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

Seismic diagnostics for transport of angular momentum in stars 2. Interpreting observed rotational splittings of slowly-rotating red giant stars