in our investigations are atomic decompositions adapted to the G-action in combination with trace theorems.

Universal commutator relations, Bogomolov multipliers, and commuting probability

Let G be a finite p-group. We prove that whenever the commuting probability of G is greater than (2p2+p−2)/p5, the unramified Brauer group of the field of G-invariant functions is trivial. Equivalently, all relations between commutators in G are consequences of some universal ones. The bound is best possible, and gives a global lower bound of 1/4 for all finite groups. The result is attained by describing the structure of groups whose Bogomolov multipliers are nontrivial, and Bogomolov multipliers of all of their proper subgroups and quotients are trivial. Applications include a classification of p-groups of minimal order that have nontrivial Bogomolov multipliers and are of nilpotency class 2, a nonprobabilistic criterion for the vanishing of the Bogomolov multiplier, and establishing a sequence of Bogomolov's absolute γ-minimal factors which are 2-groups of arbitrarily large nilpotency class, thus providing counterexamples to some of Bogomolov's claims. In relation to this, we fill a gap in the proof of triviality of Bogomolov multipliers of finite almost simple groups.

Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The G-action lifts to an action on the total space T of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on T. The Poisson algebra of G-invariant functions on T yields a Poisson structure on the space T/G of G-orbits. We relate this Poisson algebra with extensions of Lie-Rinehart algebras and derive an explicit formula for this Poisson structure in terms of differentials. We then show, for the particular case where the G-action on Q is principal, how an explicit description of the Poisson algebra derived in the literature by an ad hoc construction is essentially a special case of the formula for the corresponding extension of Lie-Rinehart algebras. By means of various examples, we also show that this kind of description breaks down when the G-action does not define a principal bundle.

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G-invariant functions of pairs ( T , s ) , where T is a hypertorus in X (of codimension 1), and s ∈ G is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic Calogero–Moser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems – an explicit construction as limits of classical Calogero–Moser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of V. Buchstaber, G. Felder and A. Veselov (1994) [BFV]), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions.

Symbolic computation of degree-three covariants for a binary form

We use elementary linear algebra to explicitly calculate a basis for, and the dimension of, the space of degree-three covariants for a binary form of arbitrary degree. We also give an explicit basis for the subspace of covariants complementary to the space of degree-three reducible covariants. The study of invariant functions was one of the main influences on the development of modern algebra. Consider the following simple example. The group GD Z acts on R by addition: g xD gC x. We define a G-invariant function to be a real-valued function f.x/ on R such that f gD f for all g2 G. In other words, f.x/D f.gC x/ for all g2 Z, x2 R. The invariant functions are precisely the real-valued functions with period one. Hence, geometric information, such as periodicity, can be recovered by studying functions with certain algebraic properties. In Section 1, we introduce the concepts of an invariant and covariant function for a binary form Q.x; y/. The problem of determining the complete set of these functions was widely worked on during the late nineteenth century. Gordan [1868] proved that the ring of invariants (and the ring of covariants) for a degree-n binary form is finitely generated. A milestone in the history of modern algebra was Hilbert’s nonconstructive proof [1890] of the following fundamental theorem. Theorem [Hilbert 1890]. The ring of invariants (and the ring of covariants) for a degree-n homogeneous polynomial in m variables is finitely generated. Hilbert’s theorem says that all invariants (resp. covariants) for a homogeneous polynomial can be expressed as polynomials in a certain finite set of invariants (resp. covariants). Hilbert [1893] subsequently gave a constructive proof of this theorem. The minimal size of the generating set is only known for a few values of .m; n/. When mD 2, this number has been determined for n 8 [Bedratyuk 2009; Bedratyuk and Bedratyuk 2008].

On covariant functions and distributions under the action of a compact group

Let G be a compact subgroup of GL n ( R ) acting linearly on a finite dimensional complex vector space E. B. Malgrange has shown that the space C ∞ ( R n , E ) G of C ∞ and G-covariant functions is a finite module over the ring C ∞ ( R n ) G of C ∞ and G-invariant functions. First, we generalize this result for the Schwartz space S ( R n , E ) G of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in [4].

Bi-Poisson Structures and Integrability of Geodesic Flow on Homogeneous Spaces

Let G=K be a semisimple orbit of the adjoint representation of a real connected reductive Lie group G. Let K1 be any closed subgroup of K containing the commutant of the identity component of K. We prove that the geodesic flow on the symplectic manifold T ⁄ (G=K1), corresponding to a G-invariant pseudo-Riemannian metric on G=K1 which is in- duced by a bi-invariant pseudo-Riemannian metric on G, is completely integrable in the class of real analytic functions, polynomial in momenta. To this end we study the Poisson geometry of the space of G-invariant functions on T ⁄ (G=K) using a one-parameter family of moment maps.

Equivariance and Multiplicity for the Scalar Curvature Problem

Given (M, g 0) a three-dimensional compact Riemannian manifold, assumed not to be conformally diffeomorphic to the standard unit 3-sphere, and G a compactsubgroup of the conformal group of (M, g 0), we first study conditions for a smooth G-invariant function f to be the scalar curvature of a G-invariant conformalmetric to g 0. Then, extending previous results of Hebeyand Vaugon, we study conditions for f to be the scalarcurvature of at least two conformal metrics to g 0.

G-majorization with applications to matrix orderings

A vector x is said to G-majorize a vector y if y lies in the convex hull of the orbit of x under a group G. The present paper contains a straightforward account with two important statements equivalent to G-majorization. In certain cases, for example when G is a finite reflection group, one equivalent condition reduces to a finite set of linear inequalities representing a cone ordering in the fundamental region of the group. The other condition is that every convex G-invariant function of y is less than the same function of x. Upper and lower weak majorizations ( GW-majorizations) are introduced by combining G with a second ordering compatible with it. The results are applied where possible to matrix orderings where A≺ G B is G-majorization when G is a subgroup of the orthogonal group O n acting by congruence, i.e. g( B)= QBQ T with Q the matrix representation for an element of O n . When G is the symmetric group of permutation matrices this defines a new ordering and generalizes a proposition of Kiefer in the design of experiments.

The geometry of spontaneous symmetry breaking

The problem of classifying the theoretically allowed patterns of spontaneous symmetry breading, in theories where the ground state is determined as a minimum of a G-invariant potential ( G a compact group of transformations), is analyzed. A detailed, complete, and rigorous justification of a recently proposed approach to the determination of the minima of G-invariant potentials (M. Abud and G. Sartori, Phys. Lett. B 104 (1981), 147) is presented. The results are obtained through an analysis of the geometry of the finite-dimensional representations of G, which leads to a complete characterization of the structure of orbit space and its partition in subsets (strata) formed by orbits with the same symmetry under G-transformations (orbit type), and to a new theorem stating that the gradients of complex analytic G-invariant functions annihilate on one-dimensional strata. Polynomial potentials in particular are studied. Conditions for instability of the residual symmetry (second-order phase transitions) are determined.

A theorem on orbit structures (strata) of compact linear Lie groups

We present a comprehensive constructive proof of a theorem characterizing the tangent space to a stratum (orbit structure) of the Euclidean space Rn, seat of an orthogonal representation of a compact group G. The characterization is made in terms of gradients of a complete set (integrity basis) of G-invariant polynomials. In a recent paper [M. Abud and G. Sartori, Phys. Lett. B 104, 147 (1981)], the theorem, which may be considered a generalization of a theorem by Michel [C. R. Acad. Sci. Ser. A 272, 433 (1971)], has been shown to be effective in the determination of the equations of the strata and in the determination of natural extrema of G-invariant functions.

On the method of symes for integrating systems of the toda type

Let G=KL and g=k+l be Lie group and Lie algebra decompositions. This identifies k o with l*. Any G-invariant function, f, on g*induces by restriction a function f|k o =l*. We prove a formula which says that the integral curve through α∈k o is obtained as b(t)α, where a(t)=exp tξ with ξ=L f (α), $$\left( * \right){\text{ }}a\left( t \right) = b\left( t \right)c\left( t \right)$$ where (*) is the KL decomposition and where L f : g* → g is the Legendre transform. This generalizes a formula of Symes for the generalized Toda lattice.

Group structure and the pointwise ergodic theorem for connected amenable groups

Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group K = G R ). We show how to explicitly construct sequences { U n } of compacta in G in terms of the structural features of G which have the following property: For any “reasonable” action G × L p ( X, μ) ↓ L p ( X, μ) on an L p space, 1 < p < ∞, and any f ∈ L p ( X, μ), the averages A nf= 1 |U n| ∫ U n T g −1fdg (|E|= left Haar measure inG) converge in L p norm, and pointwise μ-a.e. on X, to G-invariant functions f∗ in L p ( X, μ). A single sequence { U n } in G works for all L p actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.

Totally real representations and real function spaces

0* Introduction* Let X be a riemannian symmetric space of compact type and G its largest connected group of isometries. In his 1929 paper [1] on class 1 representation s, E. Cartan showed that the symmetry of X sends every uniformly closed G-invariant function space on X to its complex conjugate. Starting from the point of view of algebras, Mirkil and de Leeuw [4] showed that every rotation invariant function algebra on the sphere Sn(n ^ 2) was spanned by real-valued functions, hence (Stone-Weierstrass theorem) that such an algebra necessarily was all continuous functions on Sn, all continuous functions on real protective w-space, or just the constantsthat state of affairs is quite different from the case n = 1. When the rotation group S0(n + 1) contains the symmetry of S w, i.e. when n is even, Cartan's result mentioned above implies reality of such function algebras. The published Mirkil-de Leeuw argument rests rather on the fact that the spherical harmonics are real-valued.