Harmonic Analysis on Commutative Spaces

Wolf, Joseph A.

doi:10.1090/surv/142

Cited by 155 publications

(191 citation statements)

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“…The spherical function is independent of the choice of s λ . Using the well known theory of the spherical Fourier transform [22] or the abstract Plancherel formula it follows that for f ∈ C c (G/K) we can choose the measure μ on such that…”

Section: Bandlimited Functionsmentioning

confidence: 99%

Examples of Coorbit Spaces for Dual Pairs

Christensen

Ólafsson

2008

Acta Appl Math

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In this paper we summarize and give examples of a generalization of the coorbit space theory initiated in the 1980's by H.G. Feichtinger and K.H. Gröchenig. Coorbit theory has been a powerful tool in characterizing Banach spaces of distributions with the use of integrable representations of locally compact groups. Examples are a wavelet characterization of the Besov spaces and a characterization of some Bergman spaces by the discrete series representation of SL 2 (R).We present examples of Banach spaces which could not be covered by the previous theory, and we also provide atomic decompositions for an example related to a non-integrable representation.

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Section: Bandlimited Functionsmentioning

confidence: 99%

Examples of Coorbit Spaces for Dual Pairs

Christensen

Ólafsson

2008

Acta Appl Math

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“…p2q X is weakly symmetric. Examples of weakly symmetric spaces that are not symmetric include some circle bundles over hermitian symmetric space; see [M71, N97] and [W07,Section 12.3]. The classification of reductive weakly symmetric spaces was completed by Yakimova [Y02]; see also [Y05].…”

Section: Theorem Let the Notation Be As Above Then The Following Armentioning

confidence: 99%

Book Review: Homogeneous spaces and equivariant embeddings

Ólafsson¹

2013

Bull. Amer. Math. Soc.

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Basic conceptsGroup actions on manifolds, algebraic varieties and other sets, and geometric objects have played an important role in geometry, analysis, representation theory, and physics for a long time. The book by D. A. Timashev is a welcome survey of old and new results on actions of algebraic groups on algebraic varieties. Basic facts on algebraic groups, homogeneous spaces, and equivariant embeddings are presented. Then the discussion concentrates more on a detailed description of special and interesting classes where deeper results can be obtained. Those cases include symmetric spaces, weakly symmetric spaces, spherical varieties, spaces of lower rank and complexity, and so-called wonderful varieties. Classification of several categories of homogeneous spaces are given. The book is about the algebraic side of group actions, but to complement the book, we take a more analytic viewpoint.Let us start by recalling some basic concepts. Let G be a group, and let X be a set. A G-action on X is a map GˆX Ñ X, often written as pa, xq Þ Ñ a¨x " a pxq, such that a Þ Ñ a is a group homomorphism from G into the group of bijections on X. If a G-action on X is given, then X is said to be a G-set. For a fixed x P X the map a Þ Ñ a¨x is the orbit map and G¨x is a G-orbit. The subgroup G x " ta P G | a¨x " xu is the stabilizer of x in G. We say that X is homogeneous if X " G¨x for some point, x P X. In that case X " G¨x for all points x in X. If X and Y are two G-sets, then a map ϕ : X Ñ Y is G-equivariant, or a G-map, if ϕpa¨xq " a¨ϕpxq for all a P G and all x P X. If X is homogeneous, thenIf G is a Lie group acting smoothly on a manifold X (always assumed separable), then G x is closed for all x P X and G{G x is a manifold. The group G acts smoothly on G{G x and G{G x is isomorphic to X as a manifold and as a G-space. In the algebraic category, the variety G{G x » X if the map aG x Þ Ñ a¨x o is separable. To explain the title of the book we can now say that G{H ϕ ãÑ X is an equivariant embedding if X is a normal variety with a G-action and ϕ is G-equivariant map with an open image in X. In that case it is common to identify G{H with ϕpG{Hq Ă X. Finally, if τ : G Ñ G

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“…In this section we consider a somewhat larger class of nilpotent direct systems that satisfy these conditions. In this section we extend our Heisenberg group considerations to nilpotent Lie groups with square integrable representations, following the general lines of [16] and [18].…”

Section: Extension To Commutative Nilmanifoldsmentioning

confidence: 99%

“…In each case of Table 5.31, [16,Theorem 14.4.3] says that N n has square integrable representations. In the cases dim z > 1 of Table 5.31 we have K n = K † n · L where the big factor K † n acts trivially on z and the small factor L acts on z by its adjoint representation.…”

Section: Semidirect Product Groupsmentioning

confidence: 99%

“…These (G n /K n ) are constructed from certain basic ones that satisfy several technical conditions (indecomposable, principal, maximal and Sp(1)-saturated). See [16,[19][20][21] and [18]. The basic such direct systems, with K n reducible on n n /z n , dim z n bounded and {K n } parabolic, are tabulated in [18, Table 9.15] as follows.…”

Section: Reducible Quotientsmentioning

confidence: 99%

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