Exercise 6, § 2

Integration, VIII, p. 60

Let $G$ be a locally compact group, $\beta$ a relatively invariant, nonzero positive measure on $G$ , $\chi$ and $\chi'$ its left and right multipliers. For $f \in L^p_C(G, \beta)$ and $s \in G$ , set

(U(s)f)(x) = \chi(s)^{-1/p} f(s^{-1}x)

(V(s)f)(x) = \chi'(s)^{-1/p} f(xs)

(Sf)(x) = (\chi \chi')(x)^{-1/p} \overline{f(x^{-1})}.

Then

U

and

V

are linear representations of

G

, and

S^2 = 1,\quad \|U(s)\| = \|V(s)\| = 1

U(s)V(t) = V(t)U(s),\quad SU(s)S = V(s)

for all

s, t

G

EXERCISES

INT VIII.61

Solution

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