# Geometric multiplicities (Arkady Berenstein, University of Oregon)

25.01.2022 15:30 – 16:30

The goal of my talk (based on a joint paper with Yanpeng Li) is to introduce geometric multiplicities, which are positive varieties with potential fibered over the Cartan subgroup of a split reductive group G. They form a (unitless) monoidal category Mult_G and we construct a monoidal functor from Mult_G to the representation category 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 obtained jointly with Andrei Zelevinsky in 2001 and generalize them in several directions. In the case when our geometric multiplicity X is an algebra in Mult_G (hence, the corresponding G^\vee-module is an algebra as well), we expect that the spectrum of the latter algebra is an 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.

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### Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Séminaire "Groupes de Lie et espaces de modules"

### Organisé par

Section de mathématiques