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 $Xmapsto X^vee$ has a flavor of both the Langlands duality and mirror symmetry.
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