  We report the detection of broad absorptions due to Si IV 4088-4116 A in the
Luminous Blue Variable (LBV) AG Carinae during its last hot phase (2001-2003).
Our NLTE spectral analysis, with the radiative transfer code CMFGEN, revealed
the photospheric nature of these lines predicting, however, much narrower and
deeper absorption profiles than observed. Using a recently-developed code to
compute synthetic spectra in 2D geometry allowing for the effects of rotation,
we could match the broad absorptions with a high projected rotational velocity
of 190 +/- 30 km/s on 2001 April. Analysis of spectra obtained on 2002 March
and 2003 January, when the star was cooling, yielded to a projected rotational
velocity of 110 +/- 10 km/s and 85 +/- 10 km/s, respectively. The derived
rotational velocities are proportional to R^-1, as expected from angular
momentum conservation. We discuss the effects of such high rotation on the
spectral analysis of AG Car, and on the wind terminal velocity. Our results
show direct spectroscopic evidence, for the first time, that a LBV may rotate
at a significant fraction of its break-up velocity. Thus, AG Car (and possibly
other LBVs) is indeed close to the Gamma-Omega limit, as predicted by
theoretical studies of LBVs.
