Auto/cross correlation of a cosine in the context of an adaptive beamformer
The $90$ degree shift introduces the sine component to the cross-correlation vector $P$ at lag 1 ($\phi_{dx}(1)$).

The block should be considered to be a 'high-level' description, and does not mean that a shift of one sample is equivalent to  a $90$ degree phase shift, or anything like that. Rather it should be thought of as "shift by as many samples as necessary for a phase shift of $90$ degrees".