By Garrett P.
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Additional resources for Unitary representations of topological groups
That is, X = L2 (ZΓ\G, ω) is the completion of a direct sum of irreducibles. To see that a π-isotype is of finite multiplicity, again use compactness of the operators from Cco (G). With ϕ = ϕ∗ , R(ϕ) acts by scalar λ on X π (λ). Non-zero scalar operators are compact only for finite-dimensional spaces, so X π (λ) is finite-dimensional for λ = 0. Given π, take ϕ = ϕ∗ such that R(ϕ) is non-zero on the representation space Vπ of π, invoking the existence of approximate identities. Then Vπ (λ) = 0 for some non-zero λ.
Let π be unitary. Then 0≤| ci π(gi )v|2 = ij i ci cj π(gj−1 gi )vi , v ci cj π(gj )∗ π(gi )vi , v = ci cj π(gi )vi , π(gj )v = ij ij by unitariness. On the other hand, suppose f is positive definite. Let ci f (ggi ) E(g) = dj f (ghj ) F (g) = j i Using f (g −1 ) = f (g), ci dj f (h−1 j gi ) = E, F = dj E(h−1 j )= j ij ci F (gi−1 ) i Thus, the inner product E, F is independent of the expressions for E and F , but only depends on the functions. That is, the inner product is well-defined. That the inner product has the positive semi-definiteness property F, F ≥ 0 is immediate from the definition of positive-definiteness.
Traces, characters, central functions for Z ÒG compact Take Z\G compact with Z a closed subgroup of the center of G (so G is unimodular). From above, every irreducible unitary of G is finite dimensional, with some unitary central ω on Z. 1] Definition: Let π, V be an irreducible (finite-dimensional) of G. The character χπ of π is the function on G defined by χπ (g) = trace π(g) = π(g)ei , ei i for any orthonormal basis ei of π. 2] Remark: If π were not finite-dimensional, the character χπ of π could not be defined as a pointwise-evaluatable function, but only as some more general sort of entity.