8 The geometric expansion
Thesis §12.2–12.5: the geometric expansion \(J(f)=m_{\mathrm{geom}}(\theta _f)\) under (HYP), split into a neighbourhood of the centre (descent to the Lie algebra) and its complement (semisimple descent).
8.1 Extending \(J\) to quasi-characters
A linear form \(J_{\mathrm{qc}}\) on \(\mathrm{QC}(G,\chi ^{-1})\) that depends only on values on \(G_{\mathrm{reg}}(F)\), satisfies \(J(f)=J_{\mathrm{qc}}(\theta _f)\), is continuous, and \(J_{\mathrm{qc}}-m_{\mathrm{geom}}\) is supported in the elliptic set (Thesis Prop. 12.2.3). This differs from Thesis Prop. 12.2.3, which has \(\mathrm{Supp}(J_{\mathrm{qc}})\subset G(F)_{\mathrm{ell}}\): here \(\Gamma (G,H)\) need not lie in the elliptic set.
Assume (HYP). If \(\theta _f=\theta _{f'}\) on \(G_{\mathrm{ell,reg}}(F)\) then \(J(f)-m_{\mathrm{geom}}(\theta _f)=J(f')-m_{\mathrm{geom}}(\theta _{f'})\) (Thesis Prop. 7.4.1(ii)).
Both are defects of virtual representations with the same elliptic character.
Assume (HYP). Then \(J\) extends to some \(J_{\mathrm{qc}}\) as above (Thesis Prop. 12.2.3).
\(J_{\mathrm{qc}}(\theta )=m_{\mathrm{geom}}(\theta )+(J(f)-m_{\mathrm{geom}}(\theta _f))\) for any \(f\) with \(\theta _f=\theta \) on the elliptic set (elliptic isolation); well defined by the previous lemma.
Assume (HYP). If \(\theta _f=\theta _{f'}\) on \(G_{\mathrm{reg}}(F)\) then \(J(f)=J(f')\) (Thesis Prop. 12.2.3).
\(J=J_{\mathrm{qc}}\circ \theta \).
8.2 Away from the centre
Thesis §12.3.
For an invariant clopen \(\Omega \), \(p\in P(F)\) and \(u\in U(F)\), \(pu\in \Omega \iff p\in \Omega \).
Levi decomposition \(P=MU\) in \(\mathrm{GSpin}\) and a cocharacter of the split centre of \(M\), lifted to \(\mathrm{GSpin}\), contracting \(U\) (Thesis §5.1.2): \(a_kC_k(pu)C_k^{-1}\) and \(a_kC_kpC_k^{-1}\) (\(a_k\in A_G(F)\)) have the same limit, which lies in \(\Omega \) iff \(p\) (or \(pu\)) does, as \(\Omega \) is open and closed.
Assume (HYP). If invariant clopen sets \(\Omega _i\) are each either good (the geometric expansion holds for \(f\) supported there) or contained in the zero set of \(\theta \), and \(\theta _f\) is supported in \(\bigcup _i\Omega _i\) and in the support of \(\theta \), then \(J(f)=m_{\mathrm{geom}}(\theta _f)\) (Thesis Prop. 12.3.1, [ 5 ] §11.6).
Induction on the number of sets, cutting \(f\) by \(\Omega _i\).
For every neighbourhood \(U\) of \(1\) there is an open, closed, invariant, \(A_G(F)\)-stable \(\Omega \) containing every near-central element (\(1\) in the closure of \(A_G(F)c^{G(F)}\)), each element of which is conjugate into \(U\) modulo \(A_G(F)\).
Assume (HYP). Every \(c\) whose semisimple part is not in \(A_G(F)\) has an invariant clopen neighbourhood that either contains no elliptic regular element or on which the geometric expansion holds.
Not yet formalized. Semisimple descent to \(G_x=G'_x\cdot G''_x\): (HYP) for the smaller GGP triple \((G'_x,H'_x,\xi '_x)\) and Arthur’s local trace formula on \(G''_x\) (Thesis (12.3.3), Thm. 4.7.1).
Assume (HYP). If \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) and \(\theta _f\) vanishes near \(1\), then \(J(f)=m_{\mathrm{geom}}(\theta _f)\).
Cover the support of \(\theta _f\) (compact modulo conjugation and \(A_G\)) by finitely many good sets and glue.
8.3 Near the centre: descent to the Lie algebra
Thesis §12.4–12.5 and Ch. 11. Here \(\mathfrak g^0=\mathfrak {so}(W)\oplus \mathfrak {so}(V)\).
For \(f\in \mathcal S_{\mathrm{scusp}}(\mathfrak g^0(F))\), the quasi-character \(\theta _f\) (Thesis §4.8; [ 5 ] §5.5).
\(m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta )=\lim _{s\to 0^+}m^{\mathrm{Lie}}_{\mathrm{geom},s}(\theta )\) (Thesis Prop. 12.1.1(iii)).
(Thesis Prop. 12.4.1(i), Prop. 12.1.1(iv).) If neither \(W\) nor \(V\) is a hyperbolic plane, there is an invariant clopen \(\Omega \ni 1\) such that every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) supported in \(\Omega \) has a strongly cuspidal \(F\) on \(\mathfrak g^0\) with \(J(f)=J^{\mathrm{Lie}}(F)\) and \(m_{\mathrm{geom}}(\theta _f)=m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _F)\).
Not yet formalized. \(\Omega =A_G(F)\exp (\omega )\) for a \(G(F)\)-excellent \(\omega \), \(F=(j_G^H)^{1/2}f_\omega \).
(Thesis Thm. 11.6.1, in normalization-free form.) If neither \(W\) nor \(V\) is a hyperbolic plane, \(J^{\mathrm{Lie}}(f)\) depends only on \(\theta _f|_{\mathfrak g^0_{\mathrm{reg}}}\).
If neither \(W\) nor \(V\) is a hyperbolic plane, \(J^{\mathrm{Lie}}(f)=J^{\mathrm{Lie}}_{\mathrm{qc}}(\theta _f)\) for every strongly cuspidal \(f\) on \(\mathfrak g^0\) (Thesis Prop. 12.4.1(ii)).
From the spectral expansion of \(J^{\mathrm{Lie}}\).
Assume (HYP), and that \(J^{\mathrm{Lie}}\) factors through \(\theta _f\) by a form seeing only \(\mathfrak g^0_{\mathrm{reg}}\). Then \(J^{\mathrm{Lie}}(f)-m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)=\sum _{\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g^0)}c_{\mathcal O}\, c_{\theta _f,\mathcal O}(0)\) for some constants \(c_{\mathcal O}\).
Not yet formalized. The difference is a homogeneous continuous form, vanishing on quasi-characters with \(0\notin \mathrm{Supp}\theta \) by semisimple descent and (HYP) (Prop. 12.4.1(iv)); such a form is a combination of germs at \(0\) ( [ 5 ] Prop. 4.6.1).
Assume (HYP), neither \(W\) nor \(V\) a hyperbolic plane. Then \(J^{\mathrm{Lie}}(f)-m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)=\sum _{\mathcal O\in \mathrm{Nil}_{\mathrm{reg}}(\mathfrak g^0)}c_{\mathcal O}\, c_{\theta _f,\mathcal O}(0)\).
The test-function form with \(J^{\mathrm{Lie}}_{\mathrm{qc}}\).
(Thesis §12.5.) If \(J^{\mathrm{Lie}}(f)-m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)=\sum _{\mathcal O}c_{\mathcal O}c_{\theta _f,\mathcal O}(0)\) for every strongly cuspidal \(f\) on \(\mathfrak g^0\), then every \(c_{\mathcal O}\) vanishes.
Assume (HYP), neither \(W\) nor \(V\) a hyperbolic plane. Then \(J^{\mathrm{Lie}}(f)=m^{\mathrm{Lie}}_{\mathrm{geom}}(\theta _f)\) for strongly cuspidal \(f\) on \(\mathfrak g^0\) (Thesis Thm. 12.2.2).
Prop. 12.5.1 and the endgame.
If \(V\) is a hyperbolic plane (\(\dim W=1\), \(r=0\)), the geometric expansion holds on all of \(G(F)\).
\(G\) is a split torus, \(A_G=G\); then \(J(f)=f(1)\), \(\theta _f=f\) and \(m_{\mathrm{geom}}(\theta _f)=c_{\theta _f,0}(1)=f(1)\), using a self-dual measure for \(\hat j(0,\cdot )=1\).
Assume (HYP) and \(W\) a hyperbolic plane. There is an invariant clopen \(\Omega \ni 1\) on which the geometric expansion holds.
Not yet formalized. Here \(A_{\bar G}=\mathrm{SO}(W)\) is a nontrivial split central torus; the Lie-algebra argument must be run modulo \(A_{\bar G}\), which the cited literature does not state in this form ( [ 5 ] Ch. 10–11 modulo \(A_{\bar G}\)).
(Thesis Prop. 12.4.1 and Thm. 12.2.2.) Assume (HYP). For every \(\chi \) there is an invariant clopen \(\Omega \ni 1\) with \(J(f)=m_{\mathrm{geom}}(\theta _f)\) for every \(f\in \mathcal{C}_{\mathrm{scusp}}(A_G\backslash G,\chi ^{-1})\) supported in \(\Omega \).
Cases: \(V\) a hyperbolic plane; \(W\) a hyperbolic plane; otherwise descend to \(\mathfrak g^0\) and use \(J^{\mathrm{Lie}}=J^{\mathrm{Lie}}_{\mathrm{geom}}\).