Proposition 1.6 · Forcing criteria: vanishing eigenfunctions, multiplicity two, extra endomorphisms

Proofs.

1. (S), first half. \(\Lambda(f\ast\varphi_0)=\chi(f)\Lambda(\varphi_0)=0\) for all \(f\), so \(\Lambda\) vanishes on \(\mathcal H_K\ast\varphi_0\) but not on \(C_K\).

2. (S), converse. \(\mathcal H_K\) is a finitely generated commutative \(\mathbb C\)-algebra (Satake), hence Noetherian. If \(Q=C_K/\mathcal H_K\ast\varphi_0\neq0\) is finitely generated, pick a maximal ideal \(\mathfrak m\) with \(Q_{\mathfrak m}\neq0\); by Nakayama \(Q/\mathfrak mQ=Q_{\mathfrak m}/\mathfrak mQ_{\mathfrak m}\neq0\), and \(\mathcal H_K/\mathfrak m=\mathbb C\) (Nullstellensatz). A nonzero linear form on \(Q/\mathfrak mQ\) composed with \(C_K\to Q\) is the required \(\Lambda\), with \(\chi=\chi_{\mathfrak m}\).

3. (d). Under (SG), \(\Lambda\mapsto\Lambda(\varphi_0)\) is injective on \(\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\), so the dimension is at most 1. For \(\pi\) irreducible unramified with \(\pi^K=\mathbb Cv_0\) and character \(\chi_\pi\): restriction \(\ell\mapsto\ell|_{C_K}\) embeds \(\operatorname{Hom}_G(\mathcal S(X(F)),\pi)\) into \(\operatorname{Hom}_{\mathcal H_K}(C_K,\pi^K)\), because \(\ell(C_K)=\ell(e_K\mathcal S)=e_K\ell(\mathcal S)=\pi^K\neq0\) when \(\ell\neq0\) (\(e_K\) the projector to \(K\)-invariants; \(\ell(\mathcal S)=\pi\) by irreducibility). For \(A\in\operatorname{Hom}_G(\pi,C^\infty(X(F)))\), \(\Omega=A(v_0)\) determines \(A\), is \(K\)-invariant and satisfies \(f\ast\Omega=\chi_\pi(f)\Omega\); with an invariant measure, \(\Lambda_\Omega(\varphi)=\int_X\varphi\,\Omega\) satisfies \(\Lambda_\Omega(f\ast\varphi)=\Lambda_{f^\vee\ast\Omega}(\varphi)=\chi_\pi(f^\vee)\Lambda_\Omega(\varphi)\) (\(f^\vee(g)=f(g^{-1})\); change of variables), and \(\Omega\mapsto\Lambda_\Omega\) is injective (test on orbit indicators). Hence the bound, and \(\Lambda_\Omega(\varphi_0)=\int_{X(\mathcal O)}\Omega\neq0\) by (S).

4. (b). If \(C_K=\mathcal H_K\ast\varphi_0\cong\mathcal H_K/\mathrm{Ann}(\varphi_0)\), an \(\mathcal H_K\)-endomorphism \(D\) is determined by \(D(\varphi_0)=f_D\ast\varphi_0\), and \(D=f_D\ast-\). For the sheaf-level source: \(\sigma(V_\lambda\ast\mathrm{IC}_0)\cong V_\lambda\ast\sigma(\mathrm{IC}_0)\cong V_\lambda\ast\mathrm{IC}_0\) as Weil objects, and \(\sigma\) maps \({}^p\mathcal H^i\) to \({}^p\mathcal H^i\) and the \(\mathrm{IC}_\nu\)-isotypic part to the \(\mathrm{IC}_{s(\nu)}\)-isotypic part. If \(\sigma\) carries the canonical structure of \(\mathrm{IC}_\nu\) to \(c_\nu\) times that of \(\mathrm{IC}_{s(\nu)}\), then \([E_{s(\nu),i}(\lambda)]=[c_\nu][E_{\nu,i}(\lambda)]\) in \(R\), so \(M_{s(\nu),\lambda}=[c_\nu]M_{\nu,\lambda}\) for all \(\lambda\), and the functional \(e^*_{s(\nu)}-c_\nu^ne^*_\nu\) kills every column of \(m^{(n)}\). Its trace on \(C_{K_n}\) is an \(\mathcal H_{K_n}\)-automorphism fixing \(\varphi_0\) and moving \(\mathrm{tr}_n\mathrm{IC}_\nu\).

5. (c). The algebraic dual of \(\bigoplus_\nu\overline{\mathbb Q}_\ell\) is \(\prod_\nu\overline{\mathbb Q}_\ell\); the columns fail to span iff some nonzero functional kills them all. Each column has finite support, so each sum is finite. The finite-block version: extend by zero a nonzero \(c\in W^\perp\subset(\overline{\mathbb Q}_\ell^N)^*\).

6. Examples. In Counterexample 1.5 the eigenspaces are 2-dimensional, \(\delta_j\mapsto(-1)^j\delta_j\) is a nontrivial \(\mathcal H_K\)-automorphism fixing \(\varphi_0\), and \(c=\mathbf 1_{\text{odd}}\) is an infinite row relation. For \(\mathrm{SL}_2/T\) outside (C), Proposition J(2) of the classical page is the automorphism \(\sigma\) of (b), induced by \(\mathrm{diag}(u,1)\) with \(u\) a non-square unit.

7. Relation to Sakellaridis. His Cor. 8.0.4 (char 0, under (A1)–(A5)) derives principal generation, multiplicity one and good test vectors from surjectivity of \(r\), and his remark after it notes multiplicity \(>1\) at glued points; his Cor. 8.0.3 identifies the geometric endomorphisms with \(\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), larger than the image of \(\mathcal H_K\) when \(r\) is not onto. Here the implications from (SG) to multiplicity one and good test vectors are proved for every \(X\) and every \(F\), with no hypothesis. \(\square\)