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$_2$.
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