Arthur's Simple Trace Formula

Question

In Deligne–Kazhdan–Vigneras's "Représentations des groupes réductifs sur un corps local," they use the Simple Trace Formula to prove cases of the local Jacquet–Langlands correspondence over nonarchimedean fields.

Let's recall some setup for the Simple Trace Formula. Let $F$ be a global field, let $G$ be a connected reductive group over $F$, write $Z$ for the center of $G$, and let $\omega$ be a unitary character of $Z(F)\backslash Z(\mathbb{A})$. Write $L^2(G,\omega)$ for the space of $L^2$-functions on $G(F)\backslash G(\mathbb{A})$ where $Z(\mathbb{A})$ acts via $\omega$, and write $L^2_0(G,\omega)$ for the subspace of cusp forms.

In what follows, $v$ will range over places of $F$. Let $f=\prod_vf_v$ be a function on $G(\mathbb{A})$ on which $Z(\mathbb{A})$ acts via $\omega^{-1}$ such that every $f_v$ is a smooth function on $G(F_v)$ with compact support mod $Z(F_v)$. Such $f$ naturally yield linear operators $\rho_0(f)$ on $L^2_0(G,\omega)$. Under extra assumptions on $f$, the Simple Trace Formula says that

$$\operatorname{tr}\rho_0(f) = \sum_\gamma\operatorname{vol}(Z(\mathbb{A})G_\gamma(F)\backslash G_\gamma(\mathbb{A}))\int_{G_\gamma(\mathbb{A})\backslash G(\mathbb{A})}dg\,f(g^{-1}\gamma g),$$

where $\gamma$ ranges over elliptic regular conjugacy classes in $Z(F)\backslash G(F)$, and $G_\gamma$ denotes the centralizer of $\gamma$ in $G$.

Usually, the extra assumptions on $f$ are that $f_v$ is supercuspidal at one place $v$ and $f_{v'}$ is supported on the elliptic regular elements of $G(F_{v'})$ at one place $v'$. However, in part e. of the introduction of "Représentations des groupes réductifs sur un corps local," Deligne–Kazhdan–Vigneras say that Arthur announced the Simple Trace Formula also holds when $f_v$ and $f_w$ are supercuspidal at two places $v$ and $w$.

Question: Does the Simple Trace Formula indeed hold in this case? If so, where could I find a proof? Thank you in advance!

Answer 1

Yes, I have not known Deligne, Kazhdan and Vigneras to lie. A sketch of the proof, at least with the key details for GL(2), is given in Lecture V of

• Steve Gelbart, Lectures on the Arthur--Selberg Trace Formula

Added remarks: In that Lecture, Gelbart addresses both two kinds of simple traces formulas. The one you are asking about is essentially Prop 2.1. While Gelbart states the hypotheses in terms of vanishing of hyperbolic orbital integrals at two places, in the proof he explains how this is related to vanishing of unipotent orbital integrals, which is the condition to be a supercusp form at 2 places. As I remember, while technically he doesn't state that being a supercusp form at 2 places suffices, you should be able to work this out from what he does, at least in the case of GL(2). You'll probably need to look at papers of Arthur, Deligne, Kazhdan, Vigneras, etc for details general G. For me personally, reading Rogawski was also useful for understanding these things.