Abstract
We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p-adic group tend to zero uniformly for every noncentral semisimple element.
Highlights
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The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p-adic group tend to zero uniformly for every noncentral semisimple element
Compared with (1.1), we are averaging over π∞ ranging in the L-packet Π∞(ξ) for technical simplicity in the trace formula; this simplification does not interfere with the new phenomena at the finite prime u that we are concentrating on
Summary
Let F(ξ, σ, KS0) be the multi-set of discrete automorphic representations π, counted with multiplicity m(π) dim(πS0 )KS0 (a number occurring naturally in the limit multiplicity problem), such that π∞ ∈ Π∞(ξ), πS σ, and (πS)KS = 0. We let both ξ and σ vary, which puts discrete series representations at infinite places (grouped in Lpackets) and supercuspidal representations at finite places on an equal footing. Assuming that S0 = ∅, the global root number of π ∈ F(ξ, σ, ∅) depends only on ξ and σ This almost never happens for thicker families arising from limit multiplicity problems where the whole lattice subgroup Γj varies. The relation between formal degree and conductor is not yet established in general, this is a problem closely related to that of the depth preservation in the local Langlands correspondence [75]
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