Plancherel measures for coverings of p-adic SL2(F)

Abstract

In these notes we compute the Plancherel measures associated with genuine principal series representations of n-fold covers of p−adic SL 2 . Along the way we also compute a higher dimensional metaplectic analog of Shahidi local coefficients. Our method involves new functional equations utilizing the Tate γ-factor and a metaplectic counterpart. As an application we prove an irreducibility theorem.

On Shahidi local coefficients matrix

“…In the case where gcd(n, p) = 1 it was proven in Theorem 5.1 of [14] that for σ and χ as before, µ n (σ, s) −1 = q e(ψ)−e(χ n ) L ns, χ n L −ns, χ −n L 1 − ns, χ −n L 1 + ns, χ n . (5.14)…”

“…Namely, γ J (s, χ, ψ, a does not appear in the formulas for these covering groups. We note here that under the assumptions above, objects closely related to γ J (s, χ, ψ, a), γ J (s, χ, ψ, a) and τ L (a, b, χ, s, ψ) were computed in [14] for all n. We start by collecting some information arising from the fact that gcd(n, p) = 1.…”

“…In [14] we have studied the n ≡ 0 (mod 4) case under the assumption that gcd(p, n) = 1. In this case F * /F * d is simple enough so we were able to parameterize the genuine characters of the maximal abelian subgroup of H arising from the Lagrangian decompression given in Example 2.4, see the second part of Lemma 1.5 and Section 3.2 in [14]. As a result, an object closely related to an Slcm was computed, see Lemma 4.3 in [14].…”

“…We compute it and use it to study the Plancherel measure. The study contained in this paper is a generalization and extension of the study initiated in [14]. While we do not assume that the residual characteristic of F is relatively prime to n we do assume for most of this paper that n is not divisible by 4.…”

Subconvex equidistribution of cusp forms: Reduction to Eisenstein observables

“…, so the positivity follows from the identity 13) or equivalently, χ ψ (ξ)γ(χ ξ , 1/2, ψ) = 1, for which we refer to [51] (see also [11, App. B, (2d)] and [24]).…”