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We present the results of a thorough study of the nucleus of the Sgr dwarfspheroidal galaxy and of the bright globular cluster M54 that resides withinthe same nucleus (Sgr,N). We have obtained accurate radial velocities andmetallicity estimates for 1152 candidate Red Giant Branch stars of Sgr and M54lying within ~ 9 arcmin from the center of the galaxy, from Keck/DEIMOS andVLT/FLAMES spectra of the infrared Calcium II triplet. Using both velocity andmetallicity information we selected two samples of 425 and 321 very-likelymembers of M54 and of Sgr,N, respectively. The two considered systems displaysignificantly different velocity dispersion profiles: M54 has a steeplydecreasing profile from r=0, where sigma= 14.2 km/s, to r=3.5 arcmin where itreaches sigma=5.3 km/s, then it appears to rise again to sigma= 10 km/s at r=7arcmin. In contrast Sgr,N has a uniformly flat profile at sigma=9.6 km/s overthe whole 0 < r < 9 arcmin range. Using data from the literature we show thatthe velocity dispersion of Sgr remains constant at least out to r ~ 100 arcminand there is no sign of the transition between the outerflat-luminosity-profile core and the inner nucleus in the velocity profile.These results - together with a re-analysis of the surface brightness profileof Sgr,N and a suite of dedicated N-body simulations - provide very strongsupport for the hypothesis that the nucleus of Sgr formed independently of M54,which probably plunged to its present position, coincident with Sgr,N, becauseof significant decay of the original orbit due to dynamical friction.

The nucleus of the Sagittarius dSph galaxy and M54: a window on the process of galaxy nucleation