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Abstract: I will introduce a new paradigm for comparing relative trace formulas,
in order to prove instances of (relative) functoriality and relations
between periods of automorphic forms.
More precisely, for a spherical variety X=H\G of rank one, I will prove
that there is an explicit "transfer operator" which transforms the
orbital integrals of the relative trace formula for X x X/G to the
orbital integrals of the Kuznetsov formula for GL(2) or SL(2), equipped
with suitable non-standard test functions. The operator is determined by
the L-value associated to the square of the H-period integral, and the
proof uses a deep theory of Friedrich Knop on the cotangent bundles of
spherical varieties. This is part of an ongoing joint project with
Daniel Johnstone and Rahul Krishna, who are proving instances of the
fundamental lemma. Globally, this transfer will induce an identity of
relative trace formulas and global relative characters, translating to
an Ichino–Ikeda type formula that relates the square of the H-period to
the said L-value.
This can be viewed as part of the program of relative functoriality, a
generalization of the Langlands functoriality conjecture, predicting
relations between the automorphic spectra of two spherical varieties
when there is a map between their dual groups. The case under
consideration here is the simplest non-abelian case of this, when the
dual groups are equal and of rank one. If time permits, I will discuss
how the transfer operator here and in a few examples of higher rank
where it is known is a "deformation" of an abelian transfer operator
obtained by replacing the spherical variety by its asymptotic cone (or
boundary degeneration).