Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion

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 Setup 1.1). 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 Example 1.4.

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 Proposition 1.6. 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. \(\square\)