Proposition 1.5 · \(S_\chi\) over \(\mathbb F_q((t))\): convergence, generic uniqueness, rationality

This replaces the convergence and rationality of (3.2) and the multiplicity statement of Theorem 3.2.2, all taken from [Sa08], at the level needed for Theorem 8.0.2. The argument is the classical one: orbit filtration plus Bernstein's principle. Its two characteristic-sensitive inputs are supplied by Lemma 1.3 and Lemma 1.4.

Proof of (a).

1. Regular \(B\)-eigenfunctions generate \(F(X)^{(B)}\): the denominators of a \(B\)-eigen rational function form a non-zero \(B\)-stable ideal, which contains a \(B\)-eigenvector by Lie–Kolchin; this holds in every characteristic. Choose \(f_1,\dots,f_r\in F[X]^{(B)}\) whose weights \(\eta_i\) span \(\mathcal X(X)_{\mathbb R}\).
2. Each \(f_i\) is \(U\)-invariant, \(B\)-eigen and nowhere zero on \(X̊\). Write \(a(x)\in A_X(F)\) for the image of \(x\in X̊(F)\) under \(X̊\to X̊/U\cong A_X\), and \(\lambda(x)\in\Lambda_X\) for its class. Then \(|f_i(x)|=|f_i(x_0)|\,q^{-\langle\lambda(x),\eta_i\rangle}\).
3. On the compact set \(\operatorname{supp}\varphi\) we have \(|f_i|\le C_i\). Hence \(\lambda(x)\) lies in \(P_\varphi=\{\lambda:\langle\lambda,\eta_i\rangle\ge-c_i\ \forall i\}\), a translate of the strictly convex cone \(\mathcal T'=\{\lambda:\langle\lambda,\eta_i\rangle\ge0\}\).
4. On \(X̊\), the integrand of (3.2) is \(\varphi\) times an unramified character \(\psi_\chi(a(x))\) times the \(G\)-invariant measure \(|\omega_X|\) (the eigenmeasures differ by unramified characters, §3.2). So \(|S_\chi(\varphi)(1)|\le\|\varphi\|_\infty\sum_{\lambda\in P_\varphi}|\psi_\chi(e^\lambda)|\,v_\lambda\) with \(v_\lambda=\operatorname{vol}(\operatorname{supp}\varphi\cap\{\lambda(x)=\lambda\})\) and \(\sum v_\lambda\le\operatorname{vol}(\operatorname{supp}\varphi)<\infty\).
5. \(|\psi_\chi(e^\lambda)|=q^{-\langle\lambda,s(\chi)\rangle}\) with \(s(\chi)\) affine in \(\operatorname{Re}\chi\). This is bounded on \(P_\varphi\) when \(s(\chi)\) lies in the interior of the dual cone \(\operatorname{cone}(\eta_i)\). That interior is a non-empty open set \(R\), independent of \(\varphi\), and convergence is uniform on compacta. Smoothness and \(B\)-equivariance of \(g\mapsto S_\chi(g\varphi)(1)\) are clear, so \(S_\chi(\varphi)\in I(\chi)\).

Proof of (b).

6. Take the filtration \(X(F)=O_1\sqcup\dots\sqcup O_N\) of Lemma 1.4(c), with \(O_1=X̊(F)\) and each \(U_j=O_1\cup\dots\cup O_j\) open. The Bernstein–Zelevinsky sequences \(0\to C_c^\infty(U_j)\to C_c^\infty(U_{j+1})\to C_c^\infty(O_{j+1})\to0\) hold for any \(\ell\)-space ([BZ76, 1.8]; characteristic-free). By Lemma 1.4(a), \(O_{j+1}\cong B(F)/B_y(F)\) is a homeomorphism. So a \((B,\chi)\)-equivariant functional on \(C_c^\infty(O_{j+1})\) exists only if \(\chi\) satisfies the modular condition (3.3) on \(B_y(F)\). Call the set of such \(\chi\) \(E_y\). It is Zariski-closed in \(D'\): the condition fixes \(e^\mu(\chi)\) for \(\mu\) in the image \(\Gamma_y\) of \(B_y(F)\) in \(T(F)/T(O)\), whose rational span is \(X_*(T_y)_{\mathbb Q}\) for a maximal (split, Lemma 1.4 step 3) \(F\)-torus \(T_y\subseteq B_y\).
7. Claim: \(E_y\ne D'\) for \(y\notin X̊\). If \(E_y=D'\), every \(e^\mu\) with \(\mu\in\Gamma_y\) is constant on the translate \(D'\) of \(A_X^*\). So \(X_*(T_y)_{\mathbb Q}\subseteq X_*(T^\theta)_{\mathbb Q}\), i.e. \(T_y\subseteq(T^\theta)^\circ\) inside \(T\). On the other hand, \(\dim T_y=\dim T-\dim(T')^{-\theta_y}\ge\dim T-\dim S=\dim T^\theta\). Here \(T'\), \(\theta_y\) are as in Lemma 1.3(e), and \((T')^{-\theta_y}\), a \(\theta_y\)-split torus, is conjugate by \(g\) to a \(\theta\)-split torus, which has dimension \(\le\dim S\) by the choice of \(S\). Hence \(T_y=(T^\theta)^\circ\).

Transport \(\theta_y|_{T'}\) to an involution \(\theta'_T\) of the root datum of \((G,T)\) via \(T'\cong B/U\cong T\). It has the same \(+1\)-eigenspace as \(\theta|_T\). Both preserve a \(W\)-invariant inner product, so their \(-1\)-eigenspaces are the orthogonal complement of that space, and \(\theta'_T=\theta|_T\). By Lemma 1.3(e), \(\dim B_y=\dim T^\theta+\#\{\beta>0:\theta\beta\neq\beta,\ \theta\beta>0\}/2+\#\{\beta>0:\theta\beta=\beta,\ \theta_y|_{\mathfrak g_\beta}=1\}\). For the \(\theta\)-order positive system the middle term is 0 (if \(\beta>0\) and \(\beta\notin\Phi_L\) then \(\theta\beta<0\)), and the last term is at most \(|\Phi_L^+|\). The same count with all signs \(+1\) gives \(\dim B_{x_0}\) (Lemma 1.3(c)). So \(\dim By\ge\dim X\). Then \(By\) is open, hence equal to \(X̊\), a contradiction.

8. Put \(D_1'=D'\setminus\bigcup_{y\notin X̊}E_y\), a finite union by Lemma 1.4(c). For \(\chi\in D_1'\), induction along the filtration shows that a functional vanishing on \(C_c^\infty(O_1)\) vanishes. On \(C_c^\infty(O_1)=C_c^\infty(B(F)/B_{x_0}(F))\) the equivariant functionals form a line.

Proof of (c).

9. Bernstein's principle (pure linear algebra over \(\mathbb C(D')\); see Banks, Math. Res. Lett. 5 (1998); cited, not opened). Let \(V\) be a countable-dimensional complex vector space, \(D'\) an irreducible variety, and \(\{\ell(v_i)=c_i(\chi)\}_{i\in I}\) a system with \(c_i\in\mathbb C[D']\). If the system has a unique solution for \(\chi\) in a non-empty analytic-open set, then it has a unique solution on a Zariski-dense open set, and that solution is rational in \(\chi\).
10. Take \(V=C_c^\infty(X(F))\), which is countable-dimensional. The equations are \(\ell(b\varphi)=c_\chi(b)\ell(\varphi)\) for \(b\) in a countable set of coset representatives of \(B(F)/B(O)\) together with \(b\in B(O)\); here \(c_\chi(b)\) is a monomial in the coordinates of \(A^*\). Add the normalization \(\ell(\mathbf 1_{x_0J_0})=\operatorname{vol}\) for a small compact open \(x_0J_0\subseteq X̊(F)\).

By (b) the solution is unique on \(D_1'\), and on \(R\cap D_1'\neq\emptyset\) the integral (3.2) is a solution. Step 9 gives the rational extension and its uniqueness on a dense open \(D_0'\).