Spherical varieties, functoriality, and quantization
Yiannis Sakellaridis
Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA
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This book chapter is published open access.
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 -variety , one associates a dual group and an -value (encoded in a representation of ), which conjecturally describe the local and automorphic spectra of the variety. This sets up a problem of functoriality, for any morphism 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.