Exercise 2, § 2

Let H be a closed subgroup of G. Let $\nu$ be a quasi-invariant measure on $G/H$ (INT, VII, p. 56, § 2, n$^o5$, th. 2), and let $\varrho$ be the function on G with values in $\mathbf{R}^*_+$ associated with it. The mappings $\varrho (g)$ defined by

define a continuous linear representation of G on $\mathscr{L}^2(G/H, \nu )$, and define by passing to the quotient a unitary representation of G on $L^2(G/H, \nu )$, called the quasi-regular representation of $G/H$.