Proposition X3.2 · Support bound and top orbit for the pair model, every dominant \(\lambda\), every \(F\)

1. Top orbit. \(P_\lambda:=t^\lambda P_0t^{-\lambda}\) has entries \(\varpi^{\lambda_r-\lambda_s}(P_0)_{rs}\). In the basis \(e_i,e_{n+1-i}\) of the \(i\)-th plane (\(i\le p\)) it is the block \(\begin{pmatrix}1&0\\ \varpi^{-\varphi_i}&0\end{pmatrix}\), \(\varphi_i=\lambda_i-\lambda_{n+1-i}\ge0\), and it is \(1\) on each middle \(e_j\). The block has rank \(1\) and smallest entry valuation \(-\varphi_i\), so its Smith exponents are \(-\varphi_i,\infty\). A permutation of the basis lies in \(K\), so by Lemma X3.1(b) \(\ell(t^\lambda x_0)=(\varphi_1,\dots,\varphi_p)\). For \(\lambda=(a,0,\dots,0)\) this is Theorem 5.4(ii).

2. Reduction to minors. Let \(x=kt^\lambda y\), \(k\in K\), \(y\in X(\mathcal O)\). Then \(P_x=k\,Y\,k^{-1}\) with \(Y:=t^\lambda P_yt^{-\lambda}\) and \(P_y\in M_n(\mathcal O)\) by Lemma X3.1(a), so \(\delta_j(P_x)=\delta_j(Y)\) by Cauchy–Binet.

3. Bounding the minors. For \(|I|=|J|=j\), \(\det Y_{I,J}=\varpi^{\sum_{i\in I}\lambda_i-\sum_{s\in J}\lambda_s}\det(P_y)_{I,J}\), so

$$\operatorname{val}\det Y_{I,J}\ \ge\ (\lambda_n+\dots+\lambda_{n+1-j})-(\lambda_1+\dots+\lambda_j)\ =\ -\sum_{i\le j}\varphi_i(\lambda),$$

a sum of \(j\) entries of \(\lambda\) lying between the sum of the \(j\) smallest and the sum of the \(j\) largest. Hence \(\delta_j(P_x)\ge-\sum_{i\le j}\varphi_i\) for \(j\le p\).

4. The bound. By Lemma X3.1(b), for \(k\le p\),

$$\sum_{i\le k}\ell_i(x)=\max_{0\le j\le k}\big(-\delta_j(P_x)\big)\le\max_{0\le j\le k}\sum_{i\le j}\varphi_i=\sum_{i\le k}\varphi_i,$$

because every \(\varphi_i\ge0\). By Lemma X3.1(c) this is \(\ell(x)\le\varphi(\lambda)\), which is (ii).

5. (iii). Lemma A(2) of the classical page (\(\operatorname{vol}K=1\)): \(T_\lambda\ast\varphi_0\) is \(K\)-invariant with values in \(\mathbb Z_{\ge0}\), support \(Kt^\lambda X(\mathcal O)\), and value \(\ge1\) at \(t^\lambda x_0\). Combine with (i), (ii) and Lemma X3.1(d).

6. Relation to the run. The componentwise bound of Theorem 5.4(iii) is sharper but covers only \(\lambda=(a,0,\dots,0)\); (ii) covers every dominant \(\lambda\), lifts or not. The argument is the transfer of step 3 of Proposition 5.8 with \(\tau\) replaced by the integral embedding \(x\mapsto P_x\) (defined over \(\mathbb Z\)) and the Cartan convolution bound replaced by step 3; \(P_x\notin G(F)\), which is why minors are used. It confirms items 1–2 of Correction 10.4 by an independent derivation of the label formula (Lemma X3.1, step 2).

7. Checks. Exact random checks of (i) and (ii) over \(\mathbb Q_2\), \(\mathbb Q_3\) and \(\mathbb Q_2(\sqrt2)\) (\(e=1,0,2\)) for \((n,p)\in\{(2,1),(3,1),(4,1),(4,2),(5,1),(5,2),(6,3)\}\), 145 points per field, both the boundary case \(\ell=\varphi(\lambda)\) and strict cases occurring (pair_tau.py); the second audit's exhaustive check over \(\mathbb F_2[\![t]\!]\) (pairmodel.py) covers small \(n\) in characteristic 2.