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The Kepler space mission has made it possible to measure the rotationalsplittings of mixed modes in red giants, thereby providing an unprecedentedopportunity to probe the internal rotation of these stars. Asymmetries havebeen detected in the rotational multiplets of several red giants. This isunexpected since all the red giants whose rotation have been measured thus farare found to rotate slowly, and low rotation, in principle, producessymmetrical multiplets. Our aim here is to explain these asymmetries and find away of exploiting them to probe the internal rotation of red giants. We showthat in the cases where asymmetrical multiplets were detected, near-degeneracyeffects are expected to occur, because of the combined effects of rotation andmode mixing. Such effects have not been taken into account so far. By usingboth perturbative and non-perturbative approaches, we show that near-degeneracyeffects produce multiplet asymmetries that are very similar to theobservations. We then propose and validate a method based on the perturbativeapproach to probe the internal rotation of red giants using multipletasymmetries. We successfully apply our method to the asymmetrical $l=2$multiplets of the Kepler young red giant KIC7341231 and obtain preciseestimates of its mean rotation in the core and the envelope. The observedasymmetries are reproduced with a good statistical agreement, which confirmsthat near-degeneracy effects are very likely the cause of the detectedmultiplet asymmetries. We expect near-degeneracy effects to be important for$l=2$ mixed modes all along the red giant branch (RGB). For $l=1$ modes, theseeffects can be neglected only at the base of the RGB. They must therefore betaken into account when interpreting rotational splittings and as shown here,they can bring valuable information about the internal rotation of red giants.

Near-degeneracy effects on the frequencies of rotationally-split mixed modes in red giants