Setting, criterion and obstruction principle (Lemma A–Corollary D), the group case and the \(\mathrm{GL}_2\) examples: Iwahori pages. Left modules on \(X=G/H\); Shtotland's papers use right modules on \(H\backslash G\), and \(x\mapsto x^{-1}\) with the anti-involution \(f\mapsto f(g^{-1})\) of \(\mathcal H_I\) (which fixes every \(T_w\)) translates one into the other. \(F\) is a non-archimedean local field of residue characteristic \(\ne2\), \(\operatorname{char}F=0\) where a cited theorem needs it.
1 · Obstruction modules and witnesses
Spherical generation (SG) and Iwahori generation (IG) are statements about the basic function \(\varphi_0=\mathbf 1_{X(\mathcal O)}\); two quotient modules measure how IG can fail:
$$\text{SG}:\ \mathcal H_K\cdot\varphi_0=C_K,\qquad\text{IG}:\ \mathcal H_I\cdot\varphi_0=C_I,\qquad Q_X:=C_I\big/\mathcal H_I\cdot C_K,\qquad Q_{X,\varphi_0}:=C_I\big/\mathcal H_I\cdot\varphi_0,$$ $$0\to\frac{\mathcal H_I\cdot C_K}{\mathcal H_I\cdot\varphi_0}\to Q_{X,\varphi_0}\to Q_X\to0,\qquad e_KQ_X=0 .$$\(Q_{X,\varphi_0}=0\) is IG; \(Q_X=0\) says that \(C_K\) generates; under SG the two agree (Proposition B of the Iwahori pages). \(Q_X\ne0\) means that no \(K\)-invariant vector generates \(C_I\), whatever basic function is chosen. The rank-one note makes this distinction; it is adopted here.
Tool 1 · WitnessesProved
If an irreducible smooth \(\pi\) has \(\pi^I\ne0\), \(\pi^K=0\) and \(\operatorname{Hom}_G(\mathcal S(X),\pi)\ne0\), then \(Q_X\ne0\), so IG fails for every \(K\)-invariant basic function. For \(X=G/H\) with \(H\) reductive, the last condition says that \(\tilde\pi\) is \(H\)-distinguished.
Conversely, if \(C_I\) is finitely generated over \(\mathcal H_I\) and \(Q_X\ne0\), such a \(\pi\) exists, as a quotient of the \(G\)-subrepresentation of \(\mathcal S(X)\) generated by \(C_I\).
Proof
- Direct statement: Corollary D of the Iwahori pages gives the \(\mathcal H_I\)-linear map \(C_I\to\pi^I\), nonzero and vanishing on \(\varphi_0\); it vanishes on all of \(C_K\) because \(\pi^K=0\), so by Lemma C(2) it factors through \(Q_X\), which is therefore nonzero.
- Converse: a nonzero finitely generated module has a simple quotient; a simple quotient \(V\) of \(Q_X\) has \(e_KV=0\). By Borel's dictionary (Invent. Math. 35, Theorem 4.10 and Corollary 4.11) \(V=\pi^I\) for an irreducible \(\pi\) generated by its \(I\)-fixed vectors, \(\pi^K=e_K\pi^I=0\), and \(\operatorname{Hom}_{\mathcal H_I}(C_I,\pi^I)=\operatorname{Hom}_G(\langle C_I\rangle,\pi)\) where \(\langle C_I\rangle\subseteq\mathcal S(X)\) is generated by \(C_I\).
- Finite generation holds for every split symmetric \(X\): Shtotland, Iwahori Matsumoto presentation…, arXiv:2406.16070v3, Theorem 5.9 (\(G\) split, \(H=G^\sigma\), residue characteristic \(\ne2\); his \(\mathcal S(X)^I\) is the compactly supported module, Definition 2.37; Remark 5.10 derives it alternatively from Delorme and Aizenbud–Gourevitch–Sayag). So for the paper's symmetric families, failure of IG is always witnessed by a representation.
2 · Three sweeps
Each sweep names a representation that is Iwahori-spherical without \(K\)-fixed vectors, and a theorem deciding for which \(X\) its contragredient is distinguished. Through Tool 1, every hit is a failure of IG with \(Q_X\ne0\).
Tool 2 · The Steinberg sweepCited
Shtotland, Distinction of the Steinberg representation and the dual group of a symmetric space, arXiv:2501.01701v2. \(G\) split, \(\sigma\) an involution over \(\mathcal O\), \(H=G^\sigma\) split, \(X=G/H\). Let \(\omega:G(F)\to\Omega\) be the action on the colours of the vertices of the base alcove and \(\chi_0(g):=\operatorname{sgn}\omega(g)\) (Definition 1.4), an unramified quadratic character trivial on every parahoric. \(X\) is quasi-split if some Borel \(B\) has \(B\cap\sigma(B)\) a maximal torus (Definition 1.1); for the paper's root data this says that the Levi \(L=Z_G(S)\) of the minimal \(\sigma\)-split parabolic is a torus.
- Theorem 1.5 with \(\chi=\chi_0\), Remark 1.6: \(\mathrm{St}\otimes\chi_0\) is \(H\)-distinguished if and only if \(X\) is quasi-split and no \(\sigma\)-stable simple factor of \(G/Z\) is \(\mathrm{PGL}_{2n+1}\) with \(H\)-part \(P(\mathrm{GL}_n\times\mathrm{GL}_{n+1})\). Theorem 1.7 reads this as: the Steinberg parameter factors through the dual group \(\check G_X\).
- Quasi-splitness is necessary for any \(\mathrm{St}\otimes\chi\) to be distinguished (introduction); Proposition 1.3: for \(\mathrm{GL}_{2n+1}/(\mathrm{GL}_n\times\mathrm{GL}_{n+1})\), quasi-split, no \(\mathrm{St}\otimes\chi\) is distinguished.
- Proposition 2.7: \(\mathcal H_I\) acts on \(\mathrm{St}^I\) (one-dimensional) by \(T_s\mapsto-1\) for every affine simple \(s\) and \(T_o\mapsto\operatorname{sgn}(o)\) for \(o\in\Omega\); hence on \((\mathrm{St}\otimes\chi_0)^I\) by the affine sign character \(\varepsilon\) of the group case (\(T_o\mapsto1\)).
Tool 3 · Sweeps for \(\mathrm{GL}_n\)Cited
Matringe, arXiv:1210.4307, and Offen–Sayag. Write \(M_{p,q}=\mathrm{GL}_p\times\mathrm{GL}_q\subseteq\mathrm{GL}_n\).
- Theorem 3.1: a discrete series representation of \(\mathrm{GL}_n\) is \((M_{p,q},\mu)\)-distinguished for a character \(\mu\) only if \(n\) is even and \(p=q\). Theorem 3.2: a generic representation only if \(|p-q|\le1\).
- Theorem 6.1 (with Kewat's criterion, Proposition 6.1): \(\mathrm{St}_k(\rho)\) of \(\mathrm{GL}_{2m}\) with \(k\) even has a Shalika model iff \(L(s,\phi(\rho),\mathrm{Sym}^2)\) has a pole at \(s=0\); for \(\rho=\mathbf 1_{\mathrm{GL}_1}\), \(k=2m\), this is \(L(s,\mathbf 1)=(1-q^{-s})^{-1}\): \(\mathrm{St}_{2m}\) has a Shalika model, hence (p. 11) a linear period: it is \(M_{m,m}\)-distinguished.
- Offen–Sayag, J. Funct. Anal. 254 (2008), Theorem 1: Klyachko models are disjoint for every irreducible \(\pi\) of \(\mathrm{GL}_{2n}\); so a generic \(\pi\) is never \(\mathrm{Sp}_{2n}\)-distinguished. The Steinberg sweep is silent for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\) for two reasons.
- Matringe, p. 4, citing Prasad (Duke Math. J. 69, 1993) and Venketasubramanian (thesis): the \(\mathrm{GL}_3\)-modules \(\mathrm{St}_2\times\mathbf 1\) and \(\rho\times\mathbf 1\) (\(\rho\) cuspidal with trivial central character) are \(M_{2,1}\)-distinguished.
Tool 4 · Speh witnessesProved
\(\pi_n=U(\mathrm{St}_2,2)\times\mathbf 1_{\mathrm{GL}_2}^{\times(n-2)}\) of \(\mathrm{GL}_{2n}\), \(n\ge2\), \(\operatorname{char}F=0\): Iwahori-spherical, no \(K\)-fixed vector, \(\mathrm{Sp}_{2n}\)-distinguished and unitary (Proposition G of the Iwahori pages).
3 · Theorem A: the Steinberg sweep over the families
Theorem A · Quasi-split symmetric varietiesProved
Let \(X=G/H\) be split symmetric as in Tool 2. If \(X\) is quasi-split and has no exceptional factor \(\mathrm{PGL}_{2n+1}/P(\mathrm{GL}_n\times\mathrm{GL}_{n+1})\), then no \(K\)-invariant function generates \(C_I\):
$$\boxed{\ Q_X\ne0,\qquad\text{in particular}\qquad\mathcal H_I\cdot\varphi_0\subsetneq C_I .\ }$$The witness is \(\pi=\mathrm{St}\otimes\chi_0\), on whose \(I\)-fixed line \(\mathcal H_I\) acts by the affine sign character \(\varepsilon\).
Proof
- \(\chi_0\) is unramified and trivial on \(I\) and \(K\) (both are parahoric), so \(\pi^I=\mathrm{St}^I\ne0\) and \(\pi^K=\mathrm{St}^K=0\) (Borel 3.5; for \(\mathrm{GL}_2\) explicitly in 4). \(\tilde\pi=\widetilde{\mathrm{St}}\otimes\chi_0^{-1}=\mathrm{St}\otimes\chi_0=\pi\) is \(H\)-distinguished by Theorem 1.5 (Tool 2). Tool 1 applies, and Lemma C(2) of the Iwahori pages gives \(Q_X\ne0\), not only \(Q_{X,\varphi_0}\ne0\).
| \(X\) | \(L=Z_G(S)\) | quasi-split | Steinberg sweep |
|---|---|---|---|
| group \(H\), \(G=H\times H\), \(W\ne1\) | \(T_H\times T_H\) | yes | IG fails (also Proposition E) |
| \(\mathrm{GL}_{2p}/(\mathrm{GL}_p\times\mathrm{GL}_p)\) | \(\mathbb G_m^{2p}\) | yes | IG fails, \(Q_X\ne0\) |
| \(\mathrm{GL}_{2p+1}/(\mathrm{GL}_p\times\mathrm{GL}_{p+1})\) | \(\mathbb G_m^{2p}\times\mathrm{GL}_1\) | yes, exceptional | silent (Proposition 1.3) |
| \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(q\ge p+2\) | \(\mathbb G_m^{2p}\times\mathrm{GL}_{q-p}\) | no | silent |
| \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) | \(\mathrm{GL}_2^n\) (\(\cap\,\mathrm{SL}\)) | no | silent |
| \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\) | \(S\cdot\mathrm{Spin}_{2n-2}\) | only \(n=2\) (group \(\mathrm{SL}_2\)) | \(n=2\) fails; \(n\ge3\) silent |
| \(E_6/F_4\) | \(S\cdot\mathrm{Spin}_8\) | no | silent |
| \(\mathrm{SL}_2/T\), \(\mathrm{PGL}_2/T\), \(\mathrm{GL}_2/T\) | \(T\) | yes | IG fails, \(Q_X\ne0\) |
“Silent” means the Steinberg representation is not a witness; other witnesses may exist, and F2 finds them family by family. The quasi-split condition is the same for \(G\) and for its forms \(\mathrm{SL}\), \(\mathrm{PGL}\): it only sees \(L\).
4 · The Steinberg representation of \(\mathrm{GL}_2\), explicitly
The simplest module killed by \(e_K\), computed by hand: the witness of Tool 2 and Theorem A for \(\mathrm{GL}_2\), and of the examples \(\mathbb P^1\), \(\mathbb A^2\setminus\{0\}\) and \(\mathrm{GL}_2/T\) on the Iwahori pages. \(G=\mathrm{GL}_2(F)\) acts on \(\mathbb P^1(F)=\{[x:y]\}\) by fractional linear transformations, and on \(V:=C^\infty(\mathbb P^1(F))\) (locally constant; \(\mathbb P^1(F)\) is compact) by \((\pi(g)f)(x)=f(g^{-1}x)\). The constants are a subrepresentation:
$$\boxed{\ \mathrm{St}:=C^\infty(\mathbb P^1(F))\big/\mathbb C\mathbf 1\ },\qquad\mathrm{St}^K=0,\qquad\dim\mathrm{St}^I=1,\qquad T_s=-1\ \text{ on }\mathrm{St}^I .$$Proof
- \(K\) is transitive on \(\mathbb P^1(F)\). Scale \((x,y)\) to a primitive vector (\(x,y\in\mathcal O\), not both in \(\varpi\mathcal O\)); it completes to a basis of \(\mathcal O^2\), so some \(k\in K\) sends \((1,0)\) to it. Hence \(V^K=\mathbb C\mathbf 1\), and \(\mathrm{St}^K=V^K/\mathbb C\mathbf 1=0\) because \(K\)-invariants are exact.
- Two \(I\)-orbits. \(I=\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in K:\ c\in\varpi\mathcal O\}\). Reduction \(\mathbb P^1(F)\to\mathbb P^1(\mathbb F_q)\) is \(I\)-equivariant onto the action of \(B(\mathbb F_q)\), whose orbits are \(\{[1:0]\}\) and \(\{[x:1]\}\); the first congruence subgroup acts transitively on each fibre (for primitive \(v\), \(k\mapsto kv\) maps \(1+\varpi M_2(\mathcal O)\) onto \(v+\varpi\mathcal O^2\)). So the \(I\)-orbits are \(U_1=\{[1:y]:y\in\varpi\mathcal O\}\) and \(U_2=\{[x:1]:x\in\mathcal O\}\), \(V^I=\mathbb C\mathbf 1_{U_1}\oplus\mathbb C\mathbf 1_{U_2}\), \(\mathbf 1_{U_1}+\mathbf 1_{U_2}=\mathbf 1\), and \(\mathrm{St}^I=\mathbb C[\mathbf 1_{U_1}]\).
- Sign. \(W=\{1,s\}\), \(\mathbf 1_K=T_e+T_s\), \([K:I]=q+1\), \(e_K=(T_e+T_s)/(q+1)\), and \((T_s+1)(T_s-q)=0\). On \(V^I\), \(T_s\) has the eigenvalue \(q\) on the \(K\)-fixed line \(\mathbb C\mathbf 1\) and \(-1\) on a complement; \(e_K\) is zero on \(\mathrm{St}^I\) (its image is \(K\)-fixed), so \(T_s=-1\) there: the sign character of the group case for \(H=\mathrm{GL}_2\).
\(K\)-averaging forgets the difference between the two \(I\)-orbits; \(\mathrm{St}\) is where that difference survives after the constants are removed. (\(\mathrm{St}\) is irreducible, a classical theorem not needed here.)
Literature, as verifiedShtotland (two papers) · Broussous · Matringe · Offen–Sayag, Godement–Jacquet, Borel
- Shtotland, Iwahori Matsumoto presentation for modules of Iwahori fixed functions on symmetric spaces, arXiv:2406.16070v3 (IMRN 2026, rnag042). Checked in the text: the setting (residue characteristic \(\ne2\), \(G\) split, \(H=G^\sigma\), right action, Definition 2.37); Theorems 1.1, 3.4, 5.3, 5.9 (finite generation), 7.13, 8.5; Propositions 9.1–9.2 (\(T\backslash\mathrm{SL}_2\): \(\mathcal S(X)^I\simeq\mathbb C_{\mathrm{sgn}}\oplus\mathcal H/\mathcal H(T_{s_0}+1)\oplus\mathcal H/\mathcal H(T_{s_1}+1)\oplus\mathcal H\), orbits \(z_0,z_1,x_w,y_w\)) and 9.3 (\(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\)).
- Shtotland, Distinction of the Steinberg representation and the dual group of a symmetric space, arXiv:2501.01701v2. Checked: Definition 1.1 (quasi-split), Theorem 1.2 (finite field), Proposition 1.3, Definition 1.4 (\(\chi_0\)), Theorem 1.5, Remark 1.6, Theorem 1.7 (dual group), Proposition 1.11 (\(\dim\operatorname{Hom}_H(\mathrm{St}_{\chi_0},\mathbb C)\)), Proposition 2.7 (sign action, after Broussous §3.3). Its Galois-case references (Prasad, Matringe, Chommaux, Broussous, Courtès) concern non-split groups.
- Broussous, Orbits of a symmetric subgroup in the Bruhat–Tits building, arXiv:2406.17459: abstract checked; the Theorem 3.6 quoted in the note is not.
- Matringe, Linear and Shalika local periods for the mirabolic group, and some consequences, arXiv:1210.4307 (J. Number Theory 2014). Checked: Theorems 3.1, 3.2, 5.1, 6.1; Proposition 6.1 (Kewat); the remark on \(\mathrm{St}_2\times\mathbf 1\) (Prasad, Duke Math. J. 69 (1993); Venketasubramanian's thesis); “Shalika implies linear”. His arXiv:1602.05101 is the Galois case and does not concern \(\mathrm{GL}_p\times\mathrm{GL}_q\).
- Offen–Sayag, J. Funct. Anal. 254 (2008), Theorem 1 (Klyachko models: disjointness and uniqueness); the symplectic witnesses are in the Iwahori literature. Godement–Jacquet, Zeta functions of simple algebras, LNM 260, §3 (used on F2, Proposition E). Borel, Invent. Math. 35 (1976), 3.5, 4.10, 4.11. Kneser: \(H^1(F,\mathrm{Sp}_{2n})=1\).