Exercise 1, § 8

Let $G =\mathbf{S}\mathbf{U}(2,\mathbf{C})$, and let $\sigma$ be the identity representation of G on $\mathbf{C}^2$. a) The irreducible representations of G are the representations $\tau^n=\mathbf{S}^n\sigma$ for $n\geq 0$.

b) Let $e_1, e_2$ be the canonical basis of $\mathbf{C}^2$; show that the coefficients of $\tau^n$ in the basis $(e^i_1e^{n-i}_2)_{0\leq i\leq n}$ are the functions $\tau_{ij}^n$ such that, for $g=(-\alpha \beta )\in G$,

we have $\tau_{ij}^n(g) =\frac{(-1)^i}{j!}\alpha^{i+j-n}\beta^{j-i}P^n_{ij}(|\alpha |^2)$, with $P^n_{ij}(t) =\frac{d^j}{dt^j}[t^{n-i}(1-t)^i]$

(“Jacobi polynomials”).

c) Deduce from b) that the functions $(n+ 1)^{1/2}(\frac{j!(n-j)!}{i!(n-i)!})^{1/2}\tau_{ij}^n$, for $i, j, n$ integers with $0\leq i\leq n, 0\leq j\leq n$, form an orthonormal basis of $L^2(G)$.

d) For $g=(-\alpha \beta )\in G$, put $\alpha =t^{1/2}e^{i(\varphi-\psi)/2},\beta = (1-t)^{1/2}e^{i(\varphi+\psi)/2}$,

with $0\leq t\leq 1, 0\leq \varphi <2\pi ,-2\pi \leq \psi <2\pi$. Show that the normalized Haar measure $dg$ on G is equal to $(8\pi^2)^{-1}dtd\varphi d\psi$.

e) Let $a, b$ in $\frac{1}{2}\mathbf{Z}$ be such that $a-b\in \mathbf{Z}$. Deduce from d) that the polynomials $P^n_{n/2-a,n/2-b}(t)$, for $n$ an integer of the same parity as $2a,n\geq$ max(2$a,2b$), form an orthogonal basis of $L^2([0,1])$ for the measure $t^{-a-b}(1-t)^{a-b}dt$.