  We conjecture the equality of the numerical and Kodaira dimensions
$\nu_1^*(X)$ and $\kappa_1^*(X)$ for the cotangent bundle of compact K\"ahler
manifolds $X$, generalising the classical case of the canonical bundle. We show
or reduce it to the classical case of the canonical bundle for some peculiar
manifolds: among them, the rationally connected ones, or resolutions of
varieties with klt singularities and trivial first Chern class, in which case
we show that $\nu_1^*(X)=\kappa_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the
maximal irregularity of a finite \'etale cover of $X$. The proof rests on the
Beauville-Bogomolov decomposition, and a direct computation for smooth models
of quotients $A/G$ of complex tori by finite groups. We conjecture that these
equalities hold true, much more generally, when $X$ is `special'. The invariant
$\kappa_1^*$ was already introduced and studied by Fumio Sakai in [43], the
particular case of the preceding conjecture when $\kappa_1^*(X)=-dim(X)$ was
introduced and studied in [29].
