Computing \(\Res\)
By the Theorem, (SG) is the surjectivity of \(\Res:\C[\As]^W\to\C[\dXh\AXs]^{\WX}\), a question about symmetric polynomials. For \(\GL_n\) it is decided by power sums; for \(E_6/F_4\) it fails by exactly one dimension.
The method for \(\GL_N\)How do you test surjectivity of a restriction of symmetric functions?
For \(G=\GL_N\), \(\C[\As]^W=\C[x_1^{\pm1},\dots,x_N^{\pm1}]^{S_N}\) is generated by the Laurent power sums
\[p_{\pm k}(x):=\sum_{i=1}^Nx_i^{\pm k},\qquad k\ge1,\]since it is \(\C[x]^{S_N}[e_N^{-1}]\) and \(e_N^{-1}=e_N(x^{-1})\) is a polynomial in the \(p_{-k}\). So \(\Res\) is onto iff the restricted power sums \(\Res(p_{\pm k})\) generate the target ring.
The three examples of A* and A_X*: \(\GL_2/\SL_2\): \(\Res(p_{\pm1})=(q^{1/2}+q^{-1/2})y^{\pm1}\), onto \(\C[y^{\pm1}]\). \(\GL_2/T\): \(\Res(p_1)=y+y^{-1}\), onto \(\C[y+y^{-1}]\). The group: \(\Res(f\otimes1)=f\), onto.
\(\GL_{2n}/\Sp_{2n}\) and \(\GL_{p+q'}/(\GL_p\times\GL_{q'})\)Do the \(\GL\) families generate?
(the two \(\GL\) families) Proved here, given the Theorem
- \(X=\GL_{2n}/\Sp_{2n}\): \(\dXh\AXs=\{(q^{1/2}y_i,q^{-1/2}y_i)_{i\le n}\}\), \(\WX=S_n\), and \[\Res(p_{\pm k})=\big(q^{k/2}+q^{-k/2}\big)\,p_{\pm k}(y),\qquad\text{onto }\C[y^{\pm1}]^{S_n}.\]
- \(X=\GL_{p+q'}/(\GL_p\times\GL_{q'})\), \(p\le q'\): \(\WX=W(C_p)\) and, with constants \(c_k\in\C\), \[\Res(p_k)=\sum_{i=1}^p\big(y_i^k+y_i^{-k}\big)+c_k,\qquad\text{onto }\C[y^{\pm1}]^{W(C_p)}.\]
So under (A1)–(A5), (SG) holds for both families.
Proof
- The translates are from the δ(X) table. In (1), \(\sum_i\big((q^{1/2}y_i)^{\pm k}+(q^{-1/2}y_i)^{\pm k}\big)\) gives the formula; the \(p_{\pm k}(y)\) generate \(\C[y^{\pm1}]^{S_n}\) and \(q^{k/2}+q^{-k/2}\ne0\).
- In (2) the middle block contributes the constant \(c_k=\sum_jq^{k(\frac{q'-p+1}2-j)}\), \(1\le j\le q'-p\). The target \(\C[y^{\pm1}]^{W(C_p)}\) is \(\C[z_1,\dots,z_p]^{S_p}\), \(z_i=y_i+y_i^{-1}\), generated by \(P_k=\sum_iz_i^k\). Since \(y^k+y^{-k}=z^k+(\text{lower powers of }z)\), \(\Res(p_k)-c_k=P_k+(\text{polynomial in }P_1,\dots,P_{k-1})\), and induction on \(k\) puts every \(P_k\) in \(\im\Res\).
Both families are also decided without (A1)–(A5), over every non-archimedean \(F\), by a direct triangular system: [satake-two-levels, Prop. IV] for \(\GL_{2m}/\Sp_{2m}\) and \(\SL_{2n}/\Sp_{2n}\), and [satake-two-levels, Prop. V] for \(\GL_n/(\GL_p\times\GL_{n-p})\) and the spheres \(\Spin_{2n}/\Spin_{2n-1}\) (Characteristic 0?). The \(\SL_{2n}/\Sp_{2n}\) case also follows from \(\GL_{2n}/\Sp_{2n}\) by taking the part of \(\CK\) where the Pfaffian has valuation \(0\) (Proposition F).
Sakellaridis's counterexampleCan \(\Res\) fail to be onto under all the hypotheses?
[Sak, Ex. 8.0.1] Proved (cited)
\(X=\SL_2\times\SL_2\times\SL_2\backslash\Sp_4\times\Sp_4\), the first two \(\SL_2\) embedded as \(\Sp_2\) in the two copies of \(\Sp_4\) and the third diagonally. Its dual group is the image of \(\SL_2\times\SL_2^{\mathrm{diag}}\times\SL_2\to\SO_5\times\SO_5\); \(\dX=1\) and \(\WX=N_W(\AXs)\), but \(\WX\)-nonconjugate points of \(\AXs\) in the diagonal \(\SO_4\) are conjugate under \(W\). So \(\Res\) is not onto and (SG) fails.
\(E_6/F_4\): one dimension shortWhich orbit does \(\phiz\) miss?
Coordinates [satake-two-levels, Prop. VII]: \(\LX=\Lambda_S=\Z e_1\oplus\Z e_6\), \(\WX=S_3\), \(\dXh=q^{\rho_L}\) for the black \(D_4\); on \(\AXs\) put \(u=y(e_1)\), \(v=y(e_6)\), and \(y_1:=(q^{-5},q)\), \(\sigma y_1=(q,q^{-5})\), \(\sigma\) the diagram automorphism. \(S:=\C[\Lambda_S]^{\WX}\), \(M_\kappa\) the \(\WX\)-orbit sums.
[satake-two-levels, Prop. VI–VII] Proved in this project (audited runs)
\[\im\Res=\{g\in S:\ g(y_1)=g(\sigma y_1)\},\qquad S=\im\Res\oplus\C\,M_{3\omega_1^\vee}.\]With Sakellaridis's Theorem for the model \(G_{\Z}/G_{\Z}^{\theta}\) (\(p\) odd, \(F\) \(p\)-adic or \(\Fq(\!(t)\!)\)):
\[\CK=\HK\ast\phiz\ \oplus\ \C\,\one_{O_{3\omega_1^\vee}} .\]The reason is the Lemma on W_X: \(q^{\rho_L}y_1\) and \(q^{\rho_L}\sigma y_1\) are \(W(E_6)\)-conjugate (both to \(q^{\rho-\omega_4}\)) but \(y_1,\sigma y_1\) are not \(\WX\)-conjugate. The missed orbit is the red point of the \(E_6/F_4\) figure.
Status of (SG)Where does the basic function generate?
| \(X\) | (SG) | reason |
|---|---|---|
| \(H\) under \(H\times H\) | holds | \(\phiz\) is the unit |
| \(\GL_2/\SL_2\simeq\Gm\) | holds | \(T_1\ast\one_{\vpi^m\cO^\times}=(q+1)\one_{\vpi^{m+1}\cO^\times}\) |
| \(\GL_2/T\), \(\PGL_2/T\) | holds | adjacency on the tree |
| \(\GL_n/(\GL_p\times\GL_{n-p})\), \(\Spin_{2n}/\Spin_{2n-1}\) | holds over every \(F\) | [satake-two-levels, Prop. V] |
| \(\GL_{2m}/\Sp_{2m}\), \(\SL_{2n}/\Sp_{2n}\) | holds over every \(F\) | [satake-two-levels, Prop. IV]; also \(\Res\) onto |
| \(\SL_2/T\), \(q\) odd | fails | square classes; (A5) fails |
| \(\GL_n/\Orth_n\), \(\PGL_2/\mathrm{PO}_2\) | fails | several \(G(F)\)-orbits |
| \(\SL_2^3\backslash\Sp_4^2\) | fails | [Sak, Ex. 8.0.1] |
| \(E_6/F_4\), \(p\) odd | fails | \(\Res\) misses \(M_{3\omega_1^\vee}\) |
| \(\Aff^n\) under \(\GL_n\) | holds over every \(F\) | lattices (𝔸ⁿ) |
| \(\Mat_n\) under \(\GL_n^2\) | \(n=2\) holds; \(n\ge3\) open | L-factor and central translates |
Pattern: wherever (SG) fails, either the orbit set is too large for Hecke translates (square classes, several \(G(F)\)-orbits), or \(\iota\) glues points (Example 8.0.1, \(E_6/F_4\)).