It is known that the characteristic function $\theta_T$ of a
homogeneous contraction $T$ with an associated representation $\pi$ is of
the form
$$\theta_T(a) = \sigma_{L}(\phi_a)^{*} \theta(0) \sigma_{R}(\phi_a),$$
where, $\sigma_{L}$ and $\sigma_{R}$ are projective representation of the
M\"{o}bius group M\"{o}b with a common multiplier. We give another proof
of the ``product formula''.
Also, we prove that the projective representations $\sigma_L$ and
$\sigma_R$ for a class of multiplication operators, the two
representations $\sigma_{R}$ and $\sigma_{L}$ are unitarily equivalent to
certain known pair of representations $\sigma_{\lambda + 1}$ and
$\sigma_{\lambda - 1},$ respectively. These are described explicitly.
Let $G$ be either (i) the direct product of $n$-copies of the
bi-holomorphic automorphism group of the disc or (ii) the bi-holomorphic
automorphism group of the polydisc $\mathbb D^n.$
A commuting tuple of bounded operators $\mathsf{T} = (T_1, T_2,\ldots
,T_n)$ is said to be $G$-homogeneous if the joint spectrum of $\mathsf{T}$
lies in $\overbar{\mathbb{D}}^n$ and $\varphi(\mathsf{T}),$ defined using
the usual functional calculus, is unitarily equivalent with $\mathsf{T}$
for all $\varphi \in G.$
We show that a commuting tuple $\mathsf{T}$ in the Cowen-Douglas class of
rank $1$ is $G$ - homogeneous if and only if it is unitarily equivalent
to the tuple of the multiplication operators on either the reproducing
kernel Hilbert space with reproducing kernel $\prod_{i = 1}^{n}
\frac{1}{(1 - z_{i}\overline{w}_{i})^{\lambda_i}}$ or $\prod_{i = 1}^{n}
\frac{1}{(1 - z_{i}\overline{w}_{i})^{\lambda}},$ where $\lambda,$
$\lambda_i$, $1 \leq i \leq n,$ are positive real numbers, according as
$G$ is as in (i) or (ii).
Let $\mathsf T:=(T_1, \ldots ,T_{n-1})$ be a $G$-homogeneous $(n-1)$-tuple
of rank $1$ Cowen-Douglas class, where $G$ is the the direct product of
$n-1$-copies of the bi-holomorphic automorphism group of the disc. Let
$\hat{T}$ be an irreducible homogeneous (with respect to the
bi-holomorphic group of automorphisms of the disc) operator in the
Cowen-Douglas class on the disc of rank $2$. We show that every
irreducible $G$ - homogeneous operator, $G$ as in (i), of rank $2$ must be
of the form $$(T_1\otimes I_{\widehat{H}},\ldots , T_{n-1}\otimes
I_{\widehat{H}}, I_H \otimes \hat{T}).$$
We also show that if $G$ is chosen to be the group as in (ii), then there
are no irreducible $G$- homogeneous operators of rank $2