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By Michael Spivak

Publication by means of Michael Spivak, Spivak, Michael

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Additional info for A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition

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1] Definition: The representation π is unitarizable or pre-unitary if there is an inner product , on V (that is, a positive-definite hermitian form) so that (i) with the metric topology on V from the norm |v| = v, v continuous function G → V , and (ii) , is G-invariant, in the sense that 1/2 π(g), v, π(g)w = v, w for all v, w ∈ V and g ∈ G. 2] Proposition: In the situation just described, a unitarizable representation π can be extended (continuously) to give a unitary representation of G on the Hilbert space V˜ obtained by completing V with respect to the norm.

In particular, we can take Xo to be a minimal non-zero closed subspace of W , thus an irreducible G-space. /// Returning to the proof of the theorem, let {Xα } be a maximal set of irreducible subrepresentations such that α Xα = α Xα . ) Suppose that the closure of this sum is not the whole space X. Let X ′ be the orthogonal complement of this closure. Existence of approximate identities assures that there is ϕ ∈ Cco (G) such that R(ϕ = 0 on X ′ . Then either R(ϕ + ϕ∗ ) or R(ϕ − ϕ∗ )/i is non-zero, and yields a non-zero self-adjoint compact operator on X ′ .

Since the defining conditions are closed conditions, this is a closed subspace of L2 (Z\G, ω), though it is not generally G-stable. Of course it is stable under the conjugation action of G πconj (h)f (g) = f (h−1 gh) Also, L2 (Z\G, ω) is a unitary representation of G with the conjugation action, and so L2cen(Z\G, ω) is unitary under this action, as well. The central character under the conjugation action is trivial. Since trace is invariant under conjugation, it is immediate that each χπ is a central function.

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