  In this paper, we introduce geometric multiplicities, which are positive
varieties with potential fibered over the Cartan subgroup $H$ of a reductive
group $G$. They form a monoidal category and we construct a monoidal functor
from this category to the representations of the Langlands dual group $G^\vee$
of $G$. Using this, we explicitly compute various multiplicities in
$G^\vee$-modules in many ways. In particular, we recover the formulas for
tensor product multiplicities of Berenstein- Zelevinsky and generalize them in
several directions. In the case when our geometric multiplicity $X$ is a
monoid, i.e., the corresponding $G^\vee$ module is an algebra, we expect that
in many cases, the spectrum of this algebra is affine $G^\vee$-variety
$X^\vee$, and thus the correspondence $X\mapsto X^\vee$ has a flavor of both
the Langlands duality and mirror symmetry.
