§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:16:25Z

Computation 6.4 · \(E_6/F_4\): root data, the shift \(q^{\rho_L}\), the little Weyl group \(S_3\), and the three generators of the target

proved audit: correct

Second audit: correct. No defect. Re-derived in an independent Bourbaki R^8 model of E6: theta=-w_L reproduces the page's matrix and gives a (-1)-eigenspace of dimension 26 on g (type E IV); <rho_L,alpha_j^vee>=(-3,1,1,1,1,-3); the elements of W preserving Lambda_S(x)Q and Phi_L^+ are exactly 6 and act by s_1, s_6; w_1=w_0(D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).)w_0(D4D4 · Unitriangularity of the Satake traces tr V_λ in the basis T_λ = 1_{Kt^λK} (Zhu (5.6.2), Lemma 5.6.3; Lusztig, Kato).) has length 8 and acts as s_1; r(m_{omega_2^vee})=c_8 M_(1,1)+const recomputed. ('lengths 20+12' refers to the two factors.)

Computes the data needed to evaluate \(r\) for \(E_6/F_4\): the shifted point in coordinates, the little Weyl group \(S_3\) realized by \(\rho_L\)-fixing elements of \(W(E_6)\), and the reduction of surjectivity to a single orbit sum \(M_{(3,0)}\), at the label \(3\omega_1^\vee\) where H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. fails on the sheaf side.

Computation 6.4.

Let \(X=E_6/F_4\) with Bourbaki labels (black nodes \(2,3,4,5\) forming a \(D_4\), white nodes \(1,6\)). Write \(a\omega_1^\vee+b\omega_6^\vee\) as \((a,b)\), and put \(e_1:=\varphi(\alpha_1^\vee)=(2,-1)\), \(e_6:=\varphi(\alpha_6^\vee)=(-1,2)\), \(u:=y(e_1)\), \(v:=y(e_6)\) for \(y\in A_X^*\). For \(\kappa\in\Lambda_S^+\) let \(M_\kappa\) be the \(W_X\)-orbit sum of Lemma 6.2, and for dominant \(\mu\in\Lambda_T\) let \(m_\mu=\sum_{\nu\in W\mu}x^\nu\).

  1. \(\varphi\) kills \(\alpha_2^\vee,\dots,\alpha_5^\vee\). Of the 72 roots, 24 restrict to \(0\) on \(S\) and the others to the six vectors \(\pm(2,-1),\pm(-1,2),\pm(1,1)\), each with multiplicity 8: the restricted root system is \(A_2\).
  2. \(2\rho_L=6\alpha_2+6\alpha_3+10\alpha_4+6\alpha_5\), \(\langle\rho_L,\alpha_j^\vee\rangle=1\) for \(j=2,\dots,5\), and \(\langle\rho_L,\alpha_1^\vee\rangle=\langle\rho_L,\alpha_6^\vee\rangle=-3\). Hence \(x=q^{\rho_L}y\) satisfies
$$x^{\alpha_1^\vee}=q^{-3}u,\qquad x^{\alpha_6^\vee}=q^{-3}v,\qquad x^{\alpha_j^\vee}=q\ \ (j=2,\dots,5),\qquad x^\lambda=q^{\langle\rho_L,\lambda\rangle}u^{c_1}v^{c_6}\ \text{ for }\lambda=\textstyle\sum_jc_j\alpha_j^\vee .$$
  1. \(w_1:=w_0(D_5^{\{1,\dots,5\}})\,w_0(D_4)\) and \(w_6:=w_0(D_5^{\{2,\dots,6\}})\,w_0(D_4)\) (products of longest elements of lengths 20 and 12) fix \(\rho_L\), commute with \(\varphi\), and act on \((a,b)\) by \(s_1(a,b)=(-a,a+b)\), \(s_6(a,b)=(a+b,-b)\); on \(A_X^*\) they act by \(s_1:(u,v)\mapsto(u^{-1},uv)\), \(s_6:(u,v)\mapsto(uv,v^{-1})\). So \(W_X=S_3\), and the transported action is the restriction of the action of \(W\).
  2. \(\Lambda_S=\Z e_1\oplus\Z e_6\) is the coroot lattice of the restricted \(A_2\); the target of \(r\) is \(\C[\Lambda_S]^{S_3}=R(\PGL_3)\otimes\C\), and
$$\Lambda_S^+=\{(a,b):a,b\ge0,\ a\equiv b\bmod 3\}=\langle(1,1),(3,0),(0,3)\rangle_{\text{monoid}} .$$

So \(r\) is onto if and only if \(M_{(1,1)},M_{(3,0)},M_{(0,3)}\in\operatorname{im}r\).

  1. The diagram automorphism \(\sigma\) (\(1\leftrightarrow6\), \(3\leftrightarrow5\)) commutes with \(\theta\), fixes \(\rho_L\), normalizes \(W\) and \(W_X\), acts on \((a,b)\) by \((a,b)\mapsto(b,a)=-w_0^{A_2}(a,b)\), and satisfies \(r\circ\sigma=\sigma\circ r\). Hence \(M_{(3,0)}\in\operatorname{im}r\iff M_{(0,3)}\in\operatorname{im}r\).
  2. \(r(m_{\omega_2^\vee})=c_8(q)\,M_{(1,1)}+\mathrm{const}\) with \(c_8=q^{-3}+q^{-2}+q^{-1}+2+q+q^2+q^3\). So \(M_{(1,1)}\in\operatorname{im}r\), and the only question is whether \(M_{(3,0)}\in\operatorname{im}r\): the label \(3\omega_1^\vee\), at which H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. fails on the sheaf side.

All data are those of the \(E_6/F_4\) family page, re-verified by exact integer computation. \(G=E_6\) is simply connected, so \(\Lambda_T=\bigoplus_j\Z\alpha_j^\vee\), and \(\theta=-w_L\), where \(w_L\) is the longest element of the Weyl group of the black \(D_4\).

  1. Root data. \(\theta^2=1\), \(\theta\) is an isometry, and \(\varphi=1-\theta\) kills the black coroots, with \(\varphi\alpha_1^\vee=2\omega_1^\vee-\omega_6^\vee\) and \(\varphi\alpha_6^\vee=-\omega_1^\vee+2\omega_6^\vee\). The 24 roots of \(L\) restrict to \(0\); the other 48 fall into six classes of 8, represented in \((a,b)\)-coordinates by \(\pm(2,-1),\pm(-1,2),\pm(1,1)\). This is the restricted root system \(A_2\) with all multiplicities 8, as on the family page.
  1. The shift. The 12 positive roots of the black \(D_4\) sum to \(2\rho_L=6\alpha_2+6\alpha_3+10\alpha_4+6\alpha_5\). Since \(\rho_L\) is the \(\rho\) of \(D_4\), \(\langle\rho_L,\alpha_j^\vee\rangle=1\) for the black \(j\). Node 1 is joined only to node 3, so \(\langle2\rho_L,\alpha_1^\vee\rangle=6\langle\alpha_3,\alpha_1^\vee\rangle=-6\); likewise \(\langle2\rho_L,\alpha_6^\vee\rangle=-6\). For \(\lambda=\sum_jc_j\alpha_j^\vee\) one has \(\varphi(\lambda)=c_1e_1+c_6e_6\), which gives the formula for \(x^\lambda\).
  1. The little Weyl group. \(w_0(D_4)\) is \(-1\) on the black span, and \(w_0(D_5^{\{1,\dots,5\}})\) is minus the exchange of the two spin nodes \(2,5\) of that \(D_5\). So \(w_1\) permutes the black simple roots (\(\alpha_2\leftrightarrow\alpha_5\), with \(\alpha_3,\alpha_4\) fixed), preserves \(\Phi_L^+\) and fixes \(\rho_L\); similarly for \(w_6\). Both preserve the \((-1)\)-eigenspace of \(\theta\), commute with \(\varphi\), and act on \((a,b)\) as stated (checked on 200 random vectors). In the basis \(e_1,e_6\): \(s_1(e_1)=-e_1\) and \(s_1(e_6)=e_1+e_6\), so \((s_1y)(e_1)=y(-e_1)=u^{-1}\) and \((s_1y)(e_6)=y(e_1+e_6)=uv\); similarly for \(s_6\). Hence \(W_X=\langle s_1,s_6\rangle\simeq S_3\), and the transported action is the restriction of \(W\), as in Setup 6.1.
  1. The target. \(e_1,e_6\) lie in \(\Lambda_S=\{(a,b):a\equiv b\bmod3\}\), and \(\det\begin{pmatrix}2&-1\\-1&2\end{pmatrix}=3\) is the index of \(\Lambda_S\) in \(\Z^2\), so \(\Lambda_S=\Z e_1\oplus\Z e_6\), the coroot lattice of the restricted \(A_2\). As the character lattice of \(A_X^*\) it is the root lattice of \(\PGL_3\), whence \(\C[\Lambda_S]^{S_3}=R(\PGL_3)\otimes\C\). The \(W_X\)-chamber is \(a,b\ge0\). Every \((a,b)\in\Lambda_S^+\) with \(a\ge b\) equals \(b(1,1)+\tfrac{a-b}3(3,0)\), and symmetrically when \(a\le b\); so \((1,1),(3,0),(0,3)\) generate \(\Lambda_S^+\), and the generation criterion of Lemma 6.2 gives the stated reduction.
  1. The symmetry \(\sigma\). \(\sigma\) preserves the black \(D_4\), so it commutes with \(w_L\), hence with \(\theta\); it permutes \(\Phi_L^+\), so it fixes \(\rho_L\); and it exchanges \(\omega_1^\vee\) and \(\omega_6^\vee\). For \(f\in\C[A^*]^W\),
$$r(\sigma^*f)(y)=f\bigl(\sigma(q^{\rho_L}y)\bigr)=f\bigl(q^{\rho_L}\sigma(y)\bigr)=\sigma^*(rf)(y).$$

Since \(\sigma^*\) preserves \(\C[A^*]^W\) and maps \(M_{(3,0)}\) to \(M_{(0,3)}\), the two memberships are equivalent.

  1. The adjoint orbit. \(\omega_2^\vee\) is the highest coroot, so \(m_{\omega_2^\vee}\) is the sum of \(x^{\alpha^\vee}\) over all 72 coroots. The 24 coroots of \(L\) restrict to constants. The other 48 restrict to multiples of \(y^\kappa\) for the six \(\kappa\in W_X(1,1)=\{\pm(1,1),\pm(2,-1),\pm(-1,2)\}\), eight coroots over each, and over each \(\kappa\) the eight values of \(\langle\rho_L,\alpha^\vee\rangle\) are \(-3,-2,-1,0,0,1,2,3\). So \(r(m_{\omega_2^\vee})=c_8(q)M_{(1,1)}+\mathrm{const}\) with \(c_8(q)\ne0\).

Verified by computation: e6_setup.py, e6_wx.py, e6_restrict.py.

Builds on

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

Built on by

Original record, as the agent wrote it at 02:16 UTC

E6/F4: root data, shift, restricted A2, W_X lifts, target ring and its three generators

All from families/e6-f4.html, re-verified in e6_setup.py / e6_wx.py (exact integers): \(\theta^2=1\), \(\theta\) isometric, \(\varphi\) kills \(\alpha_2^\vee,\dots,\alpha_5^\vee\), \(\varphi\alpha_1^\vee=2\omega_1^\vee-\omega_6^\vee=:e_1\), \(\varphi\alpha_6^\vee=-\omega_1^\vee+2\omega_6^\vee=:e_6\); \(|\Phi|=72\); the restrictions \(\bar\alpha\) of roots to \(S\) are \(0\) (24 roots) and the six vectors \(\pm(2,-1),\pm(-1,2),\pm(1,1)\) in \((a,b)\)-coordinates (\(a\omega_1^\vee+b\omega_6^\vee\)), each with multiplicity \(8\): restricted root system \(A_2\), as on the page.

Shift. \(2\rho_L=6\alpha_2+6\alpha_3+10\alpha_4+6\alpha_5\); \(\langle\rho_L,\alpha_j^\vee\rangle=1\) (\(j=2..5\)), \(\langle\rho_L,\alpha_1^\vee\rangle=\langle\rho_L,\alpha_6^\vee\rangle=-3\). With \(u:=y(e_1)\), \(v:=y(e_6)\) the coordinates of \(y\in A_X^*=\operatorname{Hom}(\Lambda_S,\mathbb C^\times)\), the point \(x=q^{\rho_L}y\in A^*\) has $$x^{\alpha_1^\vee}=q^{-3}u,\quad x^{\alpha_6^\vee}=q^{-3}v,\quad x^{\alpha_j^\vee}=q\ (j=2,\dots,5),\qquad x^\lambda=q^{\langle\rho_L,\lambda\rangle}u^{\lambda_1}v^{\lambda_6}.$$

\(W_X\). \(w_1:=w_0(D_5^{\{1..5\}})\,w_0(D_4^{\{2..5\}})\) and \(w_6:=w_0(D_5^{\{2..6\}})\,w_0(D_4)\) (lengths 20+12) fix \(\rho_L\), commute with \(\varphi\), and act on \((a,b)\) by \(s_1(a,b)=(-a,a+b)\), \(s_6(a,b)=(a+b,-b)\) (checked on 200 random vectors). So \(W_X=S_3\) and the transported action is the restriction of \(W\), as in A6-1. On the basis \(e_1,e_6\): \(s_1(e_1)=-e_1\), \(s_1(e_6)=e_1+e_6\), i.e. \(s_1:(u,v)\mapsto(u^{-1},uv)\), \(s_6:(u,v)\mapsto(uv,v^{-1})\). \(\Lambda_S=\mathbb Ze_1\oplus\mathbb Ze_6\) is the root(=coroot) lattice of this \(A_2\), so the target is $$\mathbb C[\Lambda_S]^{S_3}=R(\mathrm{PGL}_3)\otimes\mathbb C,\qquad\Lambda_S^+=\{(a,b):a,b\ge0,\ a\equiv b\ (3)\}=\langle(1,1),(3,0),(0,3)\rangle_{\text{monoid}},$$ so by Lemma A: \(r\) onto \(\iff M_{(1,1)},M_{(3,0)},M_{(0,3)}\in\operatorname{im}r\). The diagram automorphism \(\sigma\) (\(1\leftrightarrow6\), \(3\leftrightarrow5\)) commutes with \(\theta\), fixes \(\rho_L\), normalizes \(W\) and \(W_X\), acts as \((a,b)\mapsto(b,a)\) (\(=-w_0^{A_2}\)) and satisfies \(r\circ\sigma=\sigma\circ r\); hence \(M_{(3,0)}\in\operatorname{im}r\iff M_{(0,3)}\in\operatorname{im}r\).

\(M_{(1,1)}\in\operatorname{im}r\). \(r(m_{\omega_2^\vee})=c_8(q)M_{(1,1)}+\text{const}\) with \(c_8=q^{-3}+q^{-2}+q^{-1}+2+q+q^{2}+q^{3}\) (the eight roots over each restricted root; e6_restrict.py). So the only question is \(M_{(3,0)}\): the same label \(3\omega_1^\vee\) at which H5 fails on the sheaf side.

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