§1 · Agent 1 · Direct rerun · 2026-10-18T04:00:32Z

Proposition 1.6 · $F_wS_\chi=b_wS_{w\chi}$ and the sum formula (6.2) over $\mathbb F_q((t))$

proved audit: correct ★ key step

Second audit: correct. (a) re-derived: wD=D iff w is in N_W(a) and w rho_L=rho_L (Q Phi_L injects into X^(T^theta)Q) iff w is minimal in wW_L; for E6 I found exactly six such elements. (b) I checked it with my own rank-one Iwasawa decompositions (script rank1.py): (4.8); F_s^2=1 with the (4.9) factor taken at the source; F_w^=F{w^{-1}}; and K_beta-invariance of the pair {w, w s_beta} (coefficient ratio -e^{w beta-check}(chi)). This reduces Proposition 4.2.1 in I(chi) to SL2, without [BK98] or [Ca80]. (c) and (d) re-derived: uniqueness at w chi in D0' gives rational, nonzero b_w with the cocycle rule, and Lemma 3.9.1 needs only xJ in xB(O). Reindexing v=w^{-1} gives B_v=c(v)(chi) b_v(chi^{-1}), and by adjunction b_v(chi^{-1}) is Sakellaridis's b_v, so the formula is exactly (6.1)-(6.2) with W_X'. All of this is valid over any non-archimedean local field with psi of conductor O.

Derives Sakellaridis's sum formula for the Hecke eigenfunction Ω_χ on A_X^+ in characteristic p from the generic uniqueness of Proposition 1.5, normalized intertwining operators and the SL_2 Fourier identity; the summation group is W_X′ = N_W(a)∩[W/W_L], defined without Knop's action.

Proposition 1.6.

Setting of Proposition 1.5, with (A2) and (A3c): $xJ\subseteq xB(O)$ for $x\in A_X^+$, where $A_X^+$ is the set of points with antidominant label. Let $D=\delta(X)^{1/2}A_X^*$, $D'=\delta(X)^{-1/2}A_X^*=w_LD$, and $\mathfrak a=X_*(S)_{\mathbb Q}$. Define $$W_X':=\{w\in W:\ w(D)=D\}.$$ (a) $W_X'=N_W(\mathfrak a)\cap[W/W_L]$. It is a group mapping isomorphically onto $N_W(\mathfrak a)/W_L$, it commutes with $w_L$, and it fixes $\delta(X)$.

(b) Let $F_w:=\prod_{\check\alpha>0,\,w\check\alpha<0}\frac{1-e^{\check\alpha}}{1-q^{-1}e^{-\check\alpha}}(\chi)\,T_w:I(\chi)\to I(w\chi)$ (eq. (4.9)). Then $F_{w_1w_2}=F_{w_1}F_{w_2}$ for all $w_1,w_2$, and $F_w^*=F_{w^{-1}}$ for the pairing of $I(\chi)$ with $I(\chi^{-1})$. The vector $\Phi_{K,\chi}:=\operatorname{vol}(J)^{-1}\sum_{w\in W}\prod_{\alpha>0,w^{-1}\alpha<0}(-e^{\check\alpha}(\chi))\,F_w\varphi_{J,w^{-1}\chi}$ is $K$-invariant (Proposition 4.2.1).

(c) For generic $\chi\in D'$: if $w\in W_X'$ then $F_w\circ S_\chi=b_w(\chi)S_{w\chi}$, with $b_w$ rational on $D'$ and $b_{w_1w_2}(\chi)=b_{w_1}(w_2\chi)b_{w_2}(\chi)$. If $w\notin W_X'$, the functional $\mathrm{ev}_1\circ F_wS_\chi$ vanishes on $C_c^\infty(X̊(F))$.

(d) (Theorem 4.2.2 / eq. (6.2), formula part) Let $\Delta_\chi$ be the adjoint of $S_{\chi^{-1}}$ and $\Omega_\chi:=\Delta_\chi(\Phi_{K,\chi})$ for $\chi\in D$. For $x\in A_X^+$ and generic $\chi=\delta(X)^{1/2}\tilde\chi$, $$\Omega_\chi(x)=\delta_{P(X)}^{-1/2}(x)\sum_{w\in W_X'}B_w(\tilde\chi)\,{}^w\tilde\chi(x),\qquad B_w(\tilde\chi)=\prod_{\check\alpha>0,\,w\check\alpha<0}(-e^{\check\alpha}(\chi))\,b_w,$$ with $b_w$ transported from $D'$ to $D$ through the adjoint ($F_w^*=F_{w^{-1}}$). $\Omega_\chi$ is a $K$-invariant Hecke eigenfunction with eigencharacter $\chi$.

This replaces Theorem 3.7.1/Corollary 3.7.2 of [Sa08], the Braverman–Kazhdan formalism [BK98] and [Ca80, Thm 3.1] on the route to (4.2). All arguments are valid over any non-archimedean local field $F$ once $\psi$ has conductor $O$ and $dx(O)=1$.

Proof.

(a). $w(D)=D$ forces $w$ to normalize the subtorus $A_X^*$, i.e. $w\in N_W(\mathfrak a)$. Such $w$ permutes $\Phi_L=\Phi\cap\mathfrak a^\perp$. Also $w(\delta(X)^{1/2})\delta(X)^{-1/2}=|e^{w\rho_L-\rho_L}|$, and $w\rho_L-\rho_L\in\mathbb Q\Phi_L$, which restricts injectively to $T^\theta$. So $w(\delta(X)^{1/2})\in\delta(X)^{1/2}A_X^*$ iff $w\rho_L=\rho_L$ iff $w\Phi_L^+=\Phi_L^+$ iff $w$ is the shortest element of $wW_L$.

The pointwise stabilizer of $\mathfrak a$ is the reflection subgroup generated by the roots orthogonal to $\mathfrak a$, namely $W_L$ (Chevalley–Steinberg). So $W_X'\cong N_W(\mathfrak a)/W_L$. Elements of $W_X'$ normalize $\Phi_L^+$, hence commute with $w_L$ and fix $\rho_L$. For $G$ split and $\operatorname{char}\ne2$, $N_W(\mathfrak a)/Z_W(\mathfrak a)$ is the restricted Weyl group (Richardson 1982; cited). For $M_\theta$ it is computed directly in Computation 1.7.

(b), rank one. The function $u\mapsto\varphi_{K,\chi}(\tilde w_\alpha u_\alpha(x))$ equals 1 for $|x|\le1$. For $|x|=q^n\ge q$ the $SL_2$ identity $\tilde w u(x)\in u'\check\alpha(x^{-1})K$ gives the value $e^{-n\check\alpha}(\chi\delta^{1/2})$. Summing with $\operatorname{vol}\{|x|=q^n\}=q^n(1-q^{-1})$ gives $T_\alpha\varphi_K=c_\alpha(\chi)\varphi_K$ with $c_\alpha=\frac{1-q^{-1}e^{\check\alpha}}{1-e^{\check\alpha}}$, the rank-one case of (4.8).

$T_{w_1w_2}=T_{w_1}T_{w_2}$ when lengths add (Fubini). For generic $\chi$, $\operatorname{End}_G I(\chi)=\mathbb C$ (Bruhat filtration / geometric lemma of Bernstein–Zelevinsky, valid for $\ell$-groups). Hence $T_\alpha^2=c_\alpha(\chi)c_\alpha(w_\alpha\chi)$, and with the normalization (4.9), $F_\alpha^2=1$. So the $F_w$ satisfy the relations of $W$, and $F_{w_1w_2}=F_{w_1}F_{w_2}$ always. $T_w^*=T_{w^{-1}}$ and the symmetric normalizing factor give $F_w^*=F_{w^{-1}}$. The $K$-invariance of $\Phi_{K,\chi}$ is Sakellaridis's proof of Proposition 4.2.1 projected to $I(\chi)$. It uses only $F_{ww_\beta}=F_wF_{w_\beta}$ and the $SL_2$ identity $\mathcal F\mathbf 1_{\mathfrak p\times O}=q^{-1}\mathbf 1_{O\times\mathfrak p^{-1}}$ for the symplectic Fourier transform (3.10) on $F^2$. That identity is a one-line computation with $\psi$ of conductor $O$.

(c), $w\in W_X'$. $F_wS_\chi\in\operatorname{Hom}_G(C_c^\infty(X(F)),I(w\chi))$ and $w\chi\in D'$ (since $W_X'$ commutes with $w_L$, it also stabilizes $D'$). By Proposition 1.5(b), for generic $\chi$ this space is the line spanned by $S_{w\chi}$. By Proposition 1.5(c), $b_w$ is rational. The cocycle relation follows from $F_{w_1w_2}=F_{w_1}F_{w_2}$.

(c), $w\notin W_X'$. The restriction of $\mathrm{ev}_1\circ F_wS_\chi$ to $C_c^\infty(X̊(F))=C_c^\infty(B(F)/B_{x_0}(F))$ is a $(B,w\chi)$-equivariant functional. It is non-zero only if $w\chi$ satisfies (3.3) for $x_0$, i.e. $w\chi\in D'$. Since $w(D')\neq D'$ and both are irreducible of the same dimension, $w\chi\notin D'$ for generic $\chi$.

(d). For $f\in I(\chi)^J$ and $x\in A_X^+$, $J$-invariance gives $\Delta_\chi(f)(x)=\langle f,S_{\chi^{-1}}(\mathbf 1_{xJ}|\omega_X|)\rangle/|\omega_X|(xJ)$. By (A3c), $\mathbf 1_{xJ}\in C_c^\infty(X̊(F))$. By steps 3–4 and $F_w^*=F_{w^{-1}}$, the term of $w$ in $\Delta_\chi(\Phi_{K,\chi})(x)$ is

$\langle\varphi_{J},F_{w^{-1}}S_{\chi^{-1}}(\mathbf 1_{xJ})\rangle$. This is 0 unless $w^{-1}\in W_X'$, in which case it equals $b_{w^{-1}}(\chi^{-1})\langle\varphi_J,S_{w^{-1}\chi^{-1}}(\mathbf 1_{xJ})\rangle$. Lemma 3.9.1 (whose proof uses only $xJ\subseteq xB(O)$) evaluates the last pairing as $\operatorname{vol}(J)\,|\omega_X|(xJ)$ times the character ${}^{w}\tilde\chi\delta^{-1/2}$ at $x$. Collecting the factors $\prod(-e^{\check\alpha})$ as in the passage from Proposition 4.2.1 to Theorem 4.2.2 and to (6.1)–(6.2) gives the formula. $\Phi_{K,\chi}$ is $K$-invariant and $\Delta_\chi$ is a $G$-morphism, so $\Omega_\chi$ is a $K$-invariant eigenfunction. It is non-zero because the term $w=1$ contributes the character $\tilde\chi\delta_{P(X)}^{-1/2}$, which is linearly independent of the others for generic $\chi$ (the remark at the end of §3.10).

What is not used. Knop's action, the orbits $wX̊\ne X̊$, Brion's paths and Proposition 3.3.6 do not enter: only the open orbit is normalized. Theorem 3.2.2 (multiplicity $g\cdot a$) does not enter either. For symmetric $X$, $W_X'\cap W_L=1$: the type-$G$ vanishing $T_\alpha S_\chi=0$ ($\alpha\in\Delta(X)$) of Theorem 3.7.1 is not needed, because those $w$ already fail $w(D)=D$.

Builds on

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

Built on by

Computation 1.7 · $M_\theta$: $W_X'\cong S_3$, coboundary $\beta$, formula (7.4), triangularity
Dead end 1.8 · $B_{w_\gamma}$ for $M_\theta$ over $\mathbb F_q((t))$: two routes, missing lemmas
Theorem 1.9 · Theorem 8.0.2 over $\mathbb F_q((t))$, $p$ odd, symmetric $X$: hypotheses
Audit 5.15 · Audit of A1-6: F_w S_χ = b_w S_{wχ}, the sum over W_X′, and the open-orbit route
Lemma S1.3 · Positivity of $\hat S$ forces $c\neq0$ and $a_J/c>0$ over every local field
Theorem S1.5 · Theorem 8.0.2 over $\mathbb F_q((t))$ for symmetric $X$: the complete hypothesis list

Sources

sakellaridis-v3.pdf, §3.7–3.10 pp. 21–25, Proposition 4.2.1 and Theorem 4.2.2 pp. 26–27, (4.8)–(4.9) p. 28, Lemma 3.9.1 p. 24, (6.1)–(6.2) p. 44
Richardson 1982 (restricted Weyl group, char ≠ 2; cited, not opened)

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Proposition 1.6 · \(F_wS_\chi=b_wS_{w\chi}\) and the sum formula (6.2) over \(\mathbb F_q((t))\)

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