6 The spectral side
Thesis Ch. 9–10 and Prop. 7.4.1: the spectral expansion of \(J\), in the elliptic form used by the induction, and the vanishing of the defect on properly induced representations.
6.1 Parabolic induction
\(i_P^G(\sigma )\): the completion of the smooth \(f:G(F)\to H_\sigma \) with \(f(pg)=\delta _P(p)^{1/2}\sigma (p)f(g)\) for \(\| f\| ^2=\int _K\| f(k)\| ^2dk\), with \(G(F)\) acting by right translation (Thesis §3.2).
\(R_{\mathrm{ind}}(G,\eta )\): the span of the virtual tempered representations whose character on \(G_{\mathrm{reg}}(F)\) is that of some \(i_P^G(\sigma )\), \(P\) proper, \(\sigma \) tempered with central character \(\eta \) (Thesis §7.4).
For \(P\) proper and \(\sigma \) tempered, \(\theta _{i_P^G\sigma }\) vanishes on \(G_{\mathrm{ell,reg}}(F)\).
If a virtual representation \(\Pi \) has the character of \(i_P^G(\sigma )\) on \(G_{\mathrm{reg}}(F)\), then \(m(\Pi )=m(i_P^G\sigma )\).
If \(\Pi \in R_{\mathrm{temp}}(G,\eta )\) has \(\theta _\Pi =0\) on \(G_{\mathrm{ell,reg}}(F)\), then \(\Pi \in R_{\mathrm{ind}}(G,\eta )\).
For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) there is a virtual tempered \(\Pi \) with central character \(\eta \) with \(\theta _\Pi =\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\).
Not verbatim in the literature (adapted from, or derived from, published results); not yet formalized. Ingredients: [ 5 ] , Prop. 5.6.1(ii), Lemma 5.4.2(i) (= Thesis Prop. 4.8.1), adapted to the \(\chi \)-equivariant setting and a finite sum.
6.2 The spectral expansion
Assume (HYP). For \(\pi =i_Q^G(\sigma )\), \(Q\) proper and \(\sigma \) tempered with central character \(\eta \), \(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\).
Not yet formalized. \(m(i_Q^G\sigma )=m(\tilde\sigma )\) for the GGP triple of the Levi (Thesis Cor. 9.5.1); \(m_{\mathrm{geom}}(\theta _{i_Q^G\sigma })=m_{\mathrm{geom}}^{\tilde G}(\theta _{\tilde\sigma })\) (Thesis Lemma 7.3.1, [ 5 ] Prop. 4.8.1(i)); conclude by (HYP).
Assume (HYP). The defect vanishes on \(R_{\mathrm{ind}}(G,\eta )\) (Thesis Prop. 7.4.1(i), extended linearly).
On a generator: \(m(\Pi )=m(i_Q^G\sigma )\), \(m_{\mathrm{geom}}(\Pi )=m_{\mathrm{geom}}(\theta _{i_Q^G\sigma })\), and these agree.
Assume (HYP). If \(\theta _\Pi =0\) on \(G_{\mathrm{ell,reg}}(F)\) then \(m(\Pi )=m_{\mathrm{geom}}(\Pi )\).
Such \(\Pi \) lie in \(R_{\mathrm{ind}}\) (Arthur), where the defect vanishes.
(Thesis Thm. 10.2.1 = Thm. 1.0.4 and Prop. 7.4.1(ii), elliptic form.) Assume (HYP). For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\), \(J(f)-m_{\mathrm{geom}}(\theta _f)=\mathrm{defect}(\Pi _f)\), where \(\Pi _f\) is a virtual tempered representation with \(\theta _{\Pi _f}=\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\).
Not yet formalized. The spectral expansion \(J(f)=\int _{\mathcal X(G,\chi )}D(\pi )\hat\theta _f(\pi )m(\bar\pi )\, d\pi \) (Thesis Thm. 10.2.1, from Thm. 10.1.1, Arthur’s local trace formula, the matricial Paley–Wiener theorem, Thm. 9.2.1 and Cor. 9.5.1); on the induced part \(m=m_{\mathrm{geom}}\) by Prop. 7.4.1(i).
Assume (HYP). For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) there is a virtual tempered \(\Pi _f\) with central character \(\eta \), \(\theta _{\Pi _f}=\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\) and \(J(f)-m_{\mathrm{geom}}(\theta _f)=m(\Pi _f)-m_{\mathrm{geom}}(\Pi _f)\).
Combine the previous theorem with the elliptic virtual character of \(\theta _f\).