Setup 1.1 · Setup: Weil Grothendieck groups over \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\)

Proof.

1. Simple Weil perverse sheaves. Let \((P,\Phi)\) be simple in \(\mathcal C^W\). The socle of \(P\) over \(k\) is carried to itself by \(\Phi\circ\mathrm{Fr}^*\), so it is a Weil subobject and equals \(P\): \(P\) is semisimple over \(k\). Under (R) every orbit \(\mathcal O_\mu\) is Frobenius-stable (its representative \(x_\mu\) is rational, IIb step 1), so \(\mathrm{Fr}^*\mathrm{IC}_\nu\simeq\mathrm{IC}_\nu\) with the canonical structure (eigenvalue \(1\) on the lowest stalk of the open stratum); under (C) the simple equivariant perverse sheaves are the \(\mathrm{IC}_\nu\) with constant coefficients (D0, D1). With \(E_\nu=\operatorname{Hom}(\mathrm{IC}_\nu,P)\), Schur's lemma (\(\operatorname{End}\mathrm{IC}_\nu=\overline{\mathbb Q}_\ell\), \(\operatorname{Hom}(\mathrm{IC}_\nu,\mathrm{IC}_{\nu'})=0\)) gives \(P\simeq\bigoplus_\nu\mathrm{IC}_\nu\otimes E_\nu\) with \(\Phi=\bigoplus_\nu\mathrm{can}_\nu\otimes A_\nu\) (as in IIb step 4). Simplicity forces a single \(\nu\) and \(E_\nu\) simple over \(\overline{\mathbb Q}_\ell[A_\nu^{\pm1}]\), hence one-dimensional: \(P=\mathrm{IC}_\nu(\alpha):=(\mathrm{IC}_\nu,\alpha\cdot\mathrm{can})\).

2. Finite length and \(K_0\). A chain of Weil subobjects is a chain of subobjects over \(k\), so every object of finite support has finite length. Jordan–Hölder gives \(K_0(\mathcal C^W)=\bigoplus_{\nu,\alpha}\mathbb Z[\mathrm{IC}_\nu(\alpha)]=\bigoplus_\nu R[\mathrm{IC}_\nu]\). A tensor product \(\mathrm{IC}_\nu\otimes(E,A)\) has a filtration by an \(A\)-stable flag with graded pieces \(\mathrm{IC}_\nu(\alpha)\), so its class is \([E,A][\mathrm{IC}_\nu]\). For \(\mathcal H^W\) the same argument runs on Schubert cells: for split \(G\) they are Frobenius-stable with connected stabilizers \(L^+G\cap t^\lambda L^+Gt^{-\lambda}\) (Zhu, §5.1).

3. The map \(\mathrm{act}\). Convolution with \(\mathrm{IC}_0\) is triangulated and commutes with \(\mathrm{Fr}^*\) (proper base change, D1). A short exact sequence of Weil perverse sheaves on \(\mathrm{Gr}\) therefore gives a Frobenius-compatible triangle on \(LX\) and a Frobenius-compatible long exact sequence of perverse cohomology, so \([F\ast\mathrm{IC}_0]\) is additive in \(F\). It is \(R\)-linear because \((F\otimes L_\alpha)\ast\mathrm{IC}_0\simeq(F\ast\mathrm{IC}_0)\otimes L_\alpha\) (projection formula; \(L_\alpha\) the rank-one Weil sheaf pulled back from the point). The formula for \(\mathrm{act}[V_\lambda]\) is the Frobenius-equivariant isotypic decomposition \({}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)\simeq\bigoplus_\nu\mathrm{IC}_\nu\otimes E_{\nu,i}(\lambda)\), \(\Phi=\bigoplus\mathrm{can}\otimes A_{\nu,i}(\lambda)\) (IIb step 4), combined with step 2. Column finiteness is the support bound of D0.

4. Traces. \(\mathrm{tr}_n\) is additive on short exact sequences and on perverse truncation triangles (Laumon (1.1.1.1)) and multiplicative under \(\otimes L_\alpha\) (Laumon (1.1.1.2)): \(\mathrm{tr}_n\mathrm{IC}_\nu(\alpha)=\alpha^n\,\mathrm{tr}_n\mathrm{IC}_\nu\). So \(\mathrm{tr}_n\) factors through \(-\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\) and sends the basis \([\mathrm{IC}_\nu]\) to the basis \(\mathrm{tr}_n\mathrm{IC}_\nu\) of \(C_{K_n}\otimes\overline{\mathbb Q}_\ell\) (IIb (i)); likewise \([V_\lambda]\mapsto\mathrm{tr}_nV_\lambda\), a basis of \(\mathcal H_{K_n}\otimes\overline{\mathbb Q}_\ell\) (IIa step 5). The square commutes on the \(R\)-basis \([V_\lambda]\) by IIa, \(\mathrm{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\mathrm{tr}_n(V_\lambda)\ast\varphi_0\), hence everywhere. Finally \(\mathrm{ev}_n(M_{\nu\lambda})=\sum_i(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)^n=m^{(n)}_{\nu\lambda}\) (IIb (ii)). \(\square\)

Remark. Chen–Yi, §8.2.1, work in the coarser ring \(\mathbb Z[\overline{\mathbb Q}_\ell^\times/\mu]\) (eigenvalues modulo roots of unity, Weil structures only for \(n\) sufficiently divisible); the difference matters, see Audit 1.9.