By Garrett P.
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Then W is Artinian as a G-space, that is, a descending chain of closed G-stable subspaces must stabilize after finitely-many steps. In particular, W contains a non-zero irreducible closed G-space. Proof: (of lemma) For closed G-subspaces W1 ⊂ W2 ⊂ W with W1 = W2 we will show that W1 (λ) = W2 (λ) Let pi be the orthogonal projection to Wi . These are G-maps, so commute with T = R(ϕ). If W1 (λ) = W2 (λ), then necessarily p1 (v) = p2 (v), so R(G) · pi (v) = pi (R(G) · v) = pi (dense subspace of W ) = dense subspace of Wi Since the Wi are closed, they are equal.
5] Definition: A topological group G is CCR (Kaplansky’s ‘completely continuous representations’) or (Dixmier’s) liminal or liminaire if for every irreducible unitary representation π, V of G the image in End(V ) of Cco (G) consists entirely of compact operators. The following result is not too hard, but we will not prove it here just now. 6] Proposition: If G is a Type I group and if π is an irreducible unitary Hilbert space representation of G × H, then π is of the form π1 ⊗ π2 where π1 is an irreducible unitary representation of G and π2 is an irreducible unitary representation of H.
Let q be the quotient map P × K → (P × K)/Θ ≈ G The lemma just below shows that this map is a homeomorphism. Compute f (q(p, k)) f (g) dµ(g) = P G since dp ∆(p) K dp dk ∆(p) is a left Haar measure on P . This is f (pk −1 ) P K dp dk = ∆(p) f (pk) P K dp dk ∆(p) since K is unimodular (being compact). On the other hand, the usual Haar integral on G also is left P invariant and right K-invariant. The uniqueness result proves that the double integral in terms of P and K must be a multiple of the usual Haar integral.
Unitary representations of topological groups by Garrett P.