Geometric Multiplicities

Abstract

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.

35 pages