Why are radial velocities measured at the core and at wings of a line profile? I mean interactive measurement -- comparing the direct and flipped images of the line profiles. What is the difference of these radial velocities (I do not mean values)? For instance, why is it better to measure radial velocities of wings for H alpha emission line for Be stars? Does it make sense to measure RVs on a blue wing and a red wing? Or are radial velocities of a core measured for absorption lines and radial velocities of wings for emission lines?
A line profile may be formed in material that has a range of line of sight velocities. The answer to your question depends on what radial velocity you are trying to measure.
For example, if the line broadening is dominated by macroscopic plasma motions, then the wavelength of any particular point in the line profile corresponds to a different line of sight velocity. The core of the line will correspond to the line of sight velocity to the region with the greatest optical depth. The wings of the line correspond to material which is blushifted or redshifted with respect to that, with correspondingly different line of sight velocities.
If you are trying to measure the line of sight velocity of the star as a whole, then you wouldn't want to include any wings of a line that are caused by inflow or outflow of material.
On the other hand, the core of the line might be corrupted by emission from a structure that doesn't correspond to the photosphere (e.g. chromospheric emission line cores in cool stars) and in which case, maybe finding the average radial velocity assuming that the wings of the line are symmetric might be the way to go.
You mention Be stars. These are usually stars that have some kind of disc. The wings of the line will reflect motion towards and away from the observer. In a Keplerian disc, this motion will be symmetric, so the line wings will be symmetrically place either side of the true velocity of the star. The line core might be produced by infalling material, so would be redshifted with respect to the true stellar velocity.