§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:07:00Z

Setup 6.1 · Conventions for Sakellaridis's restriction map \(r\), calibrated on Proposition F; the sign of the shift is irrelevant

proved audit: minor defect

Minor defect (second audit). All mathematical claims verified: L(X)=L=Z_G(S); A_X=L/L^theta identified with S by l theta(l)^{-1}, so T->A_X is phi on cocharacters and Lambda_X=phi(Lambda_T)=Lambda_S for the families at hand; delta(X)^{1/2}=q^{+-rho_L}; the formula for r; the sign lemma r_- o iota^* = iota_X^* o r_+; the transport lemma (minimal coset representatives fix rho_L). Two wording defects, neither affecting any claim: (1) the calibration cannot fix the sign, because both formulas of Proposition F are invariant under q^{1/2} <-> q^{-1/2}; the '+' convention is what Proposition F(1) literally writes, not a consequence of the calibration (harmless by the sign lemma). (2) Left versus right modules change r by precomposition with the automorphism iota^* of C[A^]^W (the Satake transform of f(g^{-1}) is iota^ of that of f); im r and hence surjectivity are unchanged, but 'does not change the ring map r' is loose.

Writes Sakellaridis's restriction map for a symmetric space as \(r(f)(y)=f(q^{\rho_L}y)\) on the dual torus of the split torus, checks the formula against Proposition F, and shows that neither the sign of the shift nor the left/right convention affects whether \(r\) is onto. This is the form of \(r\) used for \(E_6/F_4\) and the spheres.

Setup 6.1.

Let \(X=G/G^\theta\) be as in the setting, with \(B\) contained in a minimal \(\theta\)-split parabolic \(P\) whose \(\theta\)-stable Levi is \(L=Z_G(S)\); let \(\rho_L\) be the half-sum of the positive roots of \(L\), and assume \(\varphi(\Lambda_T)=\Lambda_S\) (checked on the family pages for \(E_6/F_4\) and the spheres). A point \(x\in A^*=\Hom(\Lambda_T,\C^\times)\) is the unramified character \(\lambda(\varpi)\mapsto x^\lambda\) of \(T(F)\).

  1. Sakellaridis's invariants are \(L(X)=L\), \(A_X=L/L^\theta\simeq S\), \(\Lambda_X=\varphi(\Lambda_T)=\Lambda_S\), with
$$A_X^*=\Hom(\Lambda_S,\C^\times)\hookrightarrow A^*,\qquad y\mapsto y\circ\varphi,\qquad x^\lambda=y^{\varphi(\lambda)},$$

and \(\delta(X)^{1/2}\) is the point \(q^{\pm\rho_L}\in A^*\). With the sign used on the classical page, \(\delta(X)^{1/2}(\lambda(\varpi))=q^{+\langle\rho_L,\lambda\rangle}\), the restriction map of Theorem E is $$r(f)(y)=f(q^{\rho_L}y)=\sum_\mu c_\mu\,q^{\langle\rho_L,\mu\rangle}\,y^{\varphi(\mu)}\qquad\text{for }f=\sum_\mu c_\mu x^\mu\in\C[A^*]^W,$$ and this formula reproduces Proposition F(1) and F(2).

  1. Sign irrelevance. The maps \(r_\pm(f)(y)=f(q^{\pm\rho_L}y)\) are intertwined by automorphisms of source and target, so \(r_+\) is onto if and only if \(r_-\) is. Passing from Sakellaridis's right modules on \(H\backslash G\) to the left modules of the pages (\(x\mapsto x^{-1}\)) does not change the ring map \(r\).
  2. Transport of \(W_X\). Every class of \(W_X=N_W(\Lambda_S)/W_L\) has a representative \(w\) with \(w\rho_L=\rho_L\). The \(W_X\)-action transported to \(\delta(X)^{1/2}A_X^*\) is the action of these elements of \(W\) on \(A^*\); in particular \(\operatorname{im}r\subseteq\C[\Lambda_S]^{W_X}\).

Notation is that of the setting; \(\Phi_L\subseteq\Phi\) denotes the roots of \(L\) and \(W_L\) its Weyl group, the parabolic subgroup of \(W\) generated by the simple reflections of \(L\).

  1. The Levi and the torus \(A_X\). By Sakellaridis §2.1, \(L(X)\) is the \(\theta\)-stable Levi of a minimal \(\theta\)-split parabolic. The family pages check \(-\theta\alpha\in\Phi^+\) for every \(\alpha\in\Phi^+\setminus\Phi_L\), so \(B\subseteq P\) and \(L(X)=L\). Then \(A_X=L/(L\cap H)=L/L^\theta\), and \(lL^\theta\mapsto l\theta(l)^{-1}\) identifies it with \(S\). On \(T\) this map is \(t\mapsto t\theta(t)^{-1}\), so on cocharacters the quotient \(T\twoheadrightarrow A_X\) is \(\varphi=1-\theta\). Hence \(\Lambda_X=\varphi(\Lambda_T)\), which is \(\Lambda_S\) by the index computations on the family pages, and the dual map is the embedding \(y\mapsto y\circ\varphi\).
  1. The shift. \(\delta(X)\) is the absolute value of the modular character of \(L\cap B=T\cdot U_L\) (Sakellaridis §1.7: the quotient of a right- by a left-invariant volume form). On \(T\) the modular character is \(e^{\pm2\rho_L}\), so \(\delta(X)^{1/2}\) is the point \(q^{\pm\rho_L}\), where \((q^{\pm\rho_L})^\lambda=q^{\pm\langle\rho_L,\lambda\rangle}\); the sign depends on the conventions for volume forms and for the correspondence between points and characters.
  1. The map. Restricting \(f\in\C[A^*]^W\) to \(q^{\rho_L}A_X^*\) and translating back to \(A_X^*\) gives the formula of the statement, with values in \(\C[\Lambda_S]\).
  1. Calibration on Proposition F. For \(\GL_{2n}/\Sp_{2n}\): \(L=\GL_2^n\), \(\rho_L=(\tfrac12,-\tfrac12)\) on each block and \(\varphi(e_1)=e_1+e_2\), so the points of \(A_X^*\) are \(x=(y_1,y_1,\dots,y_n,y_n)\) and the shifted points are \((q^{1/2}y_i,q^{-1/2}y_i)\) on each pair. Hence
$$r(p_{\pm k})=\sum_i\bigl((q^{1/2}y_i)^{\pm k}+(q^{-1/2}y_i)^{\pm k}\bigr)=(q^{k/2}+q^{-k/2})\,p_{\pm k}(y),$$

which is Proposition F(1). For \(\GL_n/(\GL_p\times\GL_q)\): \(L=\mathbb G_m^{2p}\times\GL_{q-p}\), and \(q^{\rho_L}\) is \(1\) on the \(y\)-coordinates and \(q^{(q-p+1)/2-j}\) on the middle block, which gives \(r(p_k)=\sum_i(y_i^k+y_i^{-k})+c_k\), Proposition F(2). So the sign convention of the classical page is \(\delta(X)^{1/2}(\lambda(\varpi))=q^{+\langle\rho_L,\lambda\rangle}\); it is used from now on.

  1. Sign irrelevance. Let \(\iota(x)=x^{-1}\) on \(A^*\). It commutes with \(W\), so \(\iota^*\) is an automorphism of \(\C[A^*]^W\), and it maps \(q^{\rho_L}A_X^*\) onto \(q^{-\rho_L}A_X^*\). Inversion on \(A_X^*\) commutes with \(W_X\), which acts linearly on \(\Lambda_S\), so it induces an automorphism \(\iota_X^*\) of \(\C[\Lambda_S]^{W_X}\). For \(f\in\C[A^*]^W\),
$$r_-(\iota^*f)(y)=f\bigl((q^{-\rho_L}y)^{-1}\bigr)=f(q^{\rho_L}y^{-1})=\iota_X^*(r_+f)(y),$$

so \(\operatorname{im}r_-=\iota_X^*(\operatorname{im}r_+)\), and \(r_+\) is onto if and only if \(r_-\) is. Likewise, Sakellaridis's right modules on \(H\backslash G\) and the left modules of the pages are exchanged by \(x\mapsto x^{-1}\), which does not change the ring map \(r\).

  1. Transport of \(W_X\). Each coset in \(N_W(\Lambda_S)/W_L\) contains a unique element \(w\) of minimal length, and it satisfies \(w(\Phi_L^+)\subseteq\Phi^+\). Since \(w\) normalizes \(\Lambda_S\), it normalizes \(L=Z_G(S)\), so \(w\Phi_L=\Phi_L\). Hence \(w\Phi_L^+=\Phi_L^+\) and \(w\rho_L=\rho_L\). Then \(w(q^{\rho_L}y)=q^{\rho_L}(wy)\): the action of \(W_X\) on \(\delta(X)^{1/2}A_X^*\) transported from \(A_X^*\) coincides with the action of these \(w\in W\) on \(A^*\), and every \(r(f)\) is \(W_X\)-invariant. For \(E_6\) this is checked explicitly in Computation 6.4.

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Original record, as the agent wrote it at 02:07 UTC

Conventions for r fixed and calibrated on Proposition F; sign of the shift is irrelevant

Setting (classical.html §4; Sakellaridis §1.7, §2.1, §8). \(A^*=\operatorname{Hom}(\Lambda_T,\mathbb C^\times)\), a point \(x\) corresponds to the unramified character \(\lambda(\varpi)\mapsto x^\lambda\). For symmetric \(X\): \(L(X)=Z_G(S)=L\) (Sakellaridis §2.1: the \(\theta\)-stable Levi of a minimal \(\theta\)-split parabolic; the family pages check \(-\theta\alpha\in\Phi^+\) for \(\alpha\in\Phi^+\setminus\Phi_L\), so \(B\subseteq P\)). \(A_X=L/(L\cap H)\simeq S\) and the quotient \(T\twoheadrightarrow A_X\) is \(\varphi=1-\theta\) on cocharacters, so \(\Lambda_X=\varphi(\Lambda_T)=\Lambda_S\) (index computations on the family pages), and $$A_X^*=\operatorname{Hom}(\Lambda_S,\mathbb C^\times)\hookrightarrow A^*,\qquad y\mapsto y\circ\varphi,\qquad x^\lambda=y^{\varphi(\lambda)}.$$ \(\delta(X)\) is the modular character of \(L\cap B=T\cdot U_L\) (Sakellaridis §1.7: quotient of right- by left-invariant volume forms), so \(\delta(X)^{1/2}\) is the point \(q^{\pm\rho_L}\in A^*\): \((q^{\pm\rho_L})^\lambda=q^{\pm\langle\rho_L,\lambda\rangle}\), \(\rho_L\) the half-sum of \(\Phi_L^+\). The restriction map is $$r(f)(y)=f(q^{\rho_L}y)=\sum_\mu c_\mu\,q^{\langle\rho_L,\mu\rangle}\,y^{\varphi(\mu)}\qquad(f=\textstyle\sum_\mu c_\mu x^\mu\in\mathbb C[A^*]^W),$$ landing in \(\mathbb C[\Lambda_S]\) after translating \(\delta(X)^{1/2}A_X^*\) back to \(A_X^*\).

Calibration on Proposition F. (1) \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\): \(L=\mathrm{GL}_2^n\), \(\rho_L=(\tfrac12,-\tfrac12)\) per block, \(q^{\rho_L}=(q^{1/2},q^{-1/2})\) per pair, \(\varphi(e_1)=e_1+e_2\) gives \(x=(y_1,y_1,\dots)\); so \(r(p_{\pm k})=\sum_i((q^{1/2}y_i)^{\pm k}+(q^{-1/2}y_i)^{\pm k})=(q^{k/2}+q^{-k/2})p_{\pm k}(y)\), exactly Proposition F(1). (2) \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\): \(L=\mathbb G_m^{2p}\times\mathrm{GL}_{q-p}\), \(q^{\rho_L}\) is \(1\) on the \(y\)-coordinates and \(q^{(q-p+1)/2-j}\) on the middle block, giving \(r(p_k)=\sum_i(y_i^k+y_i^{-k})+c_k\), Proposition F(2). So the sign convention of classical.html is \(\delta(X)^{1/2}(\lambda(\varpi))=q^{+\langle\rho_L,\lambda\rangle}\); I use it below.

Lemma (sign irrelevance). Let \(\iota(x)=x^{-1}\) on \(A^*\). \(\iota\) commutes with \(W\), maps \(q^{\rho_L}A_X^*\) onto \(q^{-\rho_L}A_X^*\) and commutes with the \(W_X\)-action (which is linear on \(\Lambda_S\)). Hence \(\iota^*\) induces automorphisms of \(\mathbb C[A^*]^W\) and of both targets intertwining \(r_{+\rho_L}\) and \(r_{-\rho_L}\): \(r_{+}\) is onto iff \(r_{-}\) is onto. (Proved.) Likewise Sakellaridis's right modules on \(H\backslash G\) versus the pages' left modules (\(x\mapsto x^{-1}\)) do not change the ring map \(r\).

Lemma (transport of the \(W_X\)-action). \(W_X=N_W(\Lambda_S)/W_L\); each class has a representative \(w\) of minimal length in \(wW_L\), and \(w\Phi_L^+=\Phi_L^+\) (as \(w\Phi_L=\Phi_L\) and \(w(\Phi_L^+)\subseteq\Phi^+\)), so \(w\rho_L=\rho_L\). Hence the transported \(W_X\)-action on \(\delta(X)^{1/2}A_X^*\) coincides with the action of these \(w\in W\) on \(A^*\); \(\operatorname{im}r\) consists of \(W_X\)-invariants automatically. (Proved; verified explicitly for \(E_6\) in A6-4.)

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