Let G be a real semisimple Lie group with an irreducible unitary representation π. The non-temperedness of π is measured by the parameter p(π), which is defined as the infimum of p≥2 such that π has matrix coefficients in Lp(G). Sarnak and Xue conjectured that for any arithmetic lattice Γ⊂G and principal congruence subgroup Γ(q)⊂Γ, the multiplicity of π in L2(G/Γ(q)) is at most O(V(q)2/p(π)+ε) where V(q) is the covolume of Γ(q). In some contexts such estimate is a decent substitute for the Ramanujan conjecture. For G of real rank 1 Sarnak and Xue translate the estimate into a Diophantine counting problem which they managed to solve for SL2(ℝ) and SL2(ℂ).
In this talk I will explain how one can get the same multiplicity bounds for families of pairwise non-commensurable lattices in G=SL2(ℝ), SL2(ℂ) given as unit groups of maximal orders of quaternion algebras over number fields (“horizontal families”). Namely: m(π,Γ)≪V2/p(π)+ε, where V is the covolume of Γ. I will also discuss similar bounds on multiplicities of representations π1×π2 of G=SL2(ℝ)2 where π1 is fixed non-tempered but π2 is allowed to vary together with the lattice.
This talk is based on joint work with Gergely Harcos, Peter Maga and Djordje Milicevic.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
