Theorem 1.2 · Theorem A: decategorification is base change of one \(R\)-matrix

Proof.

1. (ii). By IIa, \(\mathcal H_{K_n}\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\) is spanned by the \(\mathrm{tr}_n(V_\lambda\ast\mathrm{IC}_0)\), whose coordinates in the basis \(\mathrm{tr}_n\mathrm{IC}_\nu\) are the columns \(\mathrm{ev}_n(M_{\bullet\lambda})\) (Setup 1.1(c)). Tensor product is right exact: applying \(-\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\) to the exact sequence \(R^{(\Lambda_T^+)}\xrightarrow{M}R^{(\Lambda_S^+)}\to Q\to0\) gives \(\operatorname{coker}\mathrm{ev}_n(M)=Q\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\). The column criterion and the independence of the coefficient field are IIb (iii), step 7.

2. (i). (\(\Leftarrow\)) If \(M_{\mu\lambda}\neq0\) then some \(E_{\mu,i}(\lambda)\neq0\), so \(\mathrm{IC}_\mu[-i]\) is a direct summand of \(V_\lambda\ast\mathrm{IC}_0\) (D2), and Proposition I gives \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) once this holds for every \(\mu\). (\(\Rightarrow\)) Proposition I gives, for each \(\mu\), some \(\lambda,i\) with \(E_{\mu,i}(\lambda)\neq0\). \(V_\lambda\ast\mathrm{IC}_0\) is pure of some weight \(w\) (D2), so \({}^p\mathcal H^i\) is pure of weight \(w+i\) (BBD, Cor. 5.4.4), and \(\mathrm{IC}_\mu\) with its canonical structure is pure of a weight \(w_\mu\); hence every eigenvalue of \(A_{\mu,i}(\lambda)\) has weight \(w+i-w_\mu\). For different \(i\) the elements \([E_{\mu,i}(\lambda),A_{\mu,i}(\lambda)]\in R\) are supported on disjoint sets of group elements, so the alternating sum \(M_{\mu\lambda}\) is nonzero.

3. (iii). Over \(k\), \(K_0(\mathcal C_k)=\bigoplus_\nu\mathbb Z[\mathrm{IC}_\nu]\) (bounded t-structure with finite-length heart whose simples are the \(\mathrm{IC}_\nu\), by (C)), and \([V_\lambda\ast\mathrm{IC}_0]=\sum_\nu\varepsilon(M_{\nu\lambda})[\mathrm{IC}_\nu]\). Thomason's classification (Compositio Math. 105 (1997), Theorem 2.1): in an essentially small triangulated category \(\mathcal T\), dense triangulated subcategories \(\mathcal D\) (every object a direct summand of an object of \(\mathcal D\)) correspond to subgroups of \(K_0(\mathcal T)\) by \(\mathcal D\mapsto\operatorname{im}K_0(\mathcal D)\), with \(\mathcal D=\{x:[x]\in\operatorname{im}K_0(\mathcal D)\}\). The direct summands of objects of \(\mathcal D=\mathrm{Tri}\{V_\lambda\ast\mathrm{IC}_0\}\) form a thick subcategory (a cone of \(f\oplus0:x\oplus x'\to y\oplus y'\) is \(\mathrm{cone}(f)\oplus x'[1]\oplus y'\)), which is \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\); so \(\mathcal D\) is dense, and \(\mathcal D=\mathcal C_k\) iff the columns of \(\varepsilon(M)\) span \(\bigoplus\mathbb Z\), i.e. iff \(Q\otimes_\varepsilon\mathbb Z=0\).

4. (iv) is immediate from (ii).

5. The mechanism. Thick closure is closure under retracts. The class of a retract \(N\) of \(V_\lambda\ast\mathrm{IC}_0\) is in general not a combination of classes of Satake objects; condition (i) records only which retracts exist (the support of the rows), while (ii) asks that the particular combinations \(\mathrm{ev}_n(M_{\bullet\lambda})\), one per Satake object, span. In the language of Theorem 1.3: the trace sees classes, Morita invariance makes all retract classes available in \(\operatorname{Tr}(\mathrm{Fr},\mathcal C_k)\), but only the full classes lie in the image of \(\mathcal H\).

6. Independence for \(R\)-matrices. (ii) for one \(n\) gives (i) because a zero row of \(M\) stays zero under \(\mathrm{ev}_n\). Conversely, with labels \(\nu_1,\nu_2\) and every column equal to \((r,r)^{\mathsf T}\), \(r\neq0\): no row vanishes, and the columns of \(\mathrm{ev}_n(M)\) span at most a line, for every \(n\). With one label and every column in \(Rr\), \(r=[\gamma]+[-\gamma]\), the cokernel is \(R/rR\) and \(\mathrm{ev}_n(r)=\gamma^n(1+(-1)^n)\), so \((\mathrm{SG})_n\) fails exactly for odd \(n\); this pattern is realized geometrically in Counterexample 1.5.

7. Relation to Laumon. For distinct \(\alpha_1,\dots,\alpha_r\in\overline{\mathbb Q}_\ell^\times\) the functions \(n\mapsto\alpha_j^n\) are linearly independent (Artin), so \((\mathrm{ev}_n)_{n\ge1}\) is injective on \(R\otimes\overline{\mathbb Q}_\ell\); this is the content of Laumon, Theorem 1.1.2, that all \(n\) together determine a class. A single \(n\) is a single point of \(\operatorname{Spec}R\), and Laumon states explicitly before 1.1.2 that one \(n\) does not determine a class. \(\square\)