Exercise 18, § 3

Let $G =\mathbf{S}\mathbf{L}(2,\mathbf{R})$. We denote by N and P the subgroups

N = $|t\in \mathbf{R}$, P = $-_1|a\in \mathbf{R}_*, t\in \mathbf{R}$

0 1 $0a$

of G. Let $\pi$ be a unitary representation of G in a complex Hilbert space E. Suppose that there exists $x\in E$ such that $\pi (n)x=x$ for every $n\in N$. a) The homogeneous space $\mathbf{S}\mathbf{L}(2,\mathbf{R})/N$ is identified with $\mathbf{R}^2-\{0\}$ by the mapping $g\mapsto$ $ge_1$ where $e_1$ is the prime vector of the canonical basis of $\mathbf{R}^2$, and the homogeneous space $\mathbf{S}\mathbf{L}(2,\mathbf{R})/P$ is identified with the subspace $(\mathbf{R}-\{0\})\times \{0\}$ of $\mathbf{S}\mathbf{L}(2,\mathbf{R})/N$. b) Every continuous function $f$ on G such that $f(ngn') =f(g)$ for every $(n, n')\in N^2$ satisfies $f(pgp') =f(g)$ for every $(p, p')\in P^2$. (Interpret $f$ as a continuous function $\varphi$ on $\mathbf{R}^2-\{0\}$ invariant under the natural action of N and verify that it is constant on $\mathbf{S}\mathbf{L}(2,R)/P.$)

c) One has $\pi (g)x=x$ for every $x\in G$.