Dead end 2.11 · Dead end: \(G=\mathrm{GL}_2\) split, \(H=T_E\) or \(T_E^1\) non-split

(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 Lemma 2.2(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 Lemma 2.8 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.