Summation formulae and speculations on period integrals attached to triples of automorphic representations

Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces $(V_i,Q_i)$ of even dimension by the equation

$Q_1(v_1)=Q_2(v_2)=Q_3(v_3)$.

I will sketch the proof of this formula in the first portion of the talk. In the second portion, time permitting, I will discuss how these summation formulae lead to functional equations for period integrals for automorphic representations of

$GL_{n_1} \times GL_{n_2} \times \GL_{n_3}$

where the $n_i$ are arbitrary, and speculate on the relationship between these period integrals and Langlands L functions.