Matrix coefficient of a unitary representation

Depends on

• Strongly continuous unitary representations, invariant linear subspaces and intertwiners
• Real and complex inner-product spaces and their induced length
• Cauchy–Schwarz: $|\langle x,y\rangle|\le\|x\|\,\|y\|$, with equality exactly for dependent pairs
• The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane

Used by

• Normalized positive type and pointed cyclic unitary representations Corollary
• Parseval and Fourier inversion for compact groups Corollary
• The spherical complementary series converge to the trivial representation Corollary
• The unitary dual of a compact group is Fell discrete Corollary
• The unitary dual of SL2(R) is non-discrete and non-Hausdorff at the stated limits Corollary
• A limit of discrete series is not square-integrable Counterexample
• The regular representation of R is not a Hilbert direct sum of irreducibles Counterexample
• The unitary dual need not be Hausdorff Counterexample
• Almost invariant vectors for a unitary representation Definition
• Representative functions on a compact group Definition
• The Fell topology on the unitary dual Definition
• The normalized irreducible matrix coefficient family Definition
• Weak containment of unitary representations Definition
• A square-integrable discrete-series matrix coefficient Example
• Peter-Weyl for a profinite group Example
• Peter-Weyl for an infinite product of finite groups Example
• The circle: the Peter-Weyl basis is the integer characters Example
• Almost invariant vectors and normalized positive type functions Lemma
• Bounds and two-sided uniform continuity of unitary coefficients Lemma
• Continuity criteria for unitary representations Lemma
• Continuous finite-dimensional representations of a product of finite groups factor through a finite subproduct Lemma
• Continuous finite-dimensional representations of profinite groups factor through finite quotients Lemma
• Diagonal unitary coefficients have positive type Lemma
• Direct sums and tensor products of finite-dimensional unitary representations Lemma
• Dominated positive type and positive commutant contractions Lemma
• Fell continuity of the unitary principal series in the parameter Lemma
• Fell neighbourhoods of an irreducible representation are saturated under weak equivalence Lemma
• Irreducible weak containment in a family selects one coefficient Lemma
• Matrix coefficients of finite-dimensional representations separate points of a compact group Lemma
• Matrix-coefficient formulas and decay for the discrete and principal series Lemma
• Normalized coefficient approximation for irreducible weak containment Lemma
• Representative functions form a self-adjoint translation-invariant algebra Lemma
• The Fell closure of a single representation is its weak containment closure Lemma
• The L1 action of a strongly continuous unitary representation Lemma
• Translation estimates for continuous positive type functions Lemma
• Weak containment of the trivial representation and almost invariant vectors Lemma
• GNS construction for a continuous positive-type function Theorem
• Peter-Weyl decomposition of the regular representation Theorem
• Property (T) and isolation of the trivial representation in the Fell dual Theorem
• Property (T) is equivalent to the existence of a compact Kazhdan pair Theorem

…and 6 more results.

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition A.1.1, Appendix A, printed p. 305 (standard reference, not scraped)
• Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, §3.4, printed pp. 106–107 (standard reference, not scraped)