5 The multiplicity formula
Thesis Ch. 7. The multiplicity formula is proved by strong induction on \(\dim G\), together with the geometric expansion of the distribution \(J\). Test functions are taken in \(C_c^\infty \) modulo \(A_G(F)\) instead of the Harish-Chandra Schwartz space; every stated theorem of the trace formula uses this space.
5.1 Strongly cuspidal functions and the distribution \(J\)
\(\mathcal{C}_{\mathrm{scusp}}(A_G(F)\backslash G(F),\chi ^{-1})\), \(C_c^\infty \) version: locally constant \(f\) on \(G(F)\) with \(f(ag)=\chi (a)^{-1}f(g)\) for \(a\in A_G(F)\), compactly supported modulo \(A_G(F)\), and strongly cuspidal (\(\int _{U(F)}f(pu)\, du=0\) for every proper parabolic \(P=MU\) and \(p\in P(F)\)) (Thesis Ch. 1, §4.5, §7.1).
(Thesis §7.1, §10.1), for the Weil quotient measure on \(H(F)\backslash G(F)/A_G(F)\).
For every \(d{\gt}0\) there is a continuous seminorm \(\nu _d\) with \(|K(f,x)|\le \nu _d(f)\, \Xi ^{H\backslash G}(x)^2\sigma _{H\backslash G}(x)^{-d}\) for strongly cuspidal \(f\).
Not yet formalized (adapted from [ 5 ] Thm. 8.1.1(ii)).
For every compact \(C\subseteq G(F)\) with nonempty interior defining \(\Xi ^{H\backslash G}\), there is \(d{\gt}0\) with \(\int _{H(F)\backslash G(F)/A_G(F)}\Xi ^{H\backslash G}(x)^2\sigma _{H\backslash G}(x)^{-d}\, dx{\lt}\infty \).
Not yet formalized (adapted from [ 5 ] Prop. 6.7.1(iii)).
\(G(F)\) has a minimal parabolic subgroup \(P_0\) and a compact open subgroup \(K\) with \(G(F)=P_0(F)K\).
Not yet formalized (Bruhat–Tits).
For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\), \(x\mapsto K(f,x)\) is integrable on \(H(F)\backslash G(F)/A_G(F)\) (Thesis Thm. 10.1.1(ii) with Prop. 8.4.1(iii); = Thm. 1.0.3).
\(f\) is Harish-Chandra Schwartz and bi-invariant under a compact open subgroup \(K'\), so \(K(f,\cdot )\) is continuous on \(H(F)\backslash G(F)/A_G(F)\). The bound \(|K(f,x)|\le \nu (f)\, \Xi ^{H\backslash G}(x)^2\sigma (x)^{-d}\) for strongly cuspidal \(f\) and every \(d{\gt}0\), with integrability of \(\Xi ^{H\backslash G}(x)^2\sigma (x)^{-d}\) for \(d\) large, gives integrability.
\(J(f+g)=J(f)+J(g)\) for \(f,g\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) (Thesis §10.1).
Linearity of the integral, given convergence.
For strongly cuspidal \(f\), \(\theta _f\) is the function representing the orbital pairing \(f'\mapsto \int _{G(F)/A_G(F)}\int _{G(F)}f(x^{-1}gx)f'(g)\, dg\, dx\) (Thesis §4.8, [ 5 ] §5.5); \(0\) if there is none. Convention: \(\theta _f(ag)=\chi (a)^{-1}\theta _f(g)\).
Let \(f_0\in C_c^\infty (G(F))\) be strongly cuspidal. Then there is a quasi-character \(\theta \), locally integrable and locally constant on \(G_{\mathrm{reg}}(F)\), with \(\int _{G(F)}\theta f'\, dg=\int _{G(F)/A_G(F)}\int _{G(F)}f_0(x^{-1}gx)f'(g)\, dg\, dx\) for every \(f'\in C_c^\infty (G(F))\), the outer integrand being integrable.
For strongly cuspidal \(f_0\in C_c^\infty (G(F))\), the orbital pairing of its \(\chi \)-projection \(P_\chi f_0\) is represented by a quasi-character (Thesis §4.8, §7.1).
Glue central translates of the quasi-character of a cut-off of \(P_\chi f_0\) along the fibres of the Harish-Chandra map \(H_G\).
For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\), \(\theta _f\) represents the orbital pairing of \(f\) and is a quasi-character (Thesis §4.8, Prop. 4.8.1(ii)).
From the \(\chi \)-transfer and uniqueness on the open conull regular set.
Quasi-characters \(\theta \) on \(G(F)\) with \(\theta (ag)=\chi (a)^{-1}\theta (g)\) for \(a\in A_G(F)\), \(g\in G_{\mathrm{reg}}(F)\) (Thesis §4.1.1, §7.2).
For \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\), \(\theta _f\in \mathrm{QC}(G(F),\chi ^{-1})\) (Thesis Prop. 4.8.1(ii)).
From the previous proposition and central equivariance of \(f\).
If \(\pi \) is irreducible tempered with central character \(\eta \) on \(A_G(F)\) then \(\theta _\pi \in \mathrm{QC}(G(F),\eta )\) (Thesis Ch. 1, §4.4).
Harish-Chandra’s local expansion, and central translation of characters.
5.2 The induction
Thesis §7.2 and §7.4.
\(J(f)=m_{\mathrm{geom}}(\theta _f)\) for every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) (Thesis Thm. 7.2.1 = Thm. 12.2.1 = Thm. 1.0.5).
For every irreducible tempered \(\pi \) of \(G(F)\) trivial on \(A_{G,H}(F)\): \(m(\pi ){\lt}\infty \) and \(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\).
The geometric expansion (for every central character) and the multiplicity formula hold for every GGP triple \((G',H',\xi ')\) with \(\dim G'{\lt}\dim G\) (Thesis §7.2).
\(R_{\mathrm{temp}}(G,\eta )\otimes \mathbb C\): finite formal linear combinations of classes of irreducible tempered representations with central character \(\eta \) on \(A_G(F)\) (Thesis §3.4, §7.4), with the linear extensions \(\theta _\Pi \), \(m(\Pi )\), \(m_{\mathrm{geom}}(\Pi )\).
\(\Pi \mapsto m(\Pi )-m_{\mathrm{geom}}(\Pi )\), extended linearly (Thesis Prop. 7.4.1(ii)). The multiplicity formula for \(\pi \) is \(\mathrm{defect}[\pi ]=0\).
There is an open, closed, invariant, \(A_G(F)\)-stable \(\Omega \subset G(F)\) containing the elliptic regular set and compact modulo conjugation and \(A_G(F)\), such that every quasi-character supported (compactly modulo conjugation) in \(\Omega \) is \(\theta _{f_0}\) on \(G_{\mathrm{reg}}(F)\) for some strongly cuspidal \(f_0\in C_c^\infty (G(F))\) supported in \(\Omega \).
Not verbatim in the literature (adapted from, or derived from, published results); not yet formalized. Ingredients: [ 5 ] , Cor. 5.7.2(i) (= Thesis Cor. 4.8.3(i)), stated there for compact centre; adapted to a nontrivial split centre \(A_G\) by its local proof.
For every virtual tempered \(\Pi \) with central character \(\eta \) there is \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) with \(\theta _f=\theta _\Pi \) on \(G_{\mathrm{ell,reg}}(F)\) (Thesis Cor. 4.8.3(i) in \(\chi \)-equivariant form). This differs from the Thesis, which uses [ 5 ] Cor. 5.7.2(iv) and Schwartz functions.
Cut \(\theta _\Pi \) off near the elliptic set, apply elliptic isolation, and project with \(P_\chi \).
Assume (HYP). Let \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\eta )\) and \(\Pi \) virtual tempered with central character \(\eta \) such that \(\theta _\Pi =\theta _f\) on \(G_{\mathrm{ell,reg}}(F)\). Then \(J(f)-m_{\mathrm{geom}}(\theta _f)=m(\Pi )-m_{\mathrm{geom}}(\Pi )\).
The spectral expansion gives a \(\Pi _f\) with the same elliptic character and \(J(f)-m_{\mathrm{geom}}(\theta _f)=\mathrm{defect}(\Pi _f)\); \(\Pi -\Pi _f\) has vanishing elliptic character, so it lies in \(R_{\mathrm{ind}}\), where the defect vanishes (Chapter 6).
(Thesis Prop. 7.4.1(iii), direction Thm. 7.2.1 \(\Rightarrow \) Thm. 7.2.2.) Under (HYP), if the geometric expansion holds at \(G\) for all central characters, the multiplicity formula holds at \(G\).
Let \(\eta \) be the central character of \(\pi \) (Schur) and choose \(f\) with \(\theta _f=\theta _\pi \) on the elliptic set (elliptic isolation). Then \(0=J(f)-m_{\mathrm{geom}}(\theta _f)=\mathrm{defect}[\pi ]\). Finiteness is multiplicity one.
(Thesis Thm. 12.2.1 under (HYP).) If (HYP) holds for \(G\), then \(J(f)=m_{\mathrm{geom}}(\theta _f)\) for every central character \(\chi \) and every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\).
Choose the invariant clopen \(\Omega \ni 1\) of the expansion near \(1\); cut \(f=f_1+(f-f_1)\) with \(f_1\) supported in \(\Omega \) and \(\theta _{f_1}=1_\Omega \theta _f\); apply the expansion near \(1\) to \(f_1\) and the expansion away from \(1\) to \(f-f_1\), and add, using linearity of \(J\) and \(m_{\mathrm{geom}}\) (Chapter 8).
For every GGP triple, the geometric expansion (for every central character) and the multiplicity formula hold.
Strong induction on \(\dim G\): (HYP) is the induction hypothesis; apply the two preceding results.
\(m(\pi )=m_{\mathrm{geom}}(\theta _\pi )\) (and \(m(\pi ){\lt}\infty \)) for every GGP triple and every tempered \(\pi \) trivial on \(A_{G,H}(F)\) (Thesis Thm. 7.2.2 = Thm. 1.0.2).
The multiplicity-formula half of the induction.