  Asteroseismology with the space-borne missions CoRoT and Kepler provides a
powerful mean of testing the modeling of transport processes in stars.
Rotational splittings are currently measured for a large number of red giant
stars and can provide stringent constraints on the rotation profiles. The aim
of this paper is to obtain a theoretical framework for understanding the
properties of the observed rotational splittings of red giant stars with slowly
rotating cores. This allows us to establish appropriate seismic diagnostics for
rotation of these evolved stars. Rotational splittings for stochastically
excited dipolar modes are computed adopting a first-order perturbative approach
for two $1.3 M_\odot$ benchmark models assuming slowly rotating cores. For red
giant stars with slowly rotating cores, we show that the variation of the
rotational splittings of $\ell=1$ modes with frequency depends only on the
large frequency separation, the g-mode period spacing, and the ratio of the
average envelope to core rotation rates (${\cal R}$). This leds us to propose a
way to infer directly ${\cal R}$ from the observations. This method is
validated using the Kepler red giant star KIC 5356201. Finally, we provide a
theoretical support for the use of a Lorentzian profile to measure the observed
splittings for red giant stars.
