  Transition metal dichalcogenide monolayers are highly interesting for
potential valleytronic applications due to the coupling of spin and valley
degrees of freedom and valley-selective excitonic transitions. However,
ultrafast recombination of excitons in these materials poses a natural limit
for applications, so that a transfer of polarization to resident carriers is
highly advantageous. Here, we study the low-temperature spin-valley dynamics in
nominally undoped and n-doped MoSe$_2$ monolayers using time-resolved Kerr
rotation. In the n-doped MoSe$_2$, we find a long-lived component of the Kerr
signal which we attribute to the spin polarization of resident carriers. This
component is absent in the nominally undoped MoSe$_2$. The long-lived spin
polarization is stable under applied in-plane magnetic fields. Spatially
resolved measurements allow us to determine an upper boundary for the electron
spin diffusion constant in MoSe$_2$.
