§13 · Agent X3 · Pair model at 2 · 2026-10-17T20:21:16Z

Proposition X3.5 · Only \(\mathbb C^\times\varphi_0\) generates; J-model fails at residue characteristic 2

proved audit: correct ★ key step

Second audit: correct. Re-derived: A≅C_K by commutativity and (SG); the order ≺ refines dominance and is compatible with addition; φ is monotone, Σ_{i≤k}φ(κ)i=S_k+S{n−k}−|κ|, so top coefficients are positive and leading terms add; hence A is a domain, A^×=C^×, and only C^×φ0 generates. I recomputed the n=2 Hecke matrix over Z_2 and Z_3 from lattices: T_1 1_0=2·1_0+1_1, T_1 1_1=(q−1)1_0+1_2, T_1 1_d=q 1_{d−1}+1_{d+1}. So 1_{O_1}=(T_1−2)φ0 and x0^J∈O_1 over Q_2. The Remark is cited and I did not verify it against a source: I confirmed that the commutant of J̄ has dimension p²+(n−p)², but did not check Richardson's criterion in characteristic 2 or the Hensel/Lang step.

For the pair model over every \(F\), \(C_K\) is free of rank one over the image \(A\) of \(\mathcal H_K\), \(A\) is a domain with \(A^\times=\mathbb C^\times\), and a function generates \(C_K\) iff it is a nonzero multiple of \(\varphi_0\); so in residue characteristic 2 (\(\operatorname{char}F=0\)) the involution model's basic function \(\mathbf 1_{Kx_0^J}\) does not generate, and the restriction of the row to the pair model is necessary.

Proposition X3.5.

Notation of Lemma X3.1; \(F\) any non-archimedean local field; pair model; \(A\) the image of \(\mathcal H_K\) in \(\operatorname{End}_{\mathbb C}(C_K)\).

(i) \(a\mapsto a\varphi_0\) is an isomorphism of \(A\)-modules \(A\to C_K\); \(A\) is a commutative domain with \(A^\times=\mathbb C^\times\).

(ii) For \(\phi\in C_K\): \(\mathcal H_K\cdot\phi=C_K\) iff \(\phi\in\mathbb C^\times\varphi_0\). In particular \(\mathbf 1_{O_b}\) generates iff \(b=0\).

(iii) Let \(\operatorname{char}F=0\) and \(e=\operatorname{val}2\ge1\). Then \(\mathcal H_K\cdot\mathbf 1_{Kx_0^J}\ne C_K\), and \(\mathcal H_K\cdot\mathbf 1_{\tau^{-1}(K)}\ne C_K\), where \(\tau^{-1}(K)=\{x:2P_x-1\in M_n(\mathcal O)\}=\bigcup_{b_1\le e}O_b\) is the set of points whose reflection is integral. For \(n=2\): \(C_K=\mathbb C[T_1]\varphi_0\) freely and \(\mathbf 1_{Kx_0^J}=\mathbf 1_{O_e}=P_e(T_1)\varphi_0\) with \(\deg P_e=e\); for \(F=\mathbb Q_2\), \(\mathbf 1_{O_1}=(T_1-2)\varphi_0\).

Remark (cited standard, not re-proved here). For the quotient \(X_J=\mathrm{GL}_n/Z_{\mathrm{GL}_n}(J)\) over \(\mathcal O\), \(Z(J)\) is the unit group of the commutant of \(J\) in \(M_n(\mathcal O)\), free of rank \(p^2+(n-p)^2\) with the same rank on the special fibre, hence smooth with connected fibres; by Hensel and Lang \(X_J(\mathcal O)=Kx_0^J\). So (iii) says that the involution model with this integral structure fails (SG) whenever \(1\le e<\infty\). Its special fibre \(\mathrm{GL}_n/Z(\bar J)\) has a non-reductive stabilizer (the centralizer of the unipotent \(\bar J\)), so it is not affine (Richardson: a homogeneous space of a reductive group is affine only if the stabilizer is reductive); \(X_J\) is a model with bad reduction at \(2\), unlike the pair model, which is smooth and affine over \(\mathcal O\) with reductive stabilizer.

  1. Freeness. \(\mathcal H_K\) is commutative (Gelfand's argument, any \(F\)), and (SG) holds (Theorem X3.4(2)). If \(a\varphi_0=0\) then \(a(h\varphi_0)=h(a\varphi_0)=0\) for all \(h\), so \(a=0\) on \(C_K=\mathcal H_K\varphi_0\). Hence \(a\mapsto a\varphi_0\) is bijective.
  1. A monomial order. On labels put \(b\prec c\) iff \(|b|<|c|\), or \(|b|=|c|\) and \(b<_{\mathrm{lex}}c\). It refines dominance (at the first \(k\) with \(\sum_{i\le k}b_i<\sum_{i\le k}c_i\) and earlier sums equal, \(b_k<c_k\)), is strictly compatible with addition, and \(\{b\preceq c\}\) is finite.
  1. A basis with leading terms. Let \(t_b\) be the image of \(T_{\lambda(b)}\), \(\lambda(b)=(b_1,\dots,b_p,0,\dots,0)\). By Proposition X3.2 (or Theorem 5.4(ii),(iii)) \(t_b\varphi_0=N_b\mathbf 1_{O_b}+\sum_{c\prec b}(\cdots)\mathbf 1_{O_c}\), \(N_b\ge1\). So \(\{t_b\varphi_0\}\) is a basis of \(C_K\), \(\{t_b\}\) a basis of \(A\) (step 1), and for \(a=\sum\alpha_bt_b\ne0\) the \(\prec\)-largest label in the support of \(a\varphi_0\) is \(\mathrm{LT}(a):=\max_\prec\{b:\alpha_b\ne0\}\).
  1. Products. \(T_{\lambda(b)}\ast T_{\lambda(c)}=\sum_\kappa m_\kappa T_\kappa\) with \(m_\kappa=(T_{\lambda(b)}\ast T_{\lambda(c)})(t^\kappa)\in\mathbb Z_{\ge0}\), nonzero only for \(\kappa\le\lambda(b+c)\) (Cartan convolution bound, cited), and \(m_{\lambda(b+c)}\ge1\) because \(t^{\lambda(b+c)}=t^{\lambda(b)}t^{\lambda(c)}\). \(\varphi\) is monotone: \(\sum_{i\le k}\varphi(\kappa)_i=S_k(\kappa)+S_{n-k}(\kappa)-|\kappa|\) (partial sums \(S_j\)), so \(\kappa\le\kappa'\Rightarrow\varphi(\kappa)\le\varphi(\kappa')\). By Proposition X3.2 for every dominant \(\kappa\), lift or not, \(T_\kappa\ast\varphi_0\ge0\) is supported in \(\{\nu\le\varphi(\kappa)\}\subseteq\{\nu\le b+c\}\). Hence \(t_bt_c\varphi_0\) has coefficient \(\ge m_{\lambda(b+c)}N_{b+c}\ge1\) on \(O_{b+c}\) and is otherwise supported in \(\{\nu\prec b+c\}\). (This is where the support bound for all dominant \(\lambda\), not only for lifts, is needed.)
  1. Leading terms add. For \(a=\sum\alpha_bt_b\), \(a'=\sum\alpha'_ct_c\) with leading labels \(b_0,c_0\), the pair \((b_0,c_0)\) is the unique \(\prec\)-maximizer of \(b+c\) over pairs with \(\alpha_b\alpha'_c\ne0\) (step 2), so the coefficient of \(\mathbf 1_{O_{b_0+c_0}}\) in \(aa'\varphi_0\) is \(\alpha_{b_0}\alpha'_{c_0}\cdot(t_{b_0}t_{c_0}\varphi_0)(O_{b_0+c_0})\ne0\), and \(\mathrm{LT}(aa')=\mathrm{LT}(a)+\mathrm{LT}(a')\). So \(A\) is a domain. If \(uu'=1\) then \(\mathrm{LT}(u)=0\), \(u\varphi_0\in\mathbb C\varphi_0\), and \(u\in\mathbb C^\times\).
  1. (ii). \(\phi=a\varphi_0\) generates iff \(\varphi_0=a'a\varphi_0\) for some \(a'\), iff \(a\in A^\times\) (step 1), iff \(\phi\in\mathbb C^\times\varphi_0\).
  1. (iii). \(\ell(x_0^J)=(e,\dots,e)\) (Theorem X3.4, step 6), so \(\mathbf 1_{Kx_0^J}=\mathbf 1_{O_{(e,\dots,e)}}\notin\mathbb C^\times\varphi_0\). By Proposition X3.3(ii), \(\tau(x)\in K\) iff \((\ell(x)-e)^+=0\) iff \(\ell_1(x)\le e\) (the same set for \(\theta_0\) and for \(\operatorname{Ad}J\), since \(\varepsilon_0,J\in K\)); its indicator has \(\prec\)-largest label \((e,\dots,e)\ne0\). Neither is in \(\mathbb C^\times\varphi_0\).
  1. \(n=2\) by hand. On the tree (Proposition J(1), Proposition 1.2) \(T_1\mathbf 1_0=2\cdot\mathbf 1_0+\mathbf 1_1\), \(T_1\mathbf 1_1=(q-1)\mathbf 1_0+\mathbf 1_2\), \(T_1\mathbf 1_d=q\mathbf 1_{d-1}+\mathbf 1_{d+1}\) (\(d\ge2\)), the centre acts trivially and \(\mathcal H_K=\mathbb C[T_1,T_{(1,1)}^{\pm1}]\); so \(C_K\cong\mathbb C[T_1]\) with \(\varphi_0\leftrightarrow1\) and \(\mathbf 1_d\leftrightarrow P_d\), \(\deg P_d=d\). For \(e\ge1\), \(\mathcal H_K\mathbf 1_{O_e}\leftrightarrow(P_e)\ne\mathbb C[T_1]\); over \(\mathbb Q_2\), \(P_1=T_1-2\), and \(P(T_1)\varphi_0\mapsto P(2)\) is an \(\mathcal H_K\)-eigenfunctional vanishing on \(\mathcal H_K\mathbf 1_{O_1}\) but not on \(\varphi_0\).
  1. Scope. This is a statement about integral structures at residue characteristic 2 only: for \(e=0\) the two models coincide (Theorem X3.4(3)). The sheaf side and the trace dictionary of the run live over \(\mathbb C\), \(\overline{\mathbb F}_q\) and \(\mathbb F_q(\!(t)\!)\) with \(q\) odd, where \(e=0\).

Builds on

  • Proposition X3.2 · Support bound and top orbit for the pair model, every dominant \(\lambda\), every \(F\)
  • Theorem X3.4 · Corrected A5-8 for the three families, and (SG) for the pair model without \(\tau\)
  • Proposition X3.3 · Cartan label of \(\tau(x)\) is \(\mu((\ell(x)-e)^+)\), \(e=\operatorname{val}2\)
  • Lemma X3.1 · The pair model as rank-\((n-p)\) projectors; its orbit label is a polar Smith part
  • Theorem 5.4 · \(\GL_n/(\GL_p\times\GL_q)\), pair model: orbits, lifts, a componentwise support bound and \((\mathrm{SG})\) over every \(F\)
  • Proposition 1.2 · \(\SL_2/T\) versus \(\GL_2/T\): one variety, two Hecke algebras

Built on by

  • Correction X3.6 · Correction of A7-24: its two objections stand; its claim that A5-8 holds needs a new proof
  • Summary X3.7 · A7-25, A7-32 re-derived; the \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) row
  • Re-check V3 · Re-check by V3

Sources