14 - Generalities on representations of G

Summary

We first review some notions and facts about continuous representations of

in a locally complete topological vector space V. In fact, all we need is the representation by right translations of G in C^∞(G), L²(G), C^∞(H\G), or L₂(H\G) (H closed unimodular subgroup, mainly F). Much of this has been encountered earlier, implicitly or explicitly. All this is valid in a much more general framework (see e.g. [7]).

A continuous representation (π, V) of G into V is a homomorphism of G into the group of automorphisms of V that is continuous; in other words, the map (g, v) ↦ π(g).v is a continuous map of G × V into V. If V is a Hilbert space then it is said to be i>unitary if π (g) (g ∈ G) leaves the scalar product of V invariant. Then the operator norm ∥π(g)∥ is uniformly bounded (by 1), and it is known that continuity already follows from separate continuity:

(1) for every v ∈ V, the map g ↦ π(g).v of G into V is continuous (see e.g. [7, §3] or [10, §VIII.1]).

The assumption “locally complete” is made to ensure that if α ∈ C_c(G) then ∫_G α(x)π(x).v dx converges to an element of V. More generally, ∫ µ(x)π(x).v converges if µ is a compactly supported measure on G (see an important example in 14.4).