Proposition 5.7 · (II$_I$) (IG) over $\mathbb F_q((t))$ puts every simple object in $\mathcal M_I$

(II$_I$) (IG) over $\mathbb F_q((t))$ puts every simple object in $\mathcal M_I$

Proof.

1. Coefficients. $C_I$ has a $\mathbb Z$-basis of orbit indicators and the $T_w\ast\varphi_0$ are $\mathbb Z$-valued; a rational vector lies in the $\mathbb C$-span of finitely many rational vectors iff it lies in their $\mathbb Q$-span. So (IG) over $\mathbb C$ is equivalent to (IG) over $\overline{\mathbb Q}_\ell$.

2. The IC basis of $\mathcal H_I$. $\mathrm{IC}w|{IwI/I}=\overline{\mathbb Q}\ell[\ell(w)]$ and $\operatorname{supp}\mathrm{IC}_w=\bigsqcup{y\le w}IyI/I$, so $\operatorname{tr}\mathrm{IC}w=(-1)^{\ell(w)}T_w+\sum{y<w}c_{y,w}T_y$ (Lang's theorem for the connected group $I$ identifies the $\mathbb F_q$-points of the cells with the double cosets). By unitriangularity, $\operatorname{span}\{\operatorname{tr}\mathrm{IC}w\}=\mathcal H_I\otimes\overline{\mathbb Q}\ell$, and by (Tr)

3. Trace of a generator. By (P) and BBD 5.4.5, over $k$ $\mathrm{IC}w\ast\mathrm{IC}_0\simeq\bigoplus_iP_i[-i]$, $P_i={}^pH^i$ pure perverse; by BBD 5.3.8 each $P_i\simeq\bigoplus_SS\otimes E{S,i}$ over $k$. Frobenius maps the $S$-isotypic part to the $\mathrm{Fr}^*S$-isotypic part. For $\mathrm{Fr}^*S\simeq S$ it acts on $E_{S,i}=\operatorname{Hom}(S_0,P_i)$ by some $A_{S,i}$ and contributes $\operatorname{tr}(A_{S,i})\operatorname{tr}(S_0)$; a Frobenius orbit of length $>1$ is permuted off the diagonal and contributes $0$ at every $\mathbb F_q$-point. Hence

and $m_{S,w}\ne0$ only if $S$ is a shifted summand of $\mathrm{IC}_w\ast\mathrm{IC}_0$, i.e. $S\in\mathcal M_I$.

4. Conclusion. Under (IG), steps 1–3 give $C_I\otimes\overline{\mathbb Q}\ell\subseteq\operatorname{span}\{\operatorname{tr}S_0:S\in\mathcal M_I\text{ Frobenius-stable}\}$. A Frobenius-stable simple $S'$ is pulled back from a finite level and supported on some $Y_N$, so $\operatorname{tr}(S'_0)\in C_I\otimes\overline{\mathbb Q}\ell$; by (Lin) it can lie in that span only if $S'\in\mathcal M_I$. Under (W) all simples are Frobenius-stable, so $\mathcal M_I=\operatorname{Simple}(\mathcal P_I)$; (P) gives (DT), and Proposition 5.4(c) gives the box. A simple object that is not Frobenius-stable becomes stable over some $\mathbb F_{q^r}$ (finitely many simples per level), and the same argument applies over $\mathbb F_{q^r}((t))$.

5. Cancellation (checklist 6). Step 4 needs only the containment of spans. Cancellation inside $m_{S,w}$ can only make $\operatorname{span}\{\operatorname{tr}(\mathrm{IC}w\ast\mathrm{IC}_0)\}$ smaller, and (IG) says it is everything. The converse direction (sheaves ⇒ functions) is the one exposed: a summand $S\in\mathcal M_I$ with $m{S,w}=0$ for all $w$ would leave $\operatorname{tr}S_0$ outside the span, and so would generation that needs retracts.

Sufficient conditions. (W) and (Lin) hold when every $H_\alpha$-orbit on $Y_\alpha$ is defined over $\mathbb F_q$ with connected stabilizer: by Lang's theorem $o(\mathbb F_q)$ is a single nonempty $H_\alpha(\mathbb F_q)$-orbit, $\operatorname{tr}\mathrm{IC}(\bar o)0=\pm\mathbf 1{o}+(\text{terms on smaller orbits})$ at a common level, and unitriangularity gives independence. (P) holds when the convolution has the proper finite-type model of Proposition 5.4 (BBD 5.1.14 and Gabber's purity of $\mathrm{IC}_w$). (Tr) is proper base change plus the Grothendieck–Lefschetz trace formula on that model, with $H^1(\mathbb F_q,I)=1$. Field. The conclusion is over $\overline{\mathbb F}_q$; reaching the Betti category over $\mathbb C$ needs a separate comparison of summand statements (spreading out, BBD §6.1), not supplied here.