  Using the spectral theory of unitary operators and the theory of orthogonal
polynomials on the unit circle, we propose a simple matrix model for the
following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)}
\prod_{1\leq k<l\leq n}|
e^{\ii\theta_k}-e^{\ii\theta_l}|^\beta\prod_{j=1}^{n}(1-e^{-\ii\theta_j})^{\delta}
(1-e^{\ii\theta_j})^{\overline{\delta}} $$ with $\Re \delta > -1/2$. If $e$ is
a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of
the pair $(U,e)$ is well parameterized by its Verblunsky coefficients
$(\alpha_0, ..., \alpha_{n-1})$. We introduce here a deformation $(\gamma_0,
>..., \gamma_{n-1})$ of these coefficients so that the associated Hessenberg
matrix (called GGT) can be decomposed into a product $r(\gamma_0)...
r(\gamma_{n-1})$ of elementary reflections parameterized by these coefficients.
If $\gamma_0, ..., \gamma_{n-1}$ are independent random variables with some
remarkable distributions, then the eigenvalues of the GGT matrix follow the
circular Jacobi distribution above.
  These deformed Verblunsky coefficients also allow to prove that, in the
regime $\delta = \delta(n)$ with $\delta(n)/n \to \dd$, the spectral measure
and the empirical spectral distribution weakly converge to an explicit
nontrivial probability measure supported by an arc of the unit circle. We also
prove the large deviations for the empirical spectral distribution.
