9 Further statements not yet used by the main theorem
These statements are formalized but not (yet) on the path of the main theorem: the analytic estimates of Thesis Ch. 8 and 10 (the convergence of \(J\) is stated separately above), the Levi decomposition and transport of \(m_{\mathrm{geom}}\) (Thesis §7.2, §9.3), and checks of the normalization of \(\hat j\) on \(\mathfrak {sl}_2\).
9.1 Estimates
Thesis Ch. 8, §10.1.
(Thesis Prop. 8.1.1(i), existence.) There are full isotropic flags of \(W\) and \(V\) whose stabilizer \(\bar P\) satisfies \(H\cap \bar U_{\min }=\{ 1\} \), with \(H(F)\bar P(F)\) open in \(G(F)\) and of Haar-null complement.
Not yet formalized (adapted from [ 5 ] Prop. 6.4.1). “Good” is replaced by its two consequences used in the estimates.
For \(r=0\) there is \(\epsilon {\gt}0\) with \(\int _{H_0(F)/A_{G,H}(F)}\Xi ^{G_0}(h_0)e^{\epsilon \sigma (h_0)}\, dh_0{\lt}\infty \). This differs from Thesis Lemma 8.2.1(i), which integrates over \(H_0(F)/A_{H_0}(F)\).
Not yet formalized (adapted from [ 5 ] Lemma 6.5.1(i)).
There is \(d{\gt}0\) with \(\int _{H(F)/A_{G,H}(F)}\Xi ^G(h)\sigma (h)^{-d}\, dh{\lt}\infty \).
Not yet formalized (adapted from [ 5 ] Lemma 6.5.1(ii)).
For every \(\delta {\gt}0\) there is \(\epsilon {\gt}0\) with \(\int _{H(F)/A_{G,H}(F)}\Xi ^G(h)e^{\epsilon \sigma (h)}(1+|\lambda (h)|)^{-\delta }\, dh{\lt}\infty \).
Not yet formalized (adapted from [ 5 ] Lemma 6.5.1(iii)).
There are \(d{\gt}0\) and a continuous seminorm \(\nu \) on \(\mathcal C(A_G(F)\backslash G(F),\chi ^{-1})\) with \(|K(f,x)|\le \nu (f)\, \Xi ^{H\backslash G}(x)^2\sigma _{H\backslash G}(x)^{d}\).
Not yet formalized (adapted from [ 5 ] Thm. 8.1.1(i)).
9.2 Levi subgroups
For a GGP datum \(\tilde d\) realizing the Levi pair \((\tilde W,\tilde V)\) of a pair of isotropic flags, there is a surjective homomorphism with central kernel from the Levi quotient \(P/U\) onto \(\prod \mathrm{GL}(Z'_{i,W})\times \prod \mathrm{GL}(Z'_{j,V})\times \tilde G(F)\) (Thesis §5.1.2, (9.3.1)–(9.3.3)).
Not yet formalized: it needs the Clifford-algebra description of the Levi subgroup.
For an isomorphism of GGP triples \(e:G\to G'\) and a quasi-character \(\theta '\) of \(G'(F)\), \(m_{\mathrm{geom}}(\theta '\circ e)=m_{\mathrm{geom}}(\theta ')\) (Thesis §7.2, Prop. 12.1.1).
Not yet formalized: needs the transport of \(\mathcal O_x\) and \(\hat j\).
9.3 Normalization checks on \(\mathfrak {sl}_2\)
Sanity checks of the definition of \(\hat j\) for \(\mathrm{GL}_2(F)\) acting on \(\mathfrak {sl}_2(F)\) with the trace form.
The \(\mathrm{GL}_2(F)\)-invariant measure on the regular nilpotent orbit, normalized by the self-dual measure on \(\mathfrak g/\mathfrak g_E\) ( [ 5 ] §1.6, Thesis §2.4), exists.
Not yet formalized: needs the \(p\)-adic inverse function theorem and invariant measures on homogeneous spaces.
For \(X=x\, \mathrm{diag}(1,-1)\), \(x\neq 0\): \(\hat j(\mathcal O_{\mathrm{reg}},X)=2/|2x|\).
Not yet formalized (hand computation, every residue characteristic).
For \(\varepsilon \) a nonsquare and \(x\neq 0\), \(\hat j(\mathcal O_{\mathrm{reg}},x\begin{pmatrix} 0 & 1 \\ \varepsilon & 0 \end{pmatrix})=0\).
Not yet formalized: \(\mu _{\mathcal O_{\mathrm{reg}}}\) is induced from the Borel, so its Fourier transform is induced from the split torus.
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