# Claims to audit (verbatim from the agents' returns)
Each block gives the step id, its title, the claimed statement and the argument as written. Audit each one independently.


## A2-2 · Lemma: orbit and leading-term obstructions to \(\mathcal H_K\ast\varphi_0=C_K\)

**Kind/status as reported:** lemma / proved

**Builds on:** A2-1

### Statement

Let \(F\) be non-archimedean and \(X=G/H\) homogeneous, with compact open \(K\)-orbits \(\{O_\nu\}_{\nu\in I}\) and a partial order on \(I\) whose lower sets are finite. For a lower set \(L\subseteq I\) let \(C(L)=\bigoplus_{\nu\in L}\mathbb C\,\mathbf 1_{O_\nu}\).

**(a) Orbit obstruction.** Every \(f\in\mathcal H_K\ast\varphi_0\) is supported in \(G(F)\cdot X(\mathcal O)\). If \(F=\mathbb F_q(\!(t)\!)\), \(H\) is connected and \(H^1(\mathbb F_q,L^+H)=1\), then \(X(\mathcal O)=Kx_0\). So (SG) implies \(X(F)=G(F)x_0\); for instance (SG) fails when \(\ker\big(H^1(F,H)\to H^1(F,G)\big)\ne1\).

**(b) Leading-term obstruction.** Let \(\{f_j\}_{j\in J}\subset\mathcal H_K\) be such that the \(f_j\ast\varphi_0\) span \(\mathcal H_K\ast\varphi_0\). Let \(\psi:J\to I\) be **injective** with
$$\operatorname{supp}(f_j\ast\varphi_0)\subseteq\bigcup_{\nu\le\psi(j)}O_\nu,\qquad (f_j\ast\varphi_0)|_{O_{\psi(j)}}\ne0\qquad(j\in J).$$
Then for every lower set \(L\subseteq I\)
$$\mathcal H_K\ast\varphi_0\ \cap\ C(L)=\operatorname{span}\{f_j\ast\varphi_0:\psi(j)\in L\}.$$
Consequently \(\mathbf 1_{O_\mu}\in\mathcal H_K\ast\varphi_0\Rightarrow\mu\in\psi(J)\), and \(\text{(SG)}\iff\psi(J)=I\).

### Argument

**Proof of (a).**
1. \((f\ast\varphi_0)(x)=\sum_{gK}f(g)\,\varphi_0(g^{-1}x)\). This is nonzero only if \(g^{-1}x\in X(\mathcal O)\) for some \(g\), so \(x\in G(F)\cdot X(\mathcal O)\).
2. \(G\to X\) is smooth and surjective because \(H\) is smooth. Hence \(L^+G\to L^+X\) is surjective on \(k\)-points: lift modulo \(t\), then lift step by step using smoothness. So \(L^+X(k)=L^+G(k)\cdot x_0\cong L^+G/L^+H\).
3. The \(\mathbb F_q\)-points of a homogeneous space \(L^+G/L^+H\) form \(\ker\big(H^1(\mathbb F_q,L^+H)\to H^1(\mathbb F_q,L^+G)\big)\)-many \(K\)-orbits. This is the argument of IIb, step 1, with \(\mu=0\). With \(H^1(\mathbb F_q,L^+H)=1\) (D3, \(H\) connected) we get \(X(\mathcal O)=L^+X(\mathbb F_q)=Kx_0\).
4. By steps 1 and 3, \(\mathcal H_K\ast\varphi_0\) is supported in \(G(F)x_0\). The \(G(F)\)-orbits on \(X(F)\) are indexed by \(\ker(H^1(F,H)\to H^1(F,G))\). If this kernel is nontrivial, the indicator of a \(K\)-orbit outside \(G(F)x_0\) is not generated.

**Proof of (b).**
1. *Independence.* The \(f_j\ast\varphi_0\) are linearly independent. A nontrivial relation would have a term whose \(\psi(j)\) is maximal among the terms present, and by step 3 below that term would be the only one nonzero on \(O_{\psi(j)}\).
2. *Setup.* Let \(f=\sum_jc_jf_j\ast\varphi_0\) be a finite sum lying in \(C(L)\). Put \(M=\{\psi(j):c_j\ne0\}\), a finite set, and let \(m\in M\) be maximal.
3. *Top value.* \(f_j\ast\varphi_0\) vanishes on \(O_m\) unless \(m\le\psi(j)\). For \(c_j\ne0\) this forces \(\psi(j)=m\) by maximality, hence \(j=j_0:=\psi^{-1}(m)\) by injectivity. So \(f|_{O_m}=c_{j_0}(f_{j_0}\ast\varphi_0)|_{O_m}\ne0\), and therefore \(m\in L\).
4. *All terms.* Every element of the finite set \(M\) lies below a maximal one, and \(L\) is a lower set, so \(M\subseteq L\). This gives "\(\subseteq\)". The inclusion "\(\supseteq\)" is the support bound.
5. *One orbit.* Take \(f=\mathbf 1_{O_\mu}\) and \(L=\{\nu\le\mu\}\), which is finite. Step 3 applied to a maximal \(m\) gives \(\mathbf 1_{O_\mu}|_{O_m}\ne0\), so \(m=\mu\in\psi(J)\).
6. *(SG).* If (SG) holds, step 5 puts every \(\mu\) in \(\psi(J)\). Conversely, if \(\psi\) is onto, then \(f_\mu:=f_{\psi^{-1}(\mu)}\) satisfies the hypotheses of the classical page's Proposition B (support in \(\{\nu\le\mu\}\), nonzero on \(O_\mu\)), and Proposition B gives (SG).

**Remarks.**
- Part (b) needs \(\psi\) to be injective. When several Hecke labels share one leading orbit, combinations can cancel leading terms and reach labels that are not leading terms. [[A2-12]](4) shows that under (R) and (C), either \(\varphi|_{\Lambda_T^+}\) is not injective modulo cocharacters acting trivially on \(X\), or every label lifts and (SG) holds. So (b) never yields a failure in Proposition II's setting.
- In the trace setting the hypotheses of (b) hold automatically for \(\psi=\varphi\) restricted to rational labels: the support bound comes from D0 by induction on the unitriangular \(\operatorname{tr}V_\lambda\) (IIa), and the top value from the coset \(t^\lambda K\) (Lemma A(2)). Only injectivity is extra.

**Sources cited:** spherical-generation/classical.html, Lemma A(2) and Proposition B; trace-pages/trace-b.html, step 1 (cocycle and Lang); trace-pages/_hypotheses.html, D3


## A2-3 · Counterexample: \(\operatorname{Res}\mathbb G_m\) on \(\mathbb G_m\) through the norm

**Kind/status as reported:** computation / proved

**Builds on:** A2-2

### Statement

Let \(R=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m\), \(z\mapsto\bar z\) its Galois involution, \(N(z)=z\bar z\), \(\theta(z)=\bar z^{-1}\), \(H=R^\theta=\ker N\) (the norm-one torus), and \(X=R/H\cong\mathbb G_m\) via \(N\). For every prime power \(q\):

1. over \(k\), \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\);
2. over \(F=\mathbb F_q(\!(t)\!)\),
$$\mathcal H_K\ast\varphi_0=\bigoplus_{m\in\mathbb Z}\mathbb C\,\mathbf 1_{t^{2m}\mathcal O^\times}\ \subsetneq\ C_K=\bigoplus_{c\in\mathbb Z}\mathbb C\,\mathbf 1_{t^{c}\mathcal O^\times},$$
so (SG) fails; \((\mathrm{SG})_n\) fails for odd \(n\) and holds for even \(n\);
3. for the Galois twist \(X'=R/\mathbb G_m\) (involution \(z\mapsto\bar z\)), \(X'(F)=X'(\mathcal O)\) is a single \(K\)-orbit and (SG) holds.

For \(X\): (C) and D0–D3 hold. (R) holds except that \(T=R\), the only maximal torus, is not split.

### Argument

**Over \(k\).**
1. *Lattices.* \(R_k=\mathbb G_m^2\) with Frobenius \((a,b)\mapsto(b^q,a^q)\), \(\theta(a,b)=(b^{-1},a^{-1})\), \(X_k=\mathbb G_m\), \((a,b)\cdot x=abx\), and Frobenius \(x\mapsto x^q\) on \(X_k\). Hence \(\Lambda_T=\mathbb Z^2\), \(\sigma(a,b)=(b,a)\), \(\theta(a,b)=(-b,-a)\), \(\Lambda_S=\mathbb Z(1,1)\), and \(\varphi(a,b)=(a+b)(1,1)\). There are no roots, so \(\Lambda_T^+=\Lambda_T\).
2. *Orbits.* \(LX(k)=k(\!(t)\!)^\times=\bigsqcup_{c\in\mathbb Z}\mathcal O_c\) with \(\mathcal O_c=t^ck[\![t]\!]^\times\), each open and closed in \((LX)_{\mathrm{red}}\). \(\operatorname{Stab}(t^c)=\{(a,a^{-1})\}\cong L^+\mathbb G_m\), which is connected.
3. *Satake objects.* \((\operatorname{Gr}_{\mathbb G_m^2})_{\mathrm{red}}=\mathbb Z^2\) and \(V_{(a,b)}=\delta_{t^{(a,b)}}\). The map \(m\) sends \(\{t^{(a,b)}\}\mathbin{\tilde\times}L^+X\) isomorphically onto \(\mathcal O_{a+b}\), so \(V_{(a,b)}\ast\mathrm{IC}_0\cong\mathrm{IC}_{a+b}\), the constant sheaf on \(\mathcal O_{a+b}\).
4. *Generation.* An object of \(\mathcal C_k\) is the finite direct sum of its restrictions to orbits. On \(\mathcal O_c=L^+R/\operatorname{Stab}\), with connected stabilizer, the cohomology sheaves are constant, so truncation triangles place the object in \(\operatorname{Thick}\{\mathrm{IC}_c\}\) (Proposition I with H1–H3). Hence \(\mathcal C_k=\operatorname{Thick}\{\mathrm{IC}_c:c\in\mathbb Z\}=\mathcal C_k^{\mathrm{Sat}}\) by step 3.

**Over \(F\).**
5. *Points.* \(G(F)=E^\times\) with \(E=\mathbb F_{q^2}(\!(t)\!)\), \(K=\mathcal O_E^\times\), \(X(F)=F^\times\), and \(g\cdot x=N(g)x\). We have \(N(\mathcal O_E^\times)=\mathcal O^\times\) (Lang for the connected \(L^+H\), D3) and \(N(t)=t^2\). So the \(K\)-orbits are \(t^c\mathcal O^\times\), \(c\in\mathbb Z\), and there are two \(G(F)\)-orbits, distinguished by the parity of \(v\).
6. *Hecke action.* \((\Lambda_T)^\sigma=\mathbb Z(1,1)\), so \(\mathcal H_K\) has basis \(T_m=\mathbf 1_{t^m\mathcal O_E^\times}\), \(m\in\mathbb Z\). With \(\operatorname{vol}K=1\),
$$(T_m\ast\varphi_0)(x)=\varphi_0\big(N(t^m)^{-1}x\big)=\mathbf 1_{t^{2m}\mathcal O^\times}(x).$$
7. *Failure.* Statement 2 follows, and \(\mathbf 1_{t\mathcal O^\times}\notin\mathcal H_K\ast\varphi_0\) for every \(q\). This is the orbit obstruction of [[A2-2]](a): \(X(F)\ne G(F)x_0\), because \(H^1(F,H)=F^\times/NE^\times\cong\mathbb Z/2\) while \(H^1(F,R)=1\). On \(G(F)x_0=\{v\text{ even}\}\) generation holds.
8. *Other \(n\).* For odd \(n\), \(E\otimes_FF_n\) is a field, and steps 5–7 apply with \(q^n\) in place of \(q\). For even \(n\), \(R\) splits, \(\mathcal H_{K_n}\) has basis \(T_{(a,b)}\), and \(T_{(a,b)}\ast\varphi_0=\mathbf 1_{t^{a+b}\mathcal O_n^\times}\), so \((\mathrm{SG})_n\) holds.

**Galois twist.**
9. *Over \(k\).* \(X'\cong R^1\) via \(z\mapsto z/\bar z\). Over \(k\), \(X'_k=\mathbb G_m\) with \((a,b)y=ay/b\) and Frobenius \(y\mapsto y^{-q}\). Frobenius sends \(\mathcal O_c\) to \(\mathcal O_{-c}\), so only \(\mathcal O_0\) is Frobenius-stable. The sheaf level is as in step 4, with \(V_{(a,b)}\ast\mathrm{IC}_0=\mathrm{IC}_{a-b}\).
10. *Over \(F\).* \(X'(F)=E^1\subset\mathcal O_E^\times\), since \(v(N z)=2v(z)\). By Hilbert 90, \(E^1=\{u/\bar u:u\in\mathcal O_E^\times\}=K\cdot1\). So \(C_K=\mathbb C\varphi_0\), and (SG) holds.

**Trace reading.**
11. *Weil structures.* \(\mathrm{Fr}^*\delta_{t^{(a,b)}}=\delta_{t^{(b,a)}}\), so \(V_\lambda\) has a Weil structure exactly when \(\lambda=(a,a)\). The rational columns give \(\mathrm{IC}_{2a}\).
12. *Odd rows.* For odd \(c\), every \(\lambda\) with \(\varphi(\lambda)=c\) has \(\sigma\lambda\ne\lambda\). The Weil object \(V_{(a,b)}\oplus V_{(b,a)}\) (with \(a+b=c\)) has convolution \(\mathrm{IC}_c\otimes\overline{\mathbb Q}_\ell^{\,2}\), and Frobenius acts on the multiplicity space by \(\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\): trace \(0\) for odd \(n\), \(2\) for even \(n\). So row \(c\) of \(m^{(1)}\) is zero, although \(\mathrm{IC}_c=V_{(a,b)}\ast\mathrm{IC}_0\) is a summand used by thick generation.
13. *Contrast.* In \(X'\) the odd rows have no rational points at all, so they impose no condition on (SG).

**Hypotheses for \(X\).**
- (R): \(G,\theta,X\) are over \(\mathbb F_q\), and \(S=\mathbb G_m\subset R\) is split; but \(T=R\) and \(T_0=R\) are not split.
- (C): \(H=R^1\) is a connected torus and \(Z_H(S)=H\).
- D0: steps 2–3.
- D1: trivial, since \(m\) is an isomorphism onto an orbit.
- D2: step 3.
- D3: \(L^+R\) and \(L^+R^1\) are connected.

**Sources cited:** trace-pages/_definitions.html, Proposition I; trace-pages/_hypotheses.html, D3; spherical-generation/classical.html, Lemma A(2)


## A2-4 · Setup: the unitary and Galois \(\mathbb F_q\)-forms of the group case

**Kind/status as reported:** lemma / proved

**Builds on:** A2-1

### Statement

Let \(H\) be a split connected reductive group over \(\mathbb F_q\) with split \(T_H\subset B_H\). Let \(\vartheta\) be an \(\mathbb F_q\)-involution of \(H\) with \(\vartheta(s)=s^{-1}\) on \(T_H\) (a Chevalley involution). Put \(G=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H\) with Galois involution \(\sigma\), \(\theta_u=\vartheta\circ\sigma\), \(\theta_g=\sigma\), \(X_u=G/G^{\theta_u}\), and \(X_g=G/G^{\theta_g}=G/H\). Write \(E=\mathbb F_{q^2}(\!(t)\!)\).

1. **Over \(k\).** \(G_k=H\times H\), \(\sigma(g_1,g_2)=(g_2,g_1)\), and \(\mathrm{Fr}(g_1,g_2)=(\mathrm{Fr}\,g_2,\mathrm{Fr}\,g_1)\). The first coordinate of \(\tau(g)=g\theta(g)^{-1}\) identifies \(X_{u,k}\) and \(X_{g,k}\) with \(H\), with actions
$$(g_1,g_2)\cdot x=g_1x\vartheta(g_2)^{-1}\ \ (\text{unitary}),\qquad (g_1,g_2)\cdot x=g_1xg_2^{-1}\ \ (\text{Galois}),$$
base point \(x_0=1\), \(L^+X=L^+H\), and \(\mathrm{IC}_0\) the constant sheaf on \(L^+H\). Frobenius on \(X_k\) is \(x\mapsto\mathrm{Fr}(\vartheta(x))^{-1}\) (unitary) and \(x\mapsto\mathrm{Fr}(x)^{-1}\) (Galois).
2. **Over \(F\).** \(X_u(F)=\{x\in H(E):\vartheta(\sigma x)=x^{-1}\}\) with \(g\cdot x=gx\vartheta(\sigma g)^{-1}\), and \(X_g(F)=\{x\in H(E):x\,\sigma(x)=1\}\) with \(g\cdot x=gx\sigma(g)^{-1}\). In both cases \(K=H(\mathcal O_E)\).
3. **Orbits.** The \(L^+G\)-orbits on \(LX\) are \(\mathcal O_c=L^+H\,t^c\,L^+H\), \(c\in\Lambda_H^+\), with connected stabilizers \(\cong J_c:=L^+H\cap t^cL^+Ht^{-c}\). Frobenius fixes every \(\mathcal O_c\) in the unitary case and sends \(\mathcal O_c\) to \(\mathcal O_{-w_0c}\) in the Galois case.
4. **\(K\)-orbits.** Taking the Cartan type of \(x\in H(E)\) gives bijections \(K\backslash X_u(F)\cong\Lambda_H^+\) (with \(t^c\in X_u(F)\)) and \(K\backslash X_g(F)\cong\{c\in\Lambda_H^+:c=-w_0c\}\).
5. **Examples.** For \(H=\mathrm{GL}_n\) and \(\vartheta(g)=g^{-T}\): \(X_u(F)=\{x\in\mathrm{GL}_n(E):x^*=x\}\) with \(g\cdot x=gxg^*\), and \(X_g(F)\cong\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\). For \(H=\mathbb G_m\) these are the two tori of [[A2-3]]. For \(H=\mathrm{SL}_2\), \(X_u(F)\) is the set of hermitian matrices of determinant \(1\).
6. **Hypotheses.** Both forms satisfy (C) and D0–D3. (R) fails: \(T=Z_G(S)\cong\operatorname{Res}T_H\) is not split, and \(G\) has no split maximal torus \(T_0\). In the unitary form \(S\cong T_H\) is split; in the Galois form \(S\) is anisotropic.

### Argument

**1. Over \(k\).** \(\mathbb F_{q^2}\otimes k=k\times k\) gives \(G_k=H\times H\) with \(\sigma\) the swap. Frobenius of \(\operatorname{Res}\) is the swap composed with \(\mathrm{Fr}\). Since \(\vartheta\) is defined over \(\mathbb F_q\), it commutes with \(\sigma\), so \(\theta_u(g_1,g_2)=(\vartheta g_2,\vartheta g_1)\) and \(\theta_u^2=1\).
- Fixed points: \(G^{\theta_u}_k=\{(g,\vartheta g)\}\cong H\) and \(G^{\theta_g}_k=\Delta H\), both connected.
- Unitary: \(\tau(g_1,g_2)=(g_1\vartheta(g_2)^{-1},\,g_2\vartheta(g_1)^{-1})=(x,\vartheta(x)^{-1})\).
- Galois: \(\tau(g_1,g_2)=(g_1g_2^{-1},g_2g_1^{-1})=(x,x^{-1})\).

Projecting to \(x\) is \(G_k\)-equivariant for the stated actions, sends \(eH\) to \(1\), and sends \(L^+G\cdot x_0\) onto \(L^+H\). For Frobenius in the unitary case, \(x=g_1\vartheta(g_2)^{-1}\) goes to \(\mathrm{Fr}(g_2)\vartheta(\mathrm{Fr}\,g_1)^{-1}=\mathrm{Fr}\big(\vartheta(\vartheta(g_2)g_1^{-1})\big)=\mathrm{Fr}(\vartheta(x))^{-1}\). The Galois case is the same with \(\vartheta=1\).

**2. Over \(F\).** \(G(F)=H(E)\) embeds in \(G(k(\!(t)\!))=H\times H\) by \(e\mapsto(e,\sigma e)\), so the first coordinate of \(\tau\) of an \(F\)-point is the point itself. The defining equations and actions follow from 1, and the Cartan type of \(x\in H(E)\) is the label of the corresponding \(k\)-point.

**3. Orbits and stabilizers.** \(\vartheta(L^+H)=L^+H\), so the \(L^+G\)-orbit of \(x\) is \(L^+HxL^+H\), and these are the \(\mathcal O_c\) by the Cartan decomposition over \(k(\!(t)\!)\) (Zhu (2.1.2)). The condition \(g_1t^c\vartheta(g_2)^{-1}=t^c\) means \(\vartheta(g_2)=t^{-c}g_1t^c\), so \(\operatorname{Stab}(t^c)\cong J_c\), which is connected (Zhu §5.1). For the labels: \(\vartheta(L^+t^cL^+)=L^+t^{-c}L^+\) because \(\vartheta(t^c)=t^{-c}\), so \(\vartheta(x)^{-1}\) has type \(c\) again; \(\mathrm{Fr}\) preserves the double coset of the rational \(t^c\). In the Galois case \(x^{-1}\) has type \(-w_0c\).

**4. \(K\)-orbits.**
- Unitary: \(\vartheta(\sigma(t^c))=t^{-c}=(t^c)^{-1}\), so \(t^c\in X_u(F)\) is a rational point of \(\mathcal O_c\).
- Galois: a Frobenius-stable \(\mathcal O_c\) is a homogeneous space of the connected group \(L^+G\), so it has a rational point by Lang; if \(c\ne-w_0c\), \(\mathcal O_c\) has no rational point.

With connected stabilizers, the cocycle argument of IIb, step 1 (Lang, D3) shows that the rational points of each Frobenius-stable orbit form a single \(K\)-orbit. Since \(LX(\mathbb F_q)=X(F)\) is the union of the rational points of the orbits, 4 follows.

**5. Examples.**
- \(H=\mathrm{GL}_n\): \(\vartheta(\sigma x)=(x^*)^{-1}\), so the condition reads \(x^*=x\), and \(gx\vartheta(\sigma g)^{-1}=gxg^*\). In the Galois case, Hilbert 90 for \(\mathrm{GL}_n\) gives \(X_g(F)=\{g\sigma(g)^{-1}\}\cong\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\).
- \(H=\mathbb G_m\): \(\vartheta(z)=z^{-1}\), \(X_u(F)=F^\times\), and \(g\cdot x=g\bar gx\).
- \(H=\mathrm{SL}_2\), \(\vartheta(g)=g^{-T}\): hermitian matrices of determinant \(1\).

**6. Hypotheses.**
- *Unitary.* \(S=\{(s,s)\}\) is \(\theta_u\)-split, since \(\theta_u(s,s)=(s^{-1},s^{-1})\); it is defined and split over \(\mathbb F_q\), because Frobenius maps \((s,s)\) to \((\mathrm{Fr}\,s,\mathrm{Fr}\,s)\). \(Z_{G^\theta}(S)=\{(t,\vartheta t):t\in T_H\}\cong T_H\), connected.
- *Galois.* \(S=\{(s,s^{-1})\}\), on which Frobenius acts by \(-1\) on \(X_*(S)\), so \(S\) is anisotropic. \(Z_{G^\theta}(S)=\Delta T_H\), connected.
- In both cases \(Z_G(S)=T_H\times T_H=\operatorname{Res}T_H\). Frobenius swaps the two factors of \(X_*(T)\), so \(T\) is not split, and \(G\) has \(\mathbb F_q\)-rank \(\operatorname{rk}H<\operatorname{rk}G\), so it has no split maximal torus.
- D0 for the group case: Cartan decomposition; closure order given by dominance (Zhu Prop. 2.1.5); support bound from the convolution map \(\operatorname{Gr}_{\le a}\mathbin{\tilde\times}\operatorname{Gr}_{\le b}\to\operatorname{Gr}_{\le a+b}\) (Zhu, proof of Lemma 5.3.1); \(J_c=Z_H(c)\ltimes\)(pro-unipotent).
- D1 and D2: [[A2-5]].
- D3: \(L^+G\) and \(J_c\) are connected.

**Sources cited:** zhu-satake-v2.pdf, (2.1.2), Prop. 2.1.5, §5.1 (connectedness of L+G ∩ t^μ L+G t^-μ); trace-pages/trace-b.html, step 1; trace-pages/_hypotheses.html, (R), (C), D0-D3


## A2-5 · Lemma: \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\), so sheaves generate for all forms

**Kind/status as reported:** lemma / proved

**Builds on:** A2-4

### Statement

Let \(H\) be connected reductive over \(k=\overline{\mathbb F}_q\), \(p\) odd, and let \(\vartheta'\) be an automorphism of \(H\) preserving \(T_H\) (so \(\vartheta'(L^+H)=L^+H\)). Let \(G_k=H\times H\) act on \(LH\) by \((g_1,g_2)\cdot x=g_1x\vartheta'(g_2)^{-1}\), with \(x_0=1\).

1. For \(c\in\Lambda_H^+\) and \(\lambda=(c,0)\), the convolution map restricts to an isomorphism \(\operatorname{Gr}_{\le\lambda}\mathbin{\tilde\times}L^+X\cong\overline{\mathcal O_c}\), and \(V_\lambda\ast\mathrm{IC}_0\cong\mathrm{IC}_c\).
2. In general \(V_{(a,b)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\star\mathrm{IC}_{b'}\), a convolution of Satake sheaves, where \(b'\) is the type of \(\vartheta'(t^b)^{-1}\). In particular \(b'=b\) for \(\vartheta'=\vartheta\) (unitary form). This object is a pure, semisimple perverse sheaf.
3. \(\mathcal C_k^{\mathrm{Sat}}=\operatorname{Thick}\{\mathrm{IC}_c:c\in\Lambda_H^+\}=\mathcal C_k\).

### Argument

1. *The convolution space.* Let \(L^+X=L^+H=L^+G/\operatorname{Stab}(1)\), where \(\operatorname{Stab}(1)=\{(h,\vartheta'^{-1}h)\}\). A point of \(LG\times^{L^+G}L^+X\) over \(\operatorname{Gr}_{\le(c,0)}=\operatorname{Gr}_{\le c}\times\{\mathrm{pt}\}\) is a class \([(g_1,g_2),y]\) with \(g_1\in\overline{L^+Ht^cL^+H}\) and \(g_2,y\in L^+H\).
   - Using the \(L^+G\)-action, move \(y\) to \(1\).
   - Then \([(g_1,g_2),1]=[(g_1\vartheta'(h),g_2h),1]\) for \(h\in L^+H\). Taking \(h=g_2^{-1}\) gives a representative \([(g,1),1]\) with \(g=g_1\vartheta'(g_2)^{-1}\), and \(g\) is unique because the second coordinate is pinned to \(1\).

   So the space is \(LH_{\le c}=\overline{L^+Ht^cL^+H}\), and \(m([(g,1),1])=g\cdot1=g\). Thus \(m\) is an isomorphism onto \(\overline{\mathcal O_c}\).
2. *The sheaf.* \(\mathrm{IC}_0\) is constant, so \(V_\lambda\mathbin{\tilde\boxtimes}\mathrm{IC}_0\) is the pullback of \(\mathrm{IC}(\operatorname{Gr}_{\le c})\) along the \(L^+H\)-torsor \(LH_{\le c}\to\operatorname{Gr}_{\le c}\). Smooth pullback of an IC sheaf is the IC sheaf (Zhu, Lemma A.1.4(1)), so \(V_\lambda\ast\mathrm{IC}_0\cong\mathrm{IC}_c\).
3. *General \(\lambda=(a,b)\).* The same normalization identifies \(V_{(a,b)}\ast\mathrm{IC}_0\) with the convolution of \(\mathrm{IC}_a\) and \(\iota_*\mathrm{IC}_b\) on \(L^+H\backslash LH/L^+H\), where \(\iota(g)=\vartheta'(g)^{-1}\) preserves \(L^+H\) and sends type \(b\) to type \(b'\). For \(\vartheta'=\vartheta\), \(\vartheta(t^b)^{-1}=t^b\), so \(b'=b\). By Zhu, Prop. 5.1.4 and Prop. 5.1.1, a convolution of Satake sheaves is perverse and semisimple. Purity is D2 (decomposition theorem on the finite-type model). This is D1–D2 for these forms.
4. *Generation.* The stabilizers \(J_c\) are connected ([[A2-4]](3)), so H1–H3 hold (Proposition I's hypotheses, as for the family \((H\times H)/H\) on the pages). By step 2, every \(\mathrm{IC}_c\) is a direct summand, indeed an equal, of a Satake object, so \(\mathcal M=\Lambda_H^+\). Proposition I then gives \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\).
5. *Independence of the form.* Steps 1–4 use only \(G_k\), \(X_k\) and the \(G_k\)-action. For the twisted forms of [[A2-4]] the automorphism is \(\vartheta'=\vartheta\) (unitary) or \(\vartheta'=1\) (Galois and split). The statement therefore holds for all of them, including \(H=\mathbb G_m\), \(\mathrm{GL}_n\), \(\mathrm{SL}_2\).

Central elements cause no trouble here, unlike for \(\mathrm{SL}_2/\mathrm{SO}_2\). The elements acting trivially are \((z,\vartheta'z)\), \(z\in Z(H)\), and they lie in every \(J_c\). Since \(J_c\) is connected they act trivially on every equivariant local system, and Proposition I applies as stated.

**Sources cited:** zhu-satake-v2.pdf, Lemma A.1.4, Prop. 5.1.1, Prop. 5.1.4, §5.1; trace-pages/_definitions.html, Proposition I; spherical-generation/families/group.html


## A2-6 · Theorem: the unitary twist of \((H\times H)/H\) fails (SG) over \(\mathbb F_q(\!(t)\!)\)

**Kind/status as reported:** lemma / proved

**Builds on:** A2-2, A2-4, A2-5

### Statement

Let \(H\ne1\) be split connected reductive over \(\mathbb F_q\) (any \(q\), in particular odd), let \((G,\theta_u,X_u)\) be its unitary twist ([[A2-4]]), and \(F=\mathbb F_q(\!(t)\!)\). Write \(O_c=K\cdot t^c\) and \(T_a=\mathbf 1_{H(\mathcal O_E)t^aH(\mathcal O_E)}\) for \(a\in\Lambda_H^+\cong(\Lambda_T^+)^\sigma\).

1. \(\operatorname{supp}(T_a\ast\varphi_0)\subseteq\bigcup_{c\le2a}O_c\) and \((T_a\ast\varphi_0)(t^{2a})\ge1\).
2. \(\mathbf 1_{O_c}\in\mathcal H_K\ast\varphi_0\Rightarrow c\in2\Lambda_H^+\). For every finite lower set \(L\subseteq\Lambda_H^+\), \(\dim\big(\mathcal H_K\ast\varphi_0\cap C(L)\big)=\#\{a:2a\in L\}\).
3. \(\Lambda_H^+\ne2\Lambda_H^+\), so (SG) fails over \(F\). With [[A2-5]]: \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) and (SG) fails. This answers IId(1) negatively by an example satisfying (C), D0–D3 and every clause of (R) except the splitness of \(T\) and \(T_0\).
4. \((\mathrm{SG})_n\) fails for odd \(n\) and holds for even \(n\).
5. Instances:
   - \(H=\mathbb G_m\) gives [[A2-3]].
   - \(H=\mathrm{GL}_n\): \(\operatorname{Res}\mathrm{GL}_n\) acting on nondegenerate hermitian matrices by \(x\mapsto gxg^*\). Here \(X(F)\) has two \(G(F)\)-orbits, distinguished by the parity of \(v(\det x)\). For \(n\ge2\) generation also fails on the base orbit: for \(L_d=\{c\le(2d,0,\dots,0)\}\), which lies in the base orbit, \(\operatorname{codim}_{C(L_d)}(\mathcal H_K\ast\varphi_0\cap C(L_d))=p_{\le n}(2d)-p_{\le n}(d)\ge1\) for \(d\ge1\), where \(p_{\le n}\) counts partitions into at most \(n\) parts.
   - \(H=\mathrm{SL}_2\): \(\mathrm{SL}_2(E)\) acting on hermitian matrices of determinant \(1\). Here \(X(F)\) is one \(G(F)\)-orbit and \(\mathbf 1_{K\cdot\operatorname{diag}(t,t^{-1})}\notin\mathcal H_K\ast\varphi_0\).

### Argument

1. *Hecke basis.* \(G\) is unramified and \(\sigma\) swaps the factors of \(\Lambda_T=\Lambda_H\oplus\Lambda_H\). For the \(\sigma\)-stable Borel \(B_H\times B_H\) we get \((\Lambda_T^+)^\sigma=\{(a,a)\}\cong\Lambda_H^+\), and \(Kt^{(a,a)}K=H(\mathcal O_E)t^aH(\mathcal O_E)\). By the Cartan decomposition of \(H(E)\) (Zhu (2.1.2)), the \(T_a\) form a basis of \(\mathcal H_K\).
2. *Support and top value.* By Lemma A(2) of the classical page, \(\operatorname{supp}(T_a\ast\varphi_0)=Kt^aX(\mathcal O)\). For \(k\in K\) and \(y\in X(\mathcal O)\subset H(\mathcal O_E)\), using \(\sigma(t^a)=t^a\) and \(\vartheta(t^a)=t^{-a}\):
$$kt^a\cdot y=kt^a\,y\,\vartheta(\sigma(kt^a))^{-1}=k\,t^a\,y\,t^a\,\vartheta(\sigma k)^{-1}\in Kt^aKt^aK\subseteq\bigcup_{\nu\le2a}Kt^\nu K.$$
The last inclusion holds because the convolution map sends \(\operatorname{Gr}_{\le a}\mathbin{\tilde\times}\operatorname{Gr}_{\le a}\) into \(\operatorname{Gr}_{\le2a}\) (Zhu, proof of Lemma 5.3.1); for \(\mathrm{GL}_n\), Cauchy–Binet on minors gives it too. By [[A2-4]](4) the \(K\)-orbit of a point is its Cartan type, which proves the support claim. For the top value, \(t^a\cdot1=t^a\vartheta(t^a)^{-1}=t^{2a}\), so the coset \(t^aK\) contributes \(1\) at \(t^{2a}\), and all contributions are \(\ge0\).
3. *Leading terms.* Apply [[A2-2]](b) with \(I=\Lambda_H^+\) under the dominance order (finite lower sets), \(J=\Lambda_H^+\), \(f_a=T_a\) and \(\psi(a)=2a\), which is injective. This gives 2.
4. *Odd labels exist.* Suppose every dominant coweight lay in \(2\Lambda_H\). Take \(\xi=2\rho^\vee\) (or any \(\xi\) if \(H\) is a torus). For \(\lambda\in\Lambda_H\) and \(N\gg0\), both \(\lambda+2N\xi\) and \(2N\xi\) are dominant, so \(\lambda\in2\Lambda_H\). Then \(\Lambda_H=2\Lambda_H\), so \(\Lambda_H=0\) and \(H=1\). Hence some dominant \(c\notin2\Lambda_H\) exists: \(c=1\) for \(\mathbb G_m\), \(c=(1,0,\dots,0)\) for \(\mathrm{GL}_n\), \(c=\alpha^\vee\) for \(\mathrm{SL}_2\). Such \(c\) is not of the form \(2a\) with \(a\) dominant, so \(\mathbf 1_{O_c}\notin\mathcal H_K\ast\varphi_0\).
5. *Other \(n\).*
   - Odd \(n\): \(\mathbb F_{q^2}\otimes\mathbb F_{q^n}=\mathbb F_{q^{2n}}\) is a field, \(G(F_n)=H(E_n)\), and steps 1–4 apply verbatim.
   - Even \(n\): \(G_{F_n}\cong H\times H\) and \(X_u(F_n)\cong H(F_n)\) with action \(g_1x\vartheta(g_2)^{-1}\). Then \(T_{(c,0)}\ast\varphi_0=\mathbf 1_{K_Ht^cK_H}\) (classical page, Proposition D, with \(\vartheta\) on the second factor), so \((\mathrm{SG})_n\) holds.
6. *\(\mathrm{GL}_n\).*
   - Parity: \(\det(gxg^*)=N(\det g)\det x\) and \(v(N(y))=2v(y)\), so \(v(\det x)\bmod2\) is \(G(F)\)-invariant; it is even on \(X(\mathcal O)\).
   - Base orbit \(=\{v(\det)\text{ even}\}\): a diagonal \(\operatorname{diag}(t^{c_i})\) reduces by \(t^{2k}=N(t^k)\) and by \(\operatorname{diag}(t,t)=gg^*\) with \(g=\left(\begin{smallmatrix}a&1\\-1&\bar a\end{smallmatrix}\right)\), \(N(a)=t-1\), using \(gg^*=(N(a)+1)\cdot1\) and \(a\in\mathcal O_E^\times\), which exists since \(N(\mathcal O_E^\times)=\mathcal O^\times\).
   - Counting: \(c\le(2d,0,\dots,0)\) forces \(c_n\ge0\), so \(\#L_d=p_{\le n}(2d)\) and \(\#\{a:2a\in L_d\}=p_{\le n}(d)\). The partition \((2d-1,1,0,\dots)\) is not even, so the difference is \(\ge1\) for \(n\ge2\), \(d\ge1\).
7. *\(\mathrm{SL}_2\).* The labels are \(c\alpha^\vee\), \(c\ge0\); the leading labels are the even \(c\), so \(\mathbf 1_{O_{\alpha^\vee}}\) is not generated. To see that there is a single \(G(F)\)-orbit:
   - A binary hermitian form \(N(u)d_1+N(v)d_2\) is a \(4\)-dimensional quadratic form over the local field \(F\) (\(p\) odd), so it represents every element of \(F^\times\). Hence \(h_x\cong\operatorname{diag}(1,\det x)=1\).
   - The isometry can be taken in \(\mathrm{SL}_2(E)\): compose with a diagonal unitary element, using that \(\det:U(h_x)\to E^1\) is onto.
   - Example: \(\operatorname{diag}(t,t^{-1})\) represents \(1\) at \((u,v)=(t^{-1},v_0)\) with \(N(v_0)=t-1\).
8. *Hypotheses:* [[A2-4]](6) and [[A2-5]].

Remark (heuristic, not used): the counts in step 6 grow in the ratio \(p_{\le n}(2d)/p_{\le n}(d)\to2^{n-1}\) on each of the two parity classes. This matches the description of \(C_K\) as a free \(\mathcal H_K\)-module of rank \(2^n\) for \(p\)-adic hermitian forms, which I recall from Hironaka's work and have not checked against a source.

**Sources cited:** zhu-satake-v2.pdf, (2.1.2), proof of Lemma 5.3.1 (convolution into Gr_{≤μ1+μ2}); spherical-generation/classical.html, Lemma A(2), Proposition B, Proposition D; trace-pages/trace-b.html, step 1


## A2-7 · Computation: \(T_\lambda\ast\varphi_0\) on hermitian \(2\times2\) matrices

**Kind/status as reported:** computation / proved

**Builds on:** A2-6

### Statement

Let \(G=\operatorname{Res}\mathrm{GL}_2\) act on nondegenerate hermitian \(2\times2\) matrices, and let \(\mathbf 1_c\) be the indicator of the \(K\)-orbit of Cartan type \(c\). Then
$$T_{(1,0)}\ast\varphi_0=\mathbf 1_{(2,0)}+(q+1)\mathbf 1_{(1,1)},\qquad T_{(1,1)}\ast\varphi_0=\mathbf 1_{(2,2)},$$
$$T_{(2,0)}\ast\varphi_0=\mathbf 1_{(4,0)}+(q+1)\mathbf 1_{(3,1)}+q(q+1)\mathbf 1_{(2,2)},\qquad T_{(1,-1)}\ast\varphi_0=\mathbf 1_{(2,-2)}+(q+1)\mathbf 1_{(1,-1)}+q(q+1)\mathbf 1_{(0,0)},$$
and every \(T_\lambda\ast\varphi_0\) vanishes on orbits with \(c_1+c_2\) odd. The last identity is also \(T_{\alpha^\vee}\ast\varphi_0\) for \(\mathrm{SL}_2(E)\) acting on hermitian matrices of determinant \(1\). Hence \(\mathbf 1_{(1,1)},\mathbf 1_{(2,0)},\mathbf 1_{(3,1)},\mathbf 1_{(4,0)}\) (for \(\mathrm{GL}_2\)) and \(\mathbf 1_{\alpha^\vee}\) (for \(\mathrm{SL}_2\)) are not in \(\mathcal H_K\ast\varphi_0\).

### Argument

1. *Formula.* \((T_\lambda\ast\varphi_0)(x)=\#\{gK\subseteq Kt^\lambda K:\ g^{-1}xg^{-*}\in\mathrm{GL}_2(\mathcal O_E)\}\). The cosets in \(Kt^{(1,0)}K/K\) are represented by \(g=\left(\begin{smallmatrix}t&b\\0&1\end{smallmatrix}\right)\), \(b\in\mathbb F_{q^2}\), and \(g=\operatorname{diag}(1,t)\): \(q^2+1\) cosets.
2. *Value at \(x=t\cdot1\).* For the first family,
$$g^{-1}xg^{-*}=\begin{pmatrix}t^{-1}(1+N(b))&-b\\-\bar b&t\end{pmatrix}.$$
This is integral iff \(N(b)=-1\), which has \(q+1\) solutions in \(\mathbb F_{q^2}\), and its determinant is then \(1\). The coset \(\operatorname{diag}(1,t)\) gives \(\operatorname{diag}(t,t^{-1})\), which is not integral. Value: \(q+1\).
3. *Value at \(x=\operatorname{diag}(t^2,1)\).* The first family gives \(\left(\begin{smallmatrix}1+t^{-2}N(b)&-bt^{-1}\\-\bar bt^{-1}&1\end{smallmatrix}\right)\), which is integral iff \(b=0\), and then it is the identity. The second coset gives \(\operatorname{diag}(t^2,t^{-2})\). Value: \(1\).
4. *Supports.* By [[A2-6]], step 2, \(T_{(1,0)}\ast\varphi_0\) is supported on \(\{(2,0),(1,1)\}\), so the first identity is complete. The remaining identities are supported on \(\{c\le2a\}\) for the same reason. Vanishing on odd \(c_1+c_2\) is the parity invariant of [[A2-6]], step 6.
5. *Enumeration.* The script enumerates the Hermite representatives \(\left(\begin{smallmatrix}t^i&b\\0&t^j\end{smallmatrix}\right)\) of \(Kt^\lambda K/K\): \(10\) and \(90\) cosets for \(q=3\), \(26\) and \(650\) for \(q=5\). It tests integrality and unimodularity of \(g^{-1}xg^{-*}\) exactly, over \(\mathbb F_{q^2}[t,t^{-1}]\). The results are \((1,4)\), \((1,4,12)\), \((1,4,12)\) for \(q=3\) and \((1,6)\), \((1,6,30)\), \((1,6,30)\) for \(q=5\), as predicted, and \(0\) on the odd orbit \((1,0)\) for \(\lambda=(1,0),(0,0),(1,-1),(1,1)\). This checks the hand computation in steps 2–3; it is not used in the proofs.
6. *Non-generation.* On the lower set \(\{c\le(4,0)\}\cap\{c_2\ge0\}=\{(4,0),(3,1),(2,2)\}\), the Hecke span is spanned by \((1,q+1,q(q+1))\) and \((0,0,1)\). This is a plane in a \(3\)-dimensional space, containing neither \(\mathbf 1_{(3,1)}\) nor \(\mathbf 1_{(4,0)}\). On \(\{(2,0),(1,1)\}\) the span is the line through \((1,q+1)\). These are the cases \(d=1,2\) of [[A2-6]](5): codimensions \(p_{\le2}(2)-p_{\le2}(1)=1\) and \(p_{\le2}(4)-p_{\le2}(2)=1\).
7. *\(\mathrm{SL}_2\).* \(K_{\mathrm{SL}}t^{\alpha^\vee}K_{\mathrm{SL}}/K_{\mathrm{SL}}\) is the set of lattices in relative position \((1,-1)\); it is the same set as for \(\mathrm{GL}_2(E)\), since determinants can be adjusted by \(\operatorname{diag}(u,1)\), \(u\in\mathcal O_E^\times\). For \(\det x=1\) the integrality test is unchanged. So \(T_{\alpha^\vee}\ast\varphi_0=\mathbf 1_{2\alpha^\vee}+(q+1)\mathbf 1_{\alpha^\vee}+q(q+1)\mathbf 1_0\) and \(T_0\ast\varphi_0=\mathbf 1_0\). On \(\{0,\alpha^\vee,2\alpha^\vee\}\) the span is \(2\)-dimensional and omits \(\mathbf 1_{\alpha^\vee}\).

**Sources cited:** script herm2.py (enumeration, q = 3, 5)


## A2-8 · Lemma: Satake objects without a Weil structure are invisible to the trace at \(n=1\)

**Kind/status as reported:** lemma / proved

**Builds on:** A2-5, A2-6

### Statement

Let \(G\) be unramified over \(\mathbb F_q\), with Frobenius \(*\)-action \(\sigma\) on \(\Lambda_T\) preserving \(\Lambda_T^+\). Let \(X=G/H\) satisfy the trace setting of Proposition II, except that \(T\) need not be split.

1. \(\mathrm{Fr}^*V_\lambda\cong V_{\sigma^{-1}\lambda}\), so \(V_\lambda\) admits a Weil structure iff \(\sigma\lambda=\lambda\).
2. For \(\sigma^n\lambda=\lambda\), IIa holds: \(\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}_n(V_\lambda)\ast\varphi_0\) with \(\operatorname{tr}_nV_\lambda=\pm T_\lambda+\) lower terms. Hence
$$\mathcal H_{K_n}\ast\varphi_0\otimes\overline{\mathbb Q}_\ell=\operatorname{span}\{\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0):\ \sigma^n\lambda=\lambda\}.$$
3. For every Weil object \((V,\beta)\) of \(\mathrm{Sat}_{G_k}\),
$$\operatorname{tr}_1(V\ast\mathrm{IC}_0)=\sum_{\sigma\lambda=\lambda}\operatorname{tr}(B_\lambda)\,\operatorname{tr}_1(V_\lambda\ast\mathrm{IC}_0),$$
where \(B_\lambda\) is the endomorphism of \(\operatorname{Hom}(V_\lambda,V)\) induced by \(\beta\). A \(\sigma\)-orbit of length \(r\ge2\) contributes \(0\) to \(\operatorname{tr}_n\) whenever \(r\nmid n\).
4. In the unitary twist of [[A2-4]], \(V_\lambda\ast\mathrm{IC}_0\cong V_{\sigma\lambda}\ast\mathrm{IC}_0\) in \(\mathcal C_k\), so every \(V_\lambda\ast\mathrm{IC}_0\) carries Weil structures as an object of \(\mathcal C_k\). For \(\lambda=(c,0)\) with \(c\notin2\Lambda_H\), no such Weil structure is induced from a Weil structure on a Satake object, because \(\operatorname{tr}_1\) of it is a nonzero multiple of \(\operatorname{tr}_1(\mathrm{IC}_c)\notin\mathcal H_K\ast\varphi_0\).

### Argument

1. *Weil structures.* \(\mathrm{Fr}\) maps \(\operatorname{Gr}^\lambda\) onto \(\operatorname{Gr}^{\sigma\lambda}\), since \(\mathrm{Fr}(t^\lambda)=t^{\sigma\lambda}\). Pullback along the universal homeomorphism \(\mathrm{Fr}\) sends the IC sheaf of \(\operatorname{Gr}_{\le\sigma\lambda}\) to the IC sheaf of its preimage \(\operatorname{Gr}_{\le\lambda}\). So \(\mathrm{Fr}^*V_{\sigma\lambda}\cong V_\lambda\). An isomorphism \(\mathrm{Fr}^*V_\lambda\cong V_\lambda\) forces \(\operatorname{Gr}_{\le\sigma^{-1}\lambda}=\operatorname{Gr}_{\le\lambda}\), i.e. \(\sigma\lambda=\lambda\).
2. *Rational \(\lambda\).* For \(\sigma\lambda=\lambda\), \(\operatorname{Gr}_{\le\lambda}\) is defined over \(\mathbb F_q\). The proof of IIa (Lang for the connected \(L^+G\); Grothendieck–Lefschetz on the projective convolution fibres) uses nothing else. The unramified Cartan decomposition gives \(K\backslash G(F)/K=(\Lambda_T^+)^\sigma\), and the open cell of \(\operatorname{Gr}_{\le\lambda}(\mathbb F_q)\) is \(Kt^\lambda K/K\), so \(\operatorname{tr}V_\lambda\) is unitriangular in the basis \(T_\lambda\). Over \(F_n\), replace \(\sigma\) by \(\sigma^n\).
3. *Induced objects.* Write \(V\cong\bigoplus_\lambda V_\lambda\otimes M_\lambda\) (Zhu, Prop. 5.1.1). Fix isomorphisms \(\mathrm{Fr}^*V_\lambda\cong V_{\sigma^{-1}\lambda}\), canonical for \(\sigma\)-fixed \(\lambda\). Schur's lemma (\(\operatorname{Hom}(V_\lambda,V_\mu)=\delta_{\lambda\mu}\overline{\mathbb Q}_\ell\)) writes \(\beta=\bigoplus_\lambda(\mathrm{iso}\otimes B_\lambda)\) with \(B_\lambda:M_\lambda\to M_{\sigma^{-1}\lambda}\). Convolution with \(\mathrm{IC}_0\) is additive, so \(V\ast\mathrm{IC}_0=\bigoplus_\lambda(V_\lambda\ast\mathrm{IC}_0)\otimes M_\lambda\) with the induced Weil structure. At \(x\in X(F_n)\), \(\Phi^{(n)}_x\) maps the \(\lambda\)-summand of the stalk to the \(\sigma^{-n}\lambda\)-summand. When \(\sigma^n\lambda\ne\lambda\) these summands are distinct and contribute \(0\) to the trace. For \(n=1\) each \(\sigma\)-fixed \(\lambda\) contributes \(\operatorname{tr}(B_\lambda)\operatorname{tr}_1(V_\lambda\ast\mathrm{IC}_0)\).
4. *Unitary twist.* By [[A2-5]](2), \(V_{(a,b)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\star\mathrm{IC}_b\) and \(V_{(b,a)}\ast\mathrm{IC}_0\cong\mathrm{IC}_b\star\mathrm{IC}_a\); the commutativity constraint of \(\mathrm{Sat}_H\) (Zhu, Thm 5.2.9; Cor. 5.4.11) identifies them. Hence \(\mathrm{Fr}^*(V_\lambda\ast\mathrm{IC}_0)\cong V_{\sigma^{-1}\lambda}\ast\mathrm{IC}_0\cong V_\lambda\ast\mathrm{IC}_0\). Decomposing into the \(\mathrm{IC}_c\), each with its canonical Weil structure, and choosing any automorphisms of the multiplicity spaces gives Weil structures.
   For \(\lambda=(c,0)\), \(V_\lambda\ast\mathrm{IC}_0=\mathrm{IC}_c\) and \(\operatorname{End}(\mathrm{IC}_c)=\overline{\mathbb Q}_\ell\), so every Weil structure is \(u\cdot\mathrm{can}\) with \(u\ne0\). Its trace is \(u\operatorname{tr}_1(\mathrm{IC}_c)\), which is supported on \(\{c'\le c\}\) and is \(\pm u\) on \(O_c\). If it lay in \(\mathcal H_K\ast\varphi_0\), then by [[A2-2]](b) it would lie in \(\operatorname{span}\{T_a\ast\varphi_0:2a\le c\}\). Since \(c\) is not even, every such \(T_a\ast\varphi_0\) has \(2a<c\) and vanishes on \(O_c\), a contradiction. By 3, the trace of any Weil object induced from \(\mathrm{Sat}_{G_k}\) lies in \(\mathcal H_K\ast\varphi_0\). So this Weil structure is not induced.
5. *Consequence for the matrix of the pages.* At level \(n\), the columns of \(m^{(n)}\) are indexed by \((\Lambda_T^+)^{\sigma^n}\), not by \(\Lambda_T^+\). A summand \(\mathrm{IC}_\mu\) of a non-rational \(V_\lambda\ast\mathrm{IC}_0\) enters \(m^{(1)}\) only if it also occurs in a rational column. In the unitary twist the summand \(\mathrm{IC}_c=V_{(c,0)}\ast\mathrm{IC}_0\), which thick generation uses, has no rational column of its own.

**Sources cited:** zhu-satake-v2.pdf, Prop. 5.1.1, Thm 5.2.9, Cor. 5.4.11; trace-pages/trace-a.html (IIa); trace-pages/trace-b.html, steps 4-5; trace-pages/trace-dictionary.html, Table 2.1


## A2-9 · Computation: unitary \(\mathrm{SL}_2\): every row of \(m^{(1)}\) nonzero, no spanning

**Kind/status as reported:** computation / proved

**Builds on:** A2-6, A2-7, A2-8

### Statement

Let \(H=\mathrm{SL}_2\) in the unitary twist, with rows \(c\ge0\) (orbits \(O_{c\alpha^\vee}\), all Frobenius-stable, one \(K\)-orbit each) and rational columns \(a\ge0\) (the objects \(V_{(a\alpha^\vee,a\alpha^\vee)}\ast\mathrm{IC}_0\)).

1. \(m^{(1)}_{c,a}\ne0\iff c\le2a\). In particular every row of \(m^{(1)}\) is nonzero.
2. The columns of \(m^{(1)}\) do not span \(\bigoplus_{c\ge0}\overline{\mathbb Q}_\ell\): the vector \(e_1\) is not in their span, which is (SG) failing ([[A2-6]]).
3. For \(H=\mathrm{GL}_2\): every row \(c\) with \(c_1+c_2\) even (the base \(G(F)\)-orbit) is nonzero at \(n=1\); every row with \(c_1+c_2\) odd is zero at \(n=1\) and nonzero at \(n=2\); the columns do not span the even part.
4. For \(\mathrm{GL}_2\) and \(a=(1,0)\), the column is \((m_{(2,0)},m_{(1,1)})=(1,q)\) at \(n=1\) and \((1,q^2)\) at \(n=2\).

### Argument

1. *Geometric decomposition.* By [[A2-5]](2), \(V_{(a,a)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\star\mathrm{IC}_a\cong\bigoplus_c\mathrm{IC}_c\otimes W_c\), with \(W_c\) in perverse degree \(0\) (Zhu, Prop. 5.1.4) and \(\dim W_c=[V_a\otimes V_a:V_c]\) for \(\hat H\) (Zhu, Cor. 5.1.5 and Thm 5.3.21(2)). For \(\mathrm{SL}_2\), \(\hat H=\mathrm{PGL}_2\) and the label \(c\alpha^\vee\) is the irreducible representation of dimension \(2c+1\). Clebsch–Gordan gives \(V_a\otimes V_a=\bigoplus_{c=0}^{2a}V_c\), each with multiplicity one.
2. *Entries.* The Weil structure of the rational \(V_{(a,a)}\) induces \(A_{c,0}(a)\in\mathrm{GL}(W_c)=\mathrm{GL}_1\) (IIb, step 4). So \(m^{(1)}_{c,a}=\operatorname{tr}A_{c,0}(a)\ne0\) for \(c\le2a\); for \(c>2a\) the entry is \(0\) by the support bound. Row \(c\) is nonzero in column \(a=c\).
3. *No span.* Column \(a\) has its last nonzero entry at \(c=2a\). Suppose a finite combination of columns equals \(e_1\), and let \(a\) be the largest column used with \(a\ge1\). Its entry at \(2a>1\) is not cancelled by smaller columns, so its coefficient is \(0\). Descending induction leaves only column \(0\), which is \(e_0\). Hence \(e_1\) is not in the span. By IIb(iii), with columns indexed by rational \(\lambda\) ([[A2-8]](2)), this is equivalent to \(\mathbf 1_{O_{\alpha^\vee}}\notin\mathcal H_K\ast\varphi_0\), as in [[A2-6]].
4. *\(\mathrm{GL}_2\).* \(V_a\otimes V_a\) is multiplicity-free: \(\mathrm{Sym}^k\otimes\mathrm{Sym}^k\otimes\det^{2a_2}=\bigoplus_{j=0}^k\det^{2a_2+j}\mathrm{Sym}^{2k-2j}\) with \(k=a_1-a_2\), and the summands are \(c=(2a_1-j,2a_2+j)\).
   - Since \(T_{(1,1)}\) translates labels by \((2,2)\), it suffices to treat \(c_2\in\{0,1\}\). Then \(c=(2m,0)\) is the leading term of \(a=(m,0)\), and \(c=(2m+1,1)\) is the \(j=1\) summand for \(a=(m+1,0)\). So every even row is nonzero.
   - Odd rows: rational columns have \(|2a|\) even, and \(V_{(a,a)}\ast\mathrm{IC}_0\) lives on \(|c|=2|a|\), so odd rows vanish at \(n=1\). At \(n=2\) the column \((c,0)\) becomes rational and gives the unit vector \(\mathrm{IC}_c\).
   - The columns fail to span the even part by [[A2-6]](5).
5. *Explicit column.* \(\operatorname{Gr}_{\le(1,0)}^2\) is smooth of even dimension, with constant IC sheaf and Frobenius \(1\), so \(\operatorname{tr}V_{((1,0),(1,0))}=T_{(1,0)}\), and \(\operatorname{tr}(V\ast\mathrm{IC}_0)=\mathbf 1_{(2,0)}+(q+1)\mathbf 1_{(1,1)}\) ([[A2-7]]). The closure \(\operatorname{Gr}_{\le(2,0)}\supset\{t^{(1,1)}\}\) has an \(A_1\) surface singularity there; its open part is the cone of \(2\times2\) nilpotent matrices (Zhu, Example 2.1.8). It is rationally smooth, so \(\mathrm{IC}_{(2,0)}\) is the constant sheaf and \(\operatorname{tr}\mathrm{IC}_{(2,0)}=\mathbf 1_{(2,0)}+\mathbf 1_{(1,1)}\). Hence \(\operatorname{tr}(V\ast\mathrm{IC}_0)=\operatorname{tr}\mathrm{IC}_{(2,0)}+q\operatorname{tr}\mathrm{IC}_{(1,1)}\): the column is \((1,q)\). Over \(F_2\) the same Weil object gives \((1,q^2)\), the eigenvalue squared, in agreement with \(T_{(1,0)}^2=T_{(2,0)}+(q^2+1)T_{(1,1)}\) in the split group case.
6. *Reading.* At \(n=1\), the hypothesis of IIc ("every row of \(m^{(n)}\) nonzero") and \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) both hold for unitary \(\mathrm{SL}_2\), yet (SG) fails. The pages' \(2\times2\) example \(\left(\begin{smallmatrix}1&2\\1&2\end{smallmatrix}\right)\) is realised geometrically, by a lower-triangular pattern with more rows than rational columns. This happens off (R): see [[A2-12]] for why it cannot happen this way under (R).

**Sources cited:** zhu-satake-v2.pdf, Prop. 5.1.4, Cor. 5.1.5, Thm 5.3.21(2), Example 2.1.8; trace-pages/trace-b.html, steps 4-7; trace-pages/trace.html, Limits


## A2-10 · Proposition: the Galois twist \(\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\) satisfies (SG)

**Kind/status as reported:** lemma / proved

**Builds on:** A2-2, A2-4, A2-5, A2-6

### Statement

Let \(G=\operatorname{Res}_{E/F}\mathrm{GL}_n\), \(X=G/\mathrm{GL}_n\), so that \(X(F)=\{y\in\mathrm{GL}_n(E):y\sigma(y)=1\}\cong\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\), with \(F=\mathbb F_q(\!(t)\!)\). Then (SG) holds over \(F\) for all \(n\ge1\) and all \(q\).

Consequently, the three \(\mathbb F_q\)-forms of the \(k\)-pair \((\mathrm{GL}_n\times\mathrm{GL}_n,\mathrm{GL}_n)\) (the split group case, the Galois twist, the unitary twist) all satisfy \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), while (SG) over \(\mathbb F_q(\!(t)\!)\) holds, holds, and fails respectively. The Galois twist also violates (R), since \(S\) is anisotropic and \(T\) is not split. So failure of (R) is not sufficient for failure of (SG).

### Argument

1. *Labels.* \(K\backslash X(F)\cong\{c\in\mathbb Z^n_{\mathrm{dom}}:c=-w_0c\}\) by [[A2-4]](4), with the dominance order (finite lower sets).
2. *Lift.* For such \(c\), put \(\lambda(c)=(c_1,\dots,c_m,0,\dots,0)\) with \(m=\lfloor n/2\rfloor\). It is dominant because \(c_1\ge\dots\ge c_m\ge0\).
3. *A point of type \(c\) in the support.* \(w_0\in\mathrm{GL}_n(\mathbb F_q)\) satisfies \(w_0\sigma(w_0)=w_0^2=1\), so \(w_0\in X(\mathcal O)=K\cdot1\) (Lang, since \(L^+\mathrm{GL}_n\) is connected). Choose \(u\in K\) with \(u\sigma(u)^{-1}=w_0\) and put \(g=t^{\lambda}u\), \(\lambda=\lambda(c)\). Then \(g\cdot1=t^\lambda w_0t^{-\lambda}=w_0\,t^{w_0\lambda-\lambda}\), whose Cartan type is the dominant representative of \(w_0\lambda-\lambda\), namely \(c\). So \((T_\lambda\ast\varphi_0)(g\cdot1)\ge1\), coming from the coset \(gK\).
4. *Support.* \(kt^\lambda\cdot y=kt^\lambda\,y\,t^{-\lambda}\sigma(k)^{-1}\in Kt^\lambda Kt^{-\lambda}K\subseteq\bigcup_{\nu\le\lambda-w_0\lambda}Kt^\nu K\), using the convolution bound \(\operatorname{Gr}_{\le\lambda}\mathbin{\tilde\times}\operatorname{Gr}_{\le-w_0\lambda}\to\operatorname{Gr}_{\le\lambda-w_0\lambda}\). Moreover \(\lambda-w_0\lambda=(c_1,\dots,c_m,[0],-c_m,\dots,-c_1)=c\). So \(\operatorname{supp}(T_{\lambda(c)}\ast\varphi_0)\subseteq\bigcup_{c'\le c}O_{c'}\).
5. *Generation.* Proposition B of the classical page, with \(f_c=T_{\lambda(c)}\), gives (SG). Corollary C does not apply literally, because \(t^\lambda x_0=x_0\); Proposition B needs only the support bound and one nonzero value on \(O_c\).
6. *Comparison.*
   - The split group case satisfies (SG) by Proposition D of the classical page.
   - The unitary twist fails (SG) by [[A2-6]].
   - All three forms satisfy \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) by [[A2-5]].
   The sheaf statement depends only on \((G_k,X_k)\), so no property of \((G_k,X_k)\) alone can be equivalent to (SG) over \(\mathbb F_q(\!(t)\!)\). An implication needs hypotheses tying the \(\mathbb F_q\)-structure to the \(k\)-structure, such as (R).
7. *\(n=1\)* is [[A2-3]](3).
8. *Contrast with the unitary twist.* Here \(\varphi\) restricted to rational labels is \(\lambda\mapsto\lambda-w_0\lambda\), which maps onto the Frobenius-stable labels. In the unitary twist it is \(a\mapsto2a\), which does not. In both cases \(\sigma\) swaps the factors of \(\Lambda_T\); the difference is whether Frobenius twists the orbit labels by the matching \(-w_0\) or fixes them.

**Sources cited:** spherical-generation/classical.html, Proposition B, Corollary C, Proposition D; zhu-satake-v2.pdf, proof of Lemma 5.3.1


## A2-11 · Dead end: \(G=\mathrm{GL}_2\) split, \(H=T_E\) or \(T_E^1\) non-split

**Kind/status as reported:** check / proved

**Builds on:** A2-2

### Statement

Let \(q\) be odd, \(\varepsilon\in\mathbb F_q^\times\) a non-square, \(T_E=\mathbb F_q[s_0]^\times\subset\mathrm{GL}_2\) with \(s_0=\left(\begin{smallmatrix}0&\varepsilon\\1&0\end{smallmatrix}\right)\) (a non-split maximal torus, \(\cong\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m\)), and \(T_E^1=\ker(\det|_{T_E})\).

1. \(X=\mathrm{GL}_2/T_E\), symmetric for \(\theta=\mathrm{Ad}(s_0)\): (SG) holds over \(\mathbb F_q(\!(t)\!)\). Over \(k\), \(X_k=\mathrm{GL}_2/T\) with \(T\) split, and \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) holds (the pages' family \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) with \(n=2\)).
2. \(X=\mathrm{GL}_2/T_E^1\): over \(k\), \(X_k=\mathrm{GL}_2/\mathrm{SO}(J)\) with \(J=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\), and \(\mathcal C_k^{\mathrm{Sat}}\ne\mathcal C_k\). Over \(F\), \(X(F)\) has two \(G(F)\)-orbits and (SG) fails.

So \(G=\mathrm{GL}_2\) with \(H\) a non-split torus gives no counterexample.

### Argument

**(1) \(H=T_E\).**
1. *Points.* \(X\) is the conjugacy class \(\{y:\operatorname{tr}y=0,\ \det y=-\varepsilon\}\) of \(s_0\). Since \(H^1(F,T_E)=1\), \(X(F)=\{y\in M_2(F):y^2=\varepsilon\}\) is one \(G(F)\)-orbit.
2. *\(K\)-orbits.* For such \(y\), \(\mathcal O[y]=\mathcal O_E\), so the \(y\)-stable lattices are the fractional \(\mathcal O_E\)-ideals, a single homothety class \(v_y\). The \(K\)-orbit of \(y\) is determined by the distance \(c=d(L_0,v_y)\) in the Bruhat–Tits tree. The distance \(c\) is \(K\)-invariant, and each \(c\ge0\) gives exactly one \(K\)-orbit. Indeed \(K\backslash X(F)\cong E^\times\backslash\mathrm{GL}_2(F)/K\), and \(\mathcal O_E^\times\) acts transitively on the vertices at distance \(c\ge1\) from its fixed vertex, since \(|\mathcal O_E^\times/(\mathcal O+t^c\mathcal O_E)^\times|=(q+1)q^{c-1}\) is the number of such vertices. Also \(\varphi_0=f_0:=\mathbf 1_{c=0}\).
3. *Hecke action.* \(T_{(1,1)}\) acts trivially, since the centre lies in \(T_E\). \((T_{(1,0)}\ast f_j)(y)\) counts the neighbours \(L'\) of \(L_0\) with \(d(L',v_y)=j\). This gives
$$T_{(1,0)}\ast f_0=f_1,\qquad T_{(1,0)}\ast f_1=f_2+(q+1)f_0,\qquad T_{(1,0)}\ast f_j=f_{j+1}+qf_{j-1}\ (j\ge2).$$
By induction every \(f_j\) lies in \(\mathbb C[T_{(1,0)}]\varphi_0\), so (SG) holds.

**(2) \(H=T_E^1\).**
4. *The involution \(\varepsilon\).* Over \(k\), \(T_E^1\) becomes \(\{\operatorname{diag}(a,a^{-1})\}=\mathrm{SO}(J)\). The element \(-I\in\mathrm{SO}(J)\) is central in \(\mathrm{GL}_2\), so it acts trivially on \(LX\) and on \(\operatorname{Gr}\). It therefore defines an involution \(\varepsilon\) of the identity functor of \(\mathcal C_k\). By naturality, \(\operatorname{Hom}\) between the \(\varepsilon=1\) and \(\varepsilon=-1\) parts vanishes, and every object splits by \((1\pm\varepsilon)/2\). So \(\mathcal C_k=\mathcal C^+\times\mathcal C^-\), and \(\mathcal C^+\) is thick.
5. *Satake objects lie in \(\mathcal C^+\).* \(-I\) lies in the connected group \(L^+\mathbb G_m\) of scalars, which acts trivially on \(\operatorname{Gr}\). Hence \(-I\) acts trivially on every \(L^+G\)-equivariant perverse sheaf on \(\operatorname{Gr}\) (Zhu, Lemma A.1.4(2)). On \(\mathrm{IC}_0\), \(-I\) lies in the connected \(L^+\mathrm{SO}(J)\) and acts trivially on the stalk at \(x_0\). Since \(m_!\) commutes with \(\varepsilon\), \(\varepsilon=1\) on every \(V_\lambda\ast\mathrm{IC}_0\), so \(\mathcal C_k^{\mathrm{Sat}}\subseteq\mathcal C^+\).
6. *\(\mathcal C^-\ne0\).* Fix \(m\ge1\) and \(y=\operatorname{diag}(t^{2m},1)\). Since \(-t^{2m}\) is a square in \(k(\!(t)\!)\), \(y=hJh^T\) for some \(h\), and \(\operatorname{Stab}_{L^+\mathrm{GL}_2}(hx_0)=\mathrm{SO}(y)(k[\![t]\!])\). Write \(\mathrm{SO}(y)\cong\mathbb G_m\) through the isotropic forms \(v\pm it^mu\), \(i^2=-1\). An element \(z\) has matrix entries \((z+z^{-1})/2\), \((z-z^{-1})it^m/2\) and \((z-z^{-1})/(2it^m)\). These are integral iff \(z\in k[\![t]\!]^\times\) and \(z^2\equiv1\bmod t^m\), i.e. \(z\equiv\pm1\bmod t^m\) (\(p\) odd). So \(\pi_0\operatorname{Stab}=\{\pm1\}\), and \(-I\) (that is, \(z=-1\)) lies in the non-identity component. The nontrivial character of \(\pi_0\) gives an equivariant local system \(\mathcal L^-\) on this orbit with \(\varepsilon=-1\), and \(j_!\mathcal L^-\ne0\) lies in \(\mathcal C^-\). Hence \(\mathcal C_k^{\mathrm{Sat}}\ne\mathcal C_k\). This is the mechanism of the pages' \(\mathrm{SL}_2/\mathrm{SO}_2\) proposition.
7. *Functions.* \(H^1(F,T_E^1)=F^\times/NE^\times\cong\mathbb Z/2\) and \(H^1(F,\mathrm{GL}_2)=1\), so \(X(F)\supsetneq G(F)x_0\), and (SG) fails by [[A2-2]](a). Both levels fail, so this is not a counterexample. It is consistent with the direction of IIc, though (R) fails here.

**(3) Why split \(G\) is unpromising.** For split \(G\), \((\Lambda_T^+)^\sigma=\Lambda_T^+\), so the mechanism of [[A2-8]] is absent. Only Frobenius on orbit labels and \(K\)-orbit splitting (which needs disconnected stabilizers, and these tend to break the sheaf statement too, as in step 6) and Frobenius on multiplicity spaces remain. That a general split \(G\) cannot yield an example is heuristic, not proved here.

**Sources cited:** zhu-satake-v2.pdf, Lemma A.1.4(2); spherical-generation/families/gl-pq.html; satake-two-levels/verdict.html, Theorem 1 (SL2/SO2 central inertia)


## A1-5 · Counterexample outside (R): the norm torus quotient, and the torus criterion

**Kind/status as reported:** lemma / proved

**Builds on:** A1-2, A1-4

### Statement

**Theorem.** Let \(T\) be a torus over \(\mathbb F_q\), \(\sigma\) the Frobenius on \(X_*(T)\), \(H\subset T\) a subtorus, \(X=T/H\), \(F=\mathbb F_q(\!(t)\!)\).
(a) \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\); indeed \(\mathrm{Tri}\{V_\lambda\ast\mathrm{IC}_0\}=\mathcal C_k\).
(b) \(K\backslash X(F)\cong X_*(X)^\sigma\); for \(\lambda\in X_*(T)^\sigma=T(F)/T(\mathcal O)\) with representative \(t^\lambda\in T(F)\), \(T_\lambda\ast\mathbf 1_{yX(\mathcal O)}=\mathbf 1_{t^\lambda yX(\mathcal O)}\); and
$$(\mathrm{SG})\iff X_*(T)^\sigma\to X_*(X)^\sigma\ \text{is onto}\iff\ker\big(X_*(H)_\sigma\to X_*(T)_\sigma\big)=0 .$$

**Example.** \(T=R_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m\), \(\theta(z)=\sigma(z)^{-1}\), \(H=T^\theta=U_1=\ker(\mathrm{Nm})\), \(X=T/U_1\xrightarrow{\ \mathrm{Nm}\ }\mathbb G_m\). Then \(X\) is a symmetric space over \(\mathbb F_q\) (any odd \(q\)); (C) and D0–D3 hold; (R) fails, only because \(T\) (the unique maximal torus, and the torus of the unique \(\theta\)-stable Borel pair \((T,T)\)) is not split; \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\); and over \(F\), \(\mathcal H_K\ast\varphi_0\) is the space of functions supported on even valuations, so (SG) fails. \((\mathrm{SG})_n\) holds iff \(n\) is even. The split form \(\mathbb G_m^2/\{(a,a^{-1})\}\) has the same \((X_k,G_k)\), hence the same \(\mathcal C_k\) and Satake objects, and satisfies (SG): **(SG) is not an invariant of \((X_k,G_k)\), while \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) is.**

**Non-abelian instance.** \(G=R_{\mathbb F_{q^2}/\mathbb F_q}\mathrm{GL}_m\), \(\theta(g)=\sigma(g)^{-\mathsf T}\), \(G^\theta=U_m\), \(X\) = nondegenerate Hermitian matrices. Over \(k\) this is the group case, so \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\); (C), D0–D3 hold, (R) fails; over \(F\), \(\operatorname{val}\det\bmod2\) separates two \(G(F)\)-orbits and (SG) fails.

### Argument

**Proof of the theorem.**

1. *Over \(k\).* \(T_k\to X_k\) is smooth and surjective with connected kernel, so \(L^+T\to L^+X\) is surjective and \((LX)_{\mathrm{red}}=\bigsqcup_{\mu\in X_*(X)}t^\mu L^+X\), each piece one \(L^+T\)-orbit, open and closed, with stabilizer \(L^+H\) (connected). Hence \(\mathcal C_k=\bigoplus_\mu D^b_c(B\,L^+H)\), and each summand is the triangulated closure of the constant sheaf (cohomology sheaves are constant). \(\mathrm{Gr}_{T,\mathrm{red}}=X_*(T)\) is discrete, \(V_\lambda=\delta_{t^\lambda}\), and on \(\{t^\lambda\}\mathbin{\widetilde\times}L^+X\) the convolution map is the isomorphism \(y\mapsto t^\lambda y\) onto \(t^\lambda L^+X\); so \(V_\lambda\ast\mathrm{IC}_0\) is the constant sheaf on the orbit \(\bar\lambda\). The sequence \(0\to X_*(H)\to X_*(T)\to X_*(X)\to0\) is exact (dual of the exact sequence of character groups), so every orbit is a translate: \(\mathrm{Tri}=\mathrm{Thick}=\mathcal C_k\).

2. *Over \(F\).* For an unramified torus \(Y\), \(Y(F)/Y(\mathcal O)\cong X_*(Y)^\sigma\) (take \(\sigma\)-invariants in \(Y(F^{\mathrm{ur}})/Y(\mathcal O^{\mathrm{ur}})=X_*(Y)\); \(H^1(\sigma,Y(\mathcal O^{\mathrm{ur}}))=1\) by Lang). \(T(\mathcal O)\to X(\mathcal O)\) is onto (the fibre is an \(H\)-torsor over \(\mathcal O\), trivial by Hensel and Lang). So the \(K\)-orbits are the cosets of \(X(\mathcal O)\), labelled by \(X_*(X)^\sigma\). Since \(Kt^\lambda K=t^\lambda K\), \((T_\lambda\ast\varphi)(x)=\varphi(t^{-\lambda}x)\), which gives the formula, and \(\mathcal H_K\ast\varphi_0=\operatorname{span}\{\mathbf 1_{\bar\lambda}:\lambda\in X_*(T)^\sigma\}\). Hence (SG) iff the image of \(X_*(T)^\sigma\) is \(X_*(X)^\sigma\). The long exact sequence of cohomology of \(\mathbb Z=\langle\sigma\rangle\) (\(H^0\) = invariants, \(H^1\) = coinvariants) identifies the cokernel with \(\ker(X_*(H)_\sigma\to X_*(T)_\sigma)\). \(\square\)

**The example.**

3. \(X_*(T)=\mathbb Z^2\) with \(\sigma(a,b)=(b,a)\); Frobenius on \(T_k=\mathbb G_m^2\) is \((a,b)\mapsto(b^q,a^q)\), and \(\theta(a,b)=(b^{-1},a^{-1})\) commutes with it, so \(\theta\) is defined over \(\mathbb F_q\). \(G^\theta=\{(a,a^{-1})\}=U_1\) is connected; \(S=\{(a,a)\}\) is maximal \(\theta\)-split and split; \(Z_{G^\theta}(S)=U_1\) is connected: (C) holds. \(X_*(U_1)=\mathbb Z(1,-1)\) with \(\sigma=-1\), so \(X_*(U_1)_\sigma=\mathbb Z/2\); \(X_*(T)_\sigma\cong\mathbb Z\) via \((a,b)\mapsto a+b\), and \((1,-1)\mapsto0\): the kernel is \(\mathbb Z/2\neq0\).

4. Concretely, with \(F_2=\mathbb F_{q^2}(\!(t)\!)\): \(G(F)=F_2^\times\), \(X(F)=F^\times\), \(z\cdot x=\mathrm{Nm}(z)x\), \(K=\mathcal O_2^\times\). \(\mathrm{Nm}(\mathcal O_2^\times)=\mathcal O^\times\) (onto on residue fields, and on \(1+t\mathcal O\) by successive approximation since the trace \(\mathbb F_{q^2}\to\mathbb F_q\) is onto). So \(K\backslash X(F)=\mathbb Z\) by valuation, \(\varphi_0=\delta_0\), \(\mathcal H_K=\mathbb C[T_1^{\pm1}]\), and \(T_m\ast\delta_j=\delta_{j+2m}\) because \(\mathrm{Nm}(t^m)=t^{2m}\). Thus \(\delta_1\notin\mathcal H_K\ast\varphi_0\). Over \(F_n\) with \(n\) even \(T\) splits, \(G(F_n)=F_n^{\times2}\) acts by \((a,b)\cdot x=abx\), \(T_{(a,b)}\ast\delta_j=\delta_{j+a+b}\), and \((\mathrm{SG})_n\) holds; for \(n\) odd \(\mathbb F_{q^n}\cap\mathbb F_{q^2}=\mathbb F_q\) and the computation is that of \(n=1\). (Finite check on a window of 13 labels: the Hecke translates of \(\varphi_0\) span a space of dimension 7 for \(n=1\) and 13 for \(n=2\).)

5. *Hypotheses.* \(\tau(a,b)=(ab,ab)\in S\), so \(\tau(x_\mu)=t^\mu\) with \(x_{j(1,1)}=t^j\); \(\varphi(\lambda)=\lambda-\theta\lambda=(\lambda_1+\lambda_2)(1,1)\); closures are trivial; stabilizers are \(L^+U_1=U_1\ltimes\)(pro-unipotent): D0. Convolution maps are isomorphisms and each \(V_\lambda\ast\mathrm{IC}_0\) is a single shifted IC: D1, D2. \(L^+T\) and \(L^+U_1\) are connected: D3. (R) fails because \(T\) is not split. \(q\) may be taken odd.

6. *The \(R\)-matrix (non-split version of [[A1-1]]).* Weil Hecke objects are \(\delta_{(a,a)}(\alpha)\) and, for \(a\neq b\), \(P=\delta_{(a,b)}\oplus\delta_{(b,a)}\) with a Weil structure exchanging the two points (Frobenius swaps them). Then \(P\ast\mathrm{IC}_0=\mathrm{IC}_{a+b}\otimes\big(\overline{\mathbb Q}_\ell^2,\begin{pmatrix}0&\phi_2\\ \phi_1&0\end{pmatrix}\big)\), with eigenvalues \(\pm\sqrt\beta\), \(\beta=\phi_1\phi_2\), and class \(([\sqrt\beta]+[-\sqrt\beta])[\mathrm{IC}_{a+b}]\). So the image of \(\mathrm{act}\) is \(\bigoplus_{j\text{ even}}R[\mathrm{IC}_j]\oplus\bigoplus_{j\text{ odd}}I[\mathrm{IC}_j]\) with \(I=\langle[\gamma]+[-\gamma]\rangle\), the cokernel is \(\bigoplus_{j\text{ odd}}R/I\), and \(\mathrm{ev}_n(I)=0\) iff \(n\) is odd, matching step 4. The canonical \(\mathrm{IC}_1\) is a Weil direct summand of \(P\ast\mathrm{IC}_0\): even Weil-thick generation holds, and the obstruction is an ideal inside \(\ker\mathrm{ev}_1\) — the pattern (E3)+(E4) of [[A1-4]].

7. *Spherical functions.* For \(z\in\mathbb C^\times\), the \(\chi_z\)-eigenfunctionals (\(\chi_z(T_1)=z\)) are the \(\Omega:\mathbb Z\to\mathbb C\) with \(\Omega(j+2)=z\,\Omega(j)\): a 2-dimensional space, containing \(\Omega(j)=z^{(j-1)/2}\) (\(j\) odd), \(0\) (\(j\) even), which vanishes on \(X(\mathcal O)\). This is criterion (S)/(d) of [[A1-6]]. Representation-theoretically: \(\dim\operatorname{Hom}_{T(F)}(\mathcal S(F^\times),\eta\circ\mathrm{Nm})=2\), from \(\eta\) and \(\eta\cdot(-1)^{\mathrm{val}}\).

**The unitary instance.**

8. Over \(k\), \(G_k=\mathrm{GL}_m^2\), \(\theta(g_1,g_2)=(g_2^{-\mathsf T},g_1^{-\mathsf T})\), \(\tau(g)=(g_1g_2^{\mathsf T},g_2g_1^{\mathsf T})\), so \(X_k\cong\mathrm{GL}_m\) with \((a,b)\cdot h=ahb^{\mathsf T}\): the group case up to the automorphism \((a,b)\mapsto(a,b^{-\mathsf T})\), which permutes the Satake sheaves. As on the pages' group-case proposition (same proof over \(\overline{\mathbb F}_q\)), \(V_{(\lambda,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_\lambda\), so every \(\mathrm{IC}\) is a Satake object and \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\). \(G^\theta\cong\mathrm{GL}_m\) and \(Z_{G^\theta}(S)=T_H\) are connected: (C). Over \(F\): \(X(F)\) is the set of nondegenerate Hermitian \(h\) (\(\sigma(h)^{\mathsf T}=h\)), \(g\cdot h=gh\sigma(g)^{\mathsf T}\), \(\det(g\cdot h)=\mathrm{Nm}(\det g)\det h\) and \(\operatorname{val}\mathrm{Nm}(z)=2\operatorname{val}z\). Elements of \(X(\mathcal O)\) have \(\operatorname{val}\det=0\), so \(\mathcal H_K\ast\varphi_0\) is supported on \(\{\operatorname{val}\det\text{ even}\}\), while \(\mathbf 1_{K\cdot\mathrm{diag}(t,1,\dots,1)}\in C_K\). (R) fails: \(G\) has no split maximal torus. \(\square\)

**Sources cited:** classical.html (Lemma A, Proposition D, Proposition J); symmetric.html (group case and its dictionary); _hypotheses.html ((R), (C), D0-D3); checks.py (window ranks 7 and 13; eigenfunction check)