Exercise 16, § 2

Let $G =\mathbf{S}\mathbf{L}(2,\mathbf{R})$ and let $\mathbf{H}$ denote the open subset of $\mathbf{C}$ consisting of the complex numbers $z$ such that $\mathscr{I}(z)>0$. Let $\mu$ denote the Lebesgue measure on $\mathbf{C}$. For every element

of G and every $z\in \mathbf{H}$, put $j(g, z) =cz+d$.

a) The group G acts continuously on the left on $\mathbf{H}$ by

The measure $\nu =y^{-2}\cdot \mu$ is G-invariant.

b) Let $n$ be an integer such that $n\geqslant 2$. Let $E_n$ denote the vector space of holomorphic functions $f:\mathbf{H}\rightarrow \mathbf{C}$ such that the function $z\mapsto |f(z)|^2\mathscr{I}(z)^n$ belongs to $\mathscr{L}^2(\mathbf{H}, \nu )$. The space $E_n$ endowed with the sesquilinear form

is a Hilbert space of countable type and of infinite dimension.

c) For every $g\in G$ and every $f\in E_n$, put

The mapping $\pi_n$ defines a unitary representation of G in $E_n$.

d) The function $f_n$ defined by $f_n(z) = (z+i)^{-n}$ belongs to $E_n$; it satisfies $\pi_n(k)f_n=\chi_{-n}(k)f_n$ for every $k\in \mathbf{S}\mathbf{O}(\mathbf{R}^2)$, where $\chi_n$ is the character such that

$(($ cos($\theta$ ) sin($\theta$ )))$_{in\theta}$

$-$ sin($\theta$ ) cos($\theta$ )

for every $\theta \in \mathbf{R}$.

e) Let F be a subrepresentation of $E_n$. The orthogonal projector of $E_n$ whose image is the $\chi_{-n}$ isotypical component of the restriction of $\pi_n$ to $\mathbf{S}\mathbf{O}(\mathbf{R}^2)$ is given by $f\mapsto -(2i)^nf(i)f_n$. (Use Cor. 2 of V, p. 465 and the Cauchy integral formula, FVC, to appear.)

f) The representation $\pi_n$ is irreducible.

g) The matrix coefficient $g\mapsto \langle f_n|\pi_n(g)f_n\rangle$ is nonzero and belongs to $\mathscr{L}^2(G)$.

h) The representation $\pi_n$ is square-integrable and there exists a constant $c >0$ such that the formal degree of $\pi_n$ is $c(n-1)$.