Lie Groups Seminar
I will 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, one can explicitly compute various multiplicities in $G^\vee$-modules in many ways. In particular, one can 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 Langlands duality and mirror symmetry.