Lemma 2.5 · Lemma: \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\), so sheaves generate for all forms

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 (Setup 2.4(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 Setup 2.4 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.