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Topological Groups 5. Exercise Sheet

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Topological Groups 5. Exercise Sheet

Department of Mathematics Summer Term 2013

Andreas Mars 12.6.2013

Groupwork

Exercise G1 (Warming Up)

Prove or disprove: The mapφ7→R1

0 φ(x)d xis a Haar-integral onR. Exercise G2 (Unimodular Groups)

LetGbe the subgroup of upper triangular matricesSL2(R). Show thatGis not unimodular.

Exercise G3 (Applications of the Haar-integral)

LetGbe locally compact and letλbe a Haar-integral onG. Consider the spaceCcC(G):={f +ih|f,h∈Cc(G)}and the natural extensionλto this space. Show that:

The map

〈·,·〉:CcC(G)×CcC(G)→C (f,h)7→λ(f h)

is a unitary scalar product onCcC(G)and the transformation f 7→ fg is unitary for everyg∈G.

Conclude that the completion of the space(CcC(G),〈·,·〉)is a Hilbert space andGadmits a unitary representation on this space.

Exercise G4 (Character Groups) Compute the character group ofT.

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