  We analyse a n-dimensional Generalized Uncertainty Principle (GUP)
quantization framework, characterized by a non-commutative nature of the
configurational variables. First, we identify a set of states which are
maximally localized only along a single direction, at the expense of being less
localized in all the other ones. Subsequently, in order to recover information
about localization on the whole configuration space, we use the only state of
the theory which exhibits maximal localization simultaneously in every
direction to construct a satisfactory quasi-position representation, by virtue
of a suitable translational operator. The resultant quantum framework is then
applied to model the dynamics of the Bianchi I cosmology. The corresponding
Wheeler-DeWitt equation is reduced to Schr\"odinger dynamics for the two
anisotropy degrees of freedom, using a WKB representation for the volume-like
variable of the Universe, in accordance with the Vilenkin scenario. The main
result of our cosmological implementation of the constructed quantum theory
demonstrates how the dynamics of a wave packet peaked at some point in the
configuration space represented in the quasi-position variables, favours as the
most probable configuration exactly the initial one for a relatively long time,
if compared with the ordinary quantum theory. This preference arises from the
distinct behavioral dynamics exhibited by wave packets in the two quantum
theories.
