Spherical varieties, functoriality, and quantization

Abstract

YIANNIS SAKELLARIDIS A. 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.

Exotic Monoidal Structures and Abstractly Automorphic Representations for

“…The resulting L-functions admit analytic continuation, satisfy a functional equation and match up with those produced from Galois representations (where the procedure to produce them is much more straightforward). Much work has been done in order to systematize this kind of procedure via the use of spherical varieties (see, for example, [12] and [13]).…”

The positivity of local equivariant Hirzebruch class for toric varieties

“…Classical examples include the variety of complete collineations [104,108,106], which compactifies PGL n , and the variety of complete quadrics [41,106,107], which compactifies SL n /SO n . The renewed interest in spherical varieties in recent years [99,15] comes from their connections to mirror symmetry.…”

Some perspectives on Eisenstein series

“…In his overview[43], Langlands was highly critical of the Rankin-Selberg method for its perceived lack of structure. Perhaps the situation is changing now through the works of Yiannis Sakellaridis and others[64].…”