Proposition 3.2 · Thick closure versus span: twisted traces, restriction of class functions, fusion

Proof of (a).

1. By D2, over \(k\), \(V:=V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu,i}\mathrm{IC}_\nu[-i]\otimes E_{\nu,i}(\lambda)\). By D1 the Weil structure \(\Phi\) preserves the perverse truncations, so in this decomposition it is block-triangular for the perverse degree, and on \({}^p\mathcal H^i(V)\) it is \(\bigoplus_\nu\Phi^{\mathrm{can}}_\nu\otimes A_{\nu,i}(\lambda)\) (IIb, step 4).
2. Fix \(\nu\in\mathcal M\) and \(i\) with \(E=E_{\nu,i}(\lambda)\ne0\), and let \(e=\mathrm{id}_{\mathrm{IC}_\nu[-i]}\otimes\varepsilon\) on that summand and \(0\) on the others, \(\varepsilon\in\operatorname{End}(E)\). For \(x\in X(F)\) the point is Frobenius-fixed, so \((\mathrm{Fr}^*e)_x=e_x\) and \(t_{\lambda,e}(x)=\sum_j(-1)^j\operatorname{tr}(e_x\Phi_x\mid\mathcal H^j)\) by cyclicity of the trace.
3. \(e_x\Phi_x\) is the compression of \(\Phi_x\) to one summand; by step 1 only the diagonal block \((\Phi^{\mathrm{can}}_\nu)_x\otimes A_{\nu,i}(\lambda)\) survives. Hence \(t_{\lambda,e}=(-1)^i\operatorname{tr}(\varepsilon A_{\nu,i}(\lambda))\operatorname{tr}\mathrm{IC}_\nu\). With \(\varepsilon=A_{\nu,i}(\lambda)^{-1}E_{11}\) the coefficient is \(\pm1\), so \(\operatorname{tr}\mathrm{IC}_\nu\) lies in the span.
4. \(t_{\lambda,\mathrm{id}}=\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}(V_\lambda)\ast\varphi_0\), and these span \(\mathcal H_K\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\) (IIa). \(\square\)

Proof of (b). \(\overline{\mathbb Q}_\ell[\check G]\to\overline{\mathbb Q}_\ell[\check G_X]\) is onto (closed subgroup), and \(\overline{\mathbb Q}_\ell[\check G_X]\) contains every irreducible \(W\) of \(\check G_X\) (matrix coefficients). So \(W\) is a quotient, hence (complete reducibility) a summand, of the restriction of a finite-dimensional \(\check G\)-subrepresentation of \(\overline{\mathbb Q}_\ell[\check G]\). On \(K_0\otimes\overline{\mathbb Q}_\ell\), classes are characters and \(\operatorname{Res}\) is restriction of functions. If \(s,s'\in\check G_X\) are semisimple, \(\check G\)-conjugate and not \(\check G_X\)-conjugate, then every restricted character takes equal values at \(s,s'\), while characters of \(\check G_X\) separate semisimple classes; so some \([W]\) is not in the image. \(\square\)

Proof of (c). (i) By IIb (iii), (SG) is the spanning of the columns of \(m^{(1)}\); a nonzero \(\psi\) vanishing on all columns prevents it. (ii) is Lemma 3.1(a). (iii) In the perverse case only \(i=0\) occurs and \(E_{\nu,0}(\lambda)\) is the top cohomology of the fibre over \(x_\nu\), on which \(\mathrm{Fr}^n\) acts by \(q^{nf_{\nu\lambda}}\) on each rational component, \(f_{\nu\lambda}=\tfrac12(\langle2\rho,\lambda\rangle-\delta(\nu))\). So \(m^{(n)}=D_n\,c\,D'_n\) with invertible diagonal \(D_n=\operatorname{diag}(q^{-n\delta(\nu)/2})\), \(D_n'=\operatorname{diag}(q^{n\langle\rho,\lambda\rangle})\), and the columns of \(m^{(n)}\) span iff those of \(c\) do; by (b) they do not. Every row of a branching matrix of a subgroup is nonzero, so \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) by Proposition I. \(\square\)

The mechanism in words. \(\operatorname{Tr}(\mathrm{Fr},-)\) is functorial for the module functor \(V\mapsto V\ast\mathrm{IC}_0\) and turns it into \(f\mapsto f\ast\varphi_0\) (IIa). Thick generation adds retracts; the trace of a Karoubi-generating functor need not be onto, because the trace of the target receives Frobenius-twisted traces of idempotents \(e\) of \(V_\lambda\ast\mathrm{IC}_0\) that do not come from \(\operatorname{End}(V_\lambda)=\overline{\mathbb Q}_\ell\). Laumon's injectivity (Thm 1.1.2) concerns classes over all \(n\) and does not help: in case (iii) the failure is the same for every \(n\). Corollary 3.9 realizes case (iii) for Sakellaridis's \(X\).