  We compute analytically the probability density function (pdf) of the largest
eigenvalue $\lambda_{\max}$ in rotationally invariant Cauchy ensembles of
$N\times N$ matrices. We consider unitary ($\beta = 2$), orthogonal ($\beta
=1$) and symplectic ($\beta=4$) ensembles of such heavy-tailed random matrices.
We show that a central non-Gaussian regime for $\lambda_{\max} \sim
\mathcal{O}(N)$ is flanked by large deviation tails on both sides which we
compute here exactly for any value of $\beta$. By matching these tails with the
central regime, we obtain the exact leading asymptotic behaviors of the pdf in
the central regime, which generalizes the Tracy-Widom distribution known for
Gaussian ensembles, both at small and large arguments and for any $\beta$. Our
analytical results are confirmed by numerical simulations.
