Audit 5.14 · Audit of A1-5: convergence, uniqueness and rationality of S_χ over F_q((t))

(a) Convergence.

1. Write \(x=b x_0\in\mathring X(F)=x_0B(F)\) (left notation). Then \(|f_i(x)|=|f_i(x_0)|q^{-\langle\lambda(x),\eta_i\rangle}\), up to the sign convention for weights. On the compact set \(\operatorname{supp}\varphi\), boundedness of the \(f_i\) confines \(\lambda(x)\) to a translate of the pointed cone \(\mathcal T'=\operatorname{cone}(\eta_i)^\vee\).
2. The integrand of (3.2) on \(\mathring X(F)\) is \(\varphi\) times an unramified character of \(A_X(F)\) times the \(G\)-invariant \(|\omega_X|\). That measure exists for \(G/G^\theta\) (both groups unimodular) and is a Radon measure on all of \(X(F)\), so \(\sum_\lambda v_\lambda\le\operatorname{vol}(\operatorname{supp}\varphi)<\infty\).
3. \(q^{-\langle\lambda,s\rangle}\) is bounded on a translate of \(\mathcal T'\) exactly when \(s\in\mathcal T'^\vee=\operatorname{cone}(\eta_i)\). Its interior gives \(R\), with locally uniform convergence and holomorphy. Nothing here uses \(\operatorname{char}F=0\).

(b) Generic uniqueness.

4. Filtration. Let \(Z_{\ge d}\) be the union of the \(B\)-orbits of dimension \(\ge d\); it is Zariski open. Each orbit of dimension \(d\) is closed in \(Z_{\ge d}\). Inside \(Y(F)\) the finitely many \(B(F)\)-orbits are open and closed (Lemma 1.4(a),(b)). So adding them one at a time gives open sets \(U_j\) with \(U_{j+1}\smallsetminus U_j=O_{j+1}\) closed in \(U_{j+1}\). The sequences of Bernstein–Zelevinsky (ℓ-spaces) apply.
5. Modular condition. A \((B,\chi)\)-equivariant functional on \(C_c^\infty(B(F)/B_y(F))\) exists only if \(\chi\) satisfies (3.3) on \(B_y(F)\). Since \(\chi\) is unramified, this fixes \(e^\mu(\chi)\) for \(\mu\) in the image \(\Gamma_y\) of \(B_y(F)\) in \(\Lambda_T\). This image has the rational span of \(X_*(M_y^\circ)\): \(B_y(F)\to M_y(F)\) is onto because \(H^1(F,U_y)=1\) (Lemma 1.4 step 3). So \(E_y\) is Zariski closed.
6. \(E_y\neq D'\). If \(E_y=D'\), then every \(e^\mu\) with \(\mu\in\Gamma_y\) is constant on a translate of \(A_X^*\), so \(\Gamma_y\otimes\mathbb Q\subseteq X_*(T^\theta)_{\mathbb Q}\). With \(\dim T_y=\dim T'-\dim(T'^{-\theta_y})^\circ\ge\dim T^\theta\) (maximality of \(\dim S\) over \(\bar F\)), the image of \(T_y\) in \(T\) equals \((T^\theta)^\circ\). Conjugating by \(b\in B\) with \(T'=bTb^{-1}\) gives an involution \(\theta''=\operatorname{Int}(b^{-1})\theta_y\operatorname{Int}(b)\) of \(G\) that stabilizes \(T\) and has the same \(+1\)-eigenspace on \(X_*(T)_{\mathbb Q}\) as \(\theta\). By the refinement in the statement, \(\theta''\beta=\theta\beta\) for all roots \(\beta\). Hence \(B\cap\theta''B=T\cdot(U\cap L)\) (for the \(\theta\)-adapted \(B\), \(\beta>0\) and \(\theta\beta>0\) force \(\beta\in\Phi_L\)). So \(B_{b^{-1}y}=T^{\theta''}\ltimes(U\cap L)^{\theta''}\) has dimension \(\le\dim T^\theta+|\Phi_L^+|=\dim B_{x_0}\). Then \(By\) is open, i.e. \(y\in\mathring X\): a contradiction.
7. Induction along the filtration kills any equivariant functional vanishing on \(C_c^\infty(\mathring X(F))\) for \(\chi\in D_1'\). On \(\mathring X(F)=B(F)/B_{x_0}(F)\) the equivariant functionals form at most a line.

(c) Rationality.

8. \(V=C_c^\infty(X(F))\) has countable dimension. Equivariance under \(B(F)\) follows from equivariance under \(B(\mathcal O)\) and under a set of coset representatives of \(B(F)/B(\mathcal O)\), and these are countably many linear conditions with coefficients polynomial in \(\chi\). The normalization \(\ell(\mathbf 1_{x_0J_0})=\operatorname{vol}\) is attained by the integral once \(J_0\) is so small that the unramified character is \(1\) on \(x_0J_0\). It is non-degenerate on the line of (b). On the analytic open set \(R\cap D_1'\) the system has exactly one solution. Bernstein's principle (as formulated by Banks, 1998; cited, not opened; pure linear algebra) gives a rational solution, unique on a dense Zariski open \(D_0'\). No characteristic-\(0\) input appears.
9. Tacit data. The data of Lemma 1.3 (a \(\theta\)-stable split \(T\subseteq B\) containing a maximal \(\theta\)-split \(S\), \(B\) adapted, \(x_0=eH\in\mathring X\)) are taken over \(F\). For \(\mathcal M_\theta\) they are explicit (S4-11). For general \(X\) they follow from (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7). with a short argument (a \(\theta_{x_0}\)-stable maximal \(F\)-torus of \(B\cap\theta_{x_0}B\)) that the node does not give; see Objection 5.20.