Family by family

Proof of (i), (ii)

• (i) \(\mathrm{St}_{2p}^I\ne0=\mathrm{St}_{2p}^K\), \(\widetilde{\mathrm{St}}=\mathrm{St}\), and \(\mathrm{St}_{2p}\) is \(M_{p,p}\)-distinguished by Tool 3 (Theorem 6.1 with \(k=2p\), \(\rho=\mathbf 1\); Shalika implies linear). Tool 1.
• (ii) \(\pi=\operatorname{Ind}_{P_{2,1}}^{\mathrm{GL}_3}(\mathrm{St}_2\otimes\mathbf 1)\) (normalized) is irreducible (the segments \(\{\nu^{-1/2},\nu^{1/2}\}\) and \(\{\nu^0\}\) are not linked), a subquotient of \(\nu^{1/2}\times\nu^{-1/2}\times\mathbf 1\), so \(\pi^I\ne0\); \(\pi^K\simeq(\mathrm{St}_2\otimes\mathbf 1)^{M(\mathcal O)}=0\) by \(G=PK\); \(\tilde\pi=\widetilde{\mathrm{St}}_2\times\mathbf 1=\pi\); \(M_{2,1}\)-distinguished by Tool 3 (Prasad, Venketasubramanian). Tool 1.

Proof (restriction of the Speh witness, no irreducibility needed)

• \(X(F)=\mathrm{SL}_{2n}(F)/\mathrm{Sp}_{2n}(F)\) since \(H^1(F,\mathrm{Sp}_{2n})=1\) (Kneser). Let \(\pi_n\) be the Speh witness of Tool 4 for \(\mathrm{GL}_{2n}\) and \(\pi:=\pi_n|_{\mathrm{SL}_{2n}(F)}\), admissible of finite length. The \(\mathrm{Sp}_{2n}\)-invariant functional on \(\tilde\pi_n\) restricts to one on \(\tilde\pi\), so by Frobenius reciprocity (\(\mathrm{Sp}_{2n}\) reductive, \(\pi\) admissible) there is a nonzero \(\mathrm{SL}_{2n}(F)\)-map \(f:\mathcal S(X)\to\pi\); let \(\pi'\ne0\) be its image.
• \(\pi'^{I'}\ne0\) for \(I'=I\cap\mathrm{SL}_{2n}\): every irreducible constituent of \(\pi\) is a subquotient of the restriction to \(\mathrm{SL}_{2n}\) of an unramified principal series of \(\mathrm{GL}_{2n}\), which is an unramified principal series of \(\mathrm{SL}_{2n}\), so by Borel's theorem it has nonzero \(I'\)-fixed vectors; in particular the subrepresentation \(\pi'\) does. And \(f(C_{I'})=f(e_{I'}\mathcal S(X))=e_{I'}\pi'=\pi'^{I'}\ne0\).
• \(\pi^{K'}=0\) for \(K'=\mathrm{SL}_{2n}(\mathcal O)\): \(K/K'\simeq\mathcal O^\times\) via \(\det\) acts on the finite-dimensional \(\pi_n^{K'}\); a nonzero \(v\) with \(kv=\chi(\det k)v\) (\(\chi\) a character of \(\mathcal O^\times\), extended to \(\tilde\chi\) of \(F^\times\)) is a \(K\)-fixed vector of \(\pi_n\otimes\tilde\chi^{-1}\circ\det\), which is then a subquotient of an unramified principal series; but it is also a subquotient of \(\chi_1\tilde\chi^{-1}\times\dots\times\chi_{2n}\tilde\chi^{-1}\) with all \(\chi_i\) unramified, and the cuspidal support is unique up to permutation, so \(\tilde\chi\) is unramified, \(\chi=1\), and \(v\in\pi_n^K=0\).
• So \(f|_{C_{I'}}:C_{I'}\to\pi'^{I'}\) is a nonzero \(\mathcal H_{I'}\)-map with \(f(\varphi_0)\in\pi'^{K'}=0\): Lemma C(1) of the Iwahori pages.

Proof of the orbit counts and of SG

• \(K\backslash G(F)/T(F)\) is the set of \(T(F)\)-orbits on vertices. For \(\mathrm{PGL}_2\): \(T(F)/T(\mathcal O)\simeq\mathbb Z\) acts on \(A_T\) by all translations; \(T(\mathcal O)=\mathcal O^\times\) fixes \(A_T\) and acts on the vertices at distance \(d\) from \(A_T\) above a fixed apartment vertex, which are the \([x:1]\in\mathbb P^1(\mathcal O/\varpi^d)\) with \(x\) a unit, by \(x\mapsto ux\): transitively. One orbit per \(d\). The Hecke operator \(T_1=\mathbf 1_{K\operatorname{diag}(\varpi,1)K}\) on functions of \(d\) is \((T_1f)(0)=2f(0)+(q-1)f(1)\), \((T_1f)(d)=f(d-1)+qf(d+1)\) for \(d\ge1\); so \(\mathbf 1_{\{d\}}\in\mathbb C[T_1]\cdot\mathbf 1_{\{0\}}\) by induction and SG holds. (The central part of \(\mathcal H_K(\mathrm{GL}_2)\) acts trivially on \(X\).)
• For \(\mathrm{SL}_2\): \(T(F)=\{\operatorname{diag}(a,a^{-1})\}\) translates \(A_T\) by \(2\operatorname{val}a\), leaving two orbits of apartment vertices (\(k\) even, \(k\) odd), and \(T(\mathcal O)\) acts on the units \(x\) by \(x\mapsto u^2x\), with two orbits on \((\mathcal O/\varpi^d)^\times\) for \(q\) odd and every \(d\ge1\). \(X(\mathcal O)\) is the orbit \((0,0)\). Two independent reasons for the failure of SG. Parity: \(\mathrm{SL}_2(F)\) preserves the parity of vertices, so \(Kt^\lambda K\cdot p\) for an even apartment vertex \(p\) is a sphere of even radius \(2a\), whose vertices satisfy \(|k-k_p|+d=2a\); every Hecke translate of \(\varphi_0\) is supported on \(\{k\equiv d\bmod2\}\), and the orbit \((d,k)=(0,1)\) is never reached. Square classes: \(\operatorname{diag}(u,1)\in\mathrm{GL}_2(\mathcal O)\), \(u\) a non-square unit, normalizes \(K\), \(T\) and every \(Kt^\lambda K\) and fixes \(\varphi_0\), so every element of \(\mathcal H_K\cdot\varphi_0\) is \(\operatorname{diag}(u,1)\)-invariant, while \(\operatorname{diag}(u,1)\) exchanges \((d,k,+)\) and \((d,k,-)\) for \(d\ge1\).
• Dictionary with the paper: over \(\mathbb C\) there are no square classes and \(T(\mathcal O)\) is transitive on each branch, so the \(K\)-orbits on \(LX\) are \((d,k\bmod2)\leftrightarrow m=2d+(k\bmod2)\in\Lambda_S^+=\mathbb Z_{\ge0}\); the orbits with \(m\) odd are the ones the paper's \(\varphi(\Lambda_T^+)=2\mathbb Z\) misses (“misses \(\check\alpha\)”), and the parity argument above is the function-level form of that failure. The square-class splitting is a residue-field phenomenon with no sheaf counterpart.

• SG. \(K\)-orbits on \(M_n(F)\) are the elementary-divisor classes, including singular ones; \(C_K\) is spanned by their indicators. The function \(\mathbf 1_{K t^\lambda K}\ast\varphi_0\) (\(\lambda\) dominant) is supported on \(Kt^\lambda K\cdot M_n(\mathcal O)=\{x:\text{elementary divisors}\ \ge\lambda\ \text{coordinatewise}\}\), a closed union of orbits with \(t^\lambda\) on top, and is nonzero on \(Kt^\lambda K\) (constant there): by induction along the dominance order, every orbit indicator is reached. (This is the function-level shadow of the paper's \(M_n\) example.)
• IG. Godement–Jacquet (LNM 260, §3): for \(\Phi\in\mathcal S(M_n(F))\) and a matrix coefficient \(c_{v,\tilde v}(g)=\langle\pi(g)v,\tilde v\rangle\) of an irreducible \(\pi\), \(Z(s,\Phi,c)=\int_{\mathrm{GL}_n(F)}\Phi(g)c(g)|\det g|^{s+(n-1)/2}dg\) converges for \(\operatorname{Re}s\gg0\), \(Z/L(s,\pi)\in\mathbb C[q^{\pm s}]\), and these quotients span \(\mathbb C[q^{\pm s}]\) (the \(L\)-factor generates the ideal of the zeta integrals). The substitution \(g=g_1hg_2^{-1}\) gives \(Z(s,(g_1,g_2)\Phi,c_{v,\tilde v})=|\det g_1|^{s'}|\det g_2|^{-s'}Z(s,\Phi,c_{\pi(g_2^{-1})v,\tilde\pi(g_1^{-1})\tilde v})\), \(s'=s+(n-1)/2\). At \(s'=0\) the bilinear form \(B(\Phi,v\otimes\tilde v):=(Z/L)|_{s'=0}\) is nonzero and invariant for \(G\) acting on \(\tilde\pi\boxtimes\pi\) by \(\tilde\pi(g_1)\otimes\pi(g_2)\); so \(\Phi\mapsto B(\Phi,\cdot)\) is a nonzero \(G\)-map \(\mathcal S(M_n)\to\pi\boxtimes\tilde\pi\).
• Take \(\pi=\mathrm{St}_n\): the target \(\mathrm{St}_n\boxtimes\mathrm{St}_n\) is irreducible, \(I\)-fixed line \(\mathrm{St}_n^{I_n}\otimes\mathrm{St}_n^{I_n}\), no \(K\)-fixed vector. Tool 1. In the language of the group case: the sign character \(\varepsilon\otimes\varepsilon\) of \(\mathcal H_{I_n}\otimes\mathcal H_{I_n}^{\mathrm{op}}=\mathcal S(\mathrm{GL}_n)^{I}\) extends along \(\mathcal S(\mathrm{GL}_n)^I\subseteq\mathcal S(M_n)^I\) through the zeta integral, whereas the Steinberg functional of \(\mathbb A^2\setminus0\) does not extend over the origin (𝔸ⁿ, why the origin matters).