Titlebook: Lie Groups; Daniel Bump Textbook 2013Latest edition Springer Science+Business Media New York 2013 Frobenius-Schur duality.Keating-Snaith f

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Haar MeasureIf . is a locally compact group, there is, up to a constant multiple, a unique regular Borel measure .. that is invariant under left translation. Here . means that .(.) = .(.) for all measurable sets ..

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Schur OrthogonalityIn this chapter and the next two, we will consider the representation theory of compact groups. Let us begin with a few observations about this theory and its relationship to some related theories.

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Compact OperatorsIf . is a normed vector space, a linear operator . is called . if there exists a constant . such that . for all .. In this case, the smallest such . is called the . of ., and is denoted |.|.

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The Peter–Weyl TheoremIn this chapter, we assume that . is a compact group. Let .(.) be the convolution ring of continuous functions on .. It is a ring (without unit unless . is finite) under the multiplication of .:

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The Exponential MapThe exponential map, introduced for closed Lie subgroups of . in ., can be defined for a general Lie group . as a map Lie(.) → ..

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The Universal CoverIf . is a Hausdorff topological space, a . is a continuous map . : [0,1].. The path is . if the endpoints coincide : .(0) = .(1). A closed path is also called a ..