§4 · Agent 4 · Close fields · 2026-10-18T02:43:11Z
Lemma 4.2 · Lemma: truncated rings agree and valuation conditions are fixed mod \(\varpi^D\)
proved audit: minor defect
Collects the tools of every later step: the truncated rings of \(\mathbb F_q(\!(t)\!)\) and of a ramified \(p\)-adic field agree; Cartan and Iwasawa double cosets of \(G(E)\) are read off from valuations of matrix coefficients in the fundamental Weyl modules; and any count or existence statement built from such valuation conditions on points of \(G(\mathcal O)\), \(H(\mathcal O)\) is the same over \(D\)-close fields.
Lemma 4.2.
Let \(V_i\) (\(i=1,\dots,6\)) be the Weyl module of highest weight \(\omega_i\) over \(\Lambda\) (free, with a basis of \(T\)-weight vectors, \(G_\Lambda\)-stable), \(\rho_i\) the action, \(\ell_i\) the coordinate of the lowest weight vector (weight \(w_0\omega_i\)). For a matrix \(A\) over \(E\) in such a basis put \(\operatorname{val}(A)=\min\) of the valuations of its entries. For \(c,d\in\Lambda_T\otimes\mathbb Q\) write \(c\preceq d\) if \(\langle d-c,\omega_j\rangle\ge0\) for all \(j\).
(a) If \(F'\supseteq\mathbb Q_q\) has residue field \(\mathbb F_q\) and ramification index \(e\ge m\), there is a \(\Lambda\)-algebra isomorphism \(\mathcal O'/\varpi'^m\cong\mathbb F_q[t]/(t^m)\) with \(\bar\varpi'\mapsto\bar t\). So \(F'_e\) and \(F\) are \(e\)-close.
(b) For a smooth affine \(\Lambda\)-scheme \(Y\) (here \(G\), \(H\), \(X\)), \(Y(\mathcal O_E)\to Y(\mathcal O_E/\varpi^m)\) is onto, and \(Y(\mathcal O_E/\varpi^m)\) depends only on the \(\Lambda\)-algebra \(\mathcal O_E/\varpi^m\); \(K/K(m)=G(\mathcal O_E/\varpi^m)\).
(c) (Cartan detection.) For \(g\in G(E)\) and \(\nu\in\Lambda_T^+\): \(g\in Kt^\nu K\iff\operatorname{val}\rho_i(g)=\langle\nu,w_0\omega_i\rangle\) for \(i=1,\dots,6\). In particular \(g\in K\iff\operatorname{val}\rho_i(g)=0\) for all \(i\). If \(g\in Kt^aK\), \(g'\in Kt^bK\), \(gg'\in Kt^cK\) (\(a,b,c\) dominant) then \(c\preceq a+b\); the set \(\{c\in\Lambda_T^+:c\preceq d\}\) is finite.
(d) (Iwasawa detection.) For \(g\in G(E)\) and \(\nu\in\Lambda_T\): \(g\in U(E)t^\nu K\iff\operatorname{val}(\ell_i\rho_i(g))=\langle\nu,w_0\omega_i\rangle\) for all \(i\).
(e) (Finite determinacy.) Let \(\mathcal C\) be a finite Boolean combination of conditions \(\operatorname{val}M(z)\ge c\), \(\operatorname{val}M(z)=c\) on \(z\in\prod_jY_j(\mathcal O_E)\), \(Y_j\in\{G,H\}\), where the entries of \(M(z)\) are polynomials over \(\Lambda\) in \(\varpi^{\pm1}\) and in the matrix coefficients of \(\rho_i(z_j^{\pm1})\), \(\rho_i(\theta(z_j)^{\pm1})\). There is \(D=D(\mathcal C)\), independent of \(E\), such that \(\mathcal C(z)\) depends only on \(\bar z\in\prod_jY_j(\mathcal O_E/\varpi^D)\), and for \(D\)-close \(E,E'\) the subsets of \(\prod_jY_j(\mathcal O/\varpi^D)\cong\prod_jY_j(\mathcal O'/\varpi'^D)\) cut out by \(\mathcal C\) correspond. Hence counts of solutions, existence statements over \(Y_j(\mathcal O)\), and equivalence relations defined in this way agree over \(D\)-close fields.
Proof.
- (a) \(p\in\varpi'^e\mathcal O'\subseteq\varpi'^m\mathcal O'\), so \(R=\mathcal O'/\varpi'^m\) is a local Artinian \(\mathbb F_p\)-algebra with residue field \(\mathbb F_q\). The map \(\Lambda\to R\) factors through \(\Lambda/p=\mathbb F_q\); its image is a coefficient field (the Teichmüller representatives). The \(\mathbb F_q\)-algebra map \(\mathbb F_q[T]\to R\), \(T\mapsto\bar\varpi'\), is onto (every element of \(\mathcal O'\) is \(\sum_i[a_i]\varpi'^i\)) and kills \(T^m\); both sides have length \(m\), so it is an isomorphism, compatible with \(\Lambda\to\mathbb F_q\). This is the close-rings step of Lemma 1.7 (A1-7) as audited in A4-17.
- (b) Hensel's lemma for smooth schemes over a complete discrete valuation ring (EGA IV\(_4\) 18.5.17); \(Y(\mathcal O_E/\varpi^m)\) is the functor \(Y\) evaluated on that \(\Lambda\)-algebra.
- (c) \(V_i\otimes\mathcal O\) is \(G(\mathcal O)\)-stable, so \(\rho_i(K)\subseteq\mathrm{GL}_n(\mathcal O)\). For \(B,C\in\mathrm{GL}_n(\mathcal O)\), \(\operatorname{val}(BAC)\ge\operatorname{val}(A)\) and \(A=B^{-1}(BAC)C^{-1}\), so equality holds. For dominant \(\nu\), \(\rho_i(t^\nu)\) is diagonal with entries \(\varpi^{\langle\nu,\chi\rangle}\), \(\chi\) running over the weights of \(V_i\); every \(\chi-w_0\omega_i\) is a sum of positive roots, so the minimum is \(\langle\nu,w_0\omega_i\rangle\). By the Cartan decomposition \(G(E)=\bigsqcup_{\nu'\in\Lambda_T^+}Kt^{\nu'}K\) (Bruhat–Tits, valid over every non-archimedean local field; for \(\mathbb F_q(\!(t)\!)\) see also Zhu, (2.1.2)), \(\operatorname{val}\rho_i(g)=\langle\nu',w_0\omega_i\rangle\) for the \(\nu'\) of \(g\). The \(w_0\omega_i=-\omega_{i^*}\) span \(X^*(T)_{\mathbb Q}\), so the six values determine \(\nu'\). Superadditivity \(\operatorname{val}(AB)\ge\operatorname{val}A+\operatorname{val}B\) gives \(\langle c,w_0\omega_i\rangle\ge\langle a+b,w_0\omega_i\rangle\), i.e. \(\langle a+b-c,\omega_{i^*}\rangle\ge0\) for all \(i\). A dominant \(c\) with \(d-c\) in the cone of positive coroots lies in a bounded region, hence in a finite set.
- (d) Positive root groups send each weight space of \(V_i\) into the sum of strictly higher ones, so \(\ell_i\rho_i(u)=\ell_i\) for \(u\in U(E)\); \(\ell_i\rho_i(t^\nu)=\varpi^{\langle\nu,w_0\omega_i\rangle}\ell_i\) (the lowest weight space has rank one); \(\ell_i\rho_i(k)\) is a row of a matrix in \(\mathrm{GL}_n(\mathcal O)\), of valuation \(0\). The Iwasawa decomposition \(G(E)=\bigsqcup_{\nu\in\Lambda_T}U(E)t^\nu K\) and the injectivity of \(\nu\mapsto(\langle\nu,w_0\omega_i\rangle)_i\) on \(\Lambda_T\) finish.
- (e) Let \(N\) be the largest negative exponent of \(\varpi\) in \(M\); then \(\varpi^NM(z)\) has entries in \(\Lambda[\text{coordinates}][\varpi]\). "\(\operatorname{val}M(z)\ge c\)" means \(\varpi^NM(z)\equiv0\bmod\varpi^{N+c}\); "\(=c\)" adds \(\not\equiv0\bmod\varpi^{N+c+1}\). The coordinates of \(\rho_i(z^{-1})=\rho_i(z)^{-1}\) and of \(\rho_i(\theta(z))\) are polynomials over \(\Lambda\) in those of \(z\) (unit determinant; \(\theta\) is defined over \(\Lambda\)). So replacing \(z_j\) by \(z_j'\equiv z_j\bmod\varpi^D\) changes \(\varpi^NM(z)\) by a matrix in \(\varpi^DM_n(\mathcal O)\); \(D:=\max(N+c+1)\) over the finitely many conditions works. A \(\Lambda\)-algebra isomorphism \(\mathcal O/\varpi^D\cong\mathcal O'/\varpi'^D\) with \(\bar\varpi\mapsto\bar\varpi'\) carries these reduced polynomials to each other and, by (b), \(Y_j(\mathcal O/\varpi^D)\) to \(Y_j(\mathcal O'/\varpi'^D)\); quantifiers over \(Y_j(\mathcal O)\) become quantifiers over \(Y_j(\mathcal O/\varpi^D)\) by (b). \(\square\)
Why it suffices. Every later count (orbits in \(Kt^\lambda X(\mathcal O)\), structure constants, Satake coefficients) will be written as a count of solutions of conditions of type (e) in \(G(\mathcal O/\varpi^D)\) and \(H(\mathcal O/\varpi^D)\); membership in \(K\), in a Cartan double coset or in an Iwasawa cell is of that type by (c), (d). No closed embedding of \(X\) is needed for this, unlike Lemma 1.7.
Builds on
- Strategy 4.1 · Strategy: move Theorem 8.0.2 from \(\mathbb Q_q(p^{1/e})\) to \(\mathbb F_q((t))\)
Built on by
- Audit 4.3 · Check: \(\tau\) is a closed immersion over \(W(\mathbb F_q)\), giving (H-int)
- Proposition 4.5 · Proposition: \(K\backslash X(\mathbb F_q((t)))\simeq\Lambda_S^+\) by counting orbits
- Proposition 4.6 · Proposition: all structure constants and Satake coefficients agree over close fields
- Audit 5.1 · Audit of A4-2: truncated rings and finite determinacy
Sources
- satake-converse/nodes/A1-7.html (close rings, cosets) and A4-17.html
- zhu-satake-v2.pdf, (2.1.2) Cartan decomposition and §5.6
- EGA IV_4 18.5.17 (Hensel for smooth schemes; standard, not opened)
- Weyl modules over Z (Jantzen, Representations of Algebraic Groups, II.8; standard, not opened)
- Bruhat–Tits I (Cartan and Iwasawa decompositions over any non-archimedean local field; not opened)