  We consider an $n$-fold Brylinski-Deligne cover of a reductive group over a
$p$-adic field. Since the space of Whittaker functionals of an irreducible
genuine representation of such a cover is not one-dimensional, one can consider
a local coefficients matrix arising from an intertwining operator, which is the
natural analogue of the local coefficients in the linear case. In this paper,
we concentrate on genuine principal series and establish some fundamental
properties of such a local coefficients matrix, including the investigation of
its arithmetic invariants. As a consequence, we prove a form of the
Casselman-Shalika formula which could be viewed as a natural analogue for
linear algebraic groups. We also investigate in some depth the behaviour of the
local coefficients matrix with respect to the restriction of genuine principal
series from covers of ${\rm GL}_2$ to ${\rm SL}_2$. In particular, some further
relations are unveiled between local coefficients matrices and gamma factors or
metaplectic-gamma factors.
