Spherical varieties, functoriality, and quantization

2021/11/04 by Sakellaridis, Yiannis
#11F70 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2111.03004

Abstract

We discuss generalizations of the Langlands program, from reductive groups to the local and automorphic spectra of spherical varieties, and to more general representations arising as "quantizations" of suitable Hamiltonian spaces. To a spherical G-variety X, one associates a dual group ^LG_X and an L-value (encoded in a representation of ^LG_X), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism ^LG_X→ ^LG_Y of dual groups. We review, and generalize, Langlands' "beyond endoscopy" approach to this problem. Then, we describe the cotangent bundles of quotient stacks of the relative trace formula, and show that transfer operators of functoriality between relative trace formulas in rank 1 can be interpreted as a change of "geometric quantization" for these cotangent stacks.

Related

• Fundamental Lemma for Rank One Spherical Varieties of Classical Types
• Spherical varieties and Langlands duality
• Local Geometric Langlands Correspondence: the Spherical Case
• Local and global questions "beyond endoscopy"
• Introduction to the relative Langlands program