Geometric Multiplicities

TL;DR

This paper introduces geometric multiplicities as positive varieties fibered over the Cartan subgroup, establishing a monoidal category and functor to Langlands dual group representations, leading to new formulas for tensor product multiplicities.

Contribution

It constructs a monoidal functor from geometric multiplicities to dual group representations and generalizes tensor product multiplicity formulas, linking Langlands duality and mirror symmetry.

Findings

Explicit computation of multiplicities in $G^) modules.

Recovery and generalization of Berenstein-Zelevinsky formulas.

Identification of spectrum of algebraic geometric multiplicities with affine $G^) varieties.

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.