Proof workspace · Iwahori level · functions

The classical analogue

Explained slowly, with the spherical and Iwahori levels side by side →

Archived 2026-10-19 — the maintained version is the Iwahori level of the basic-function pages. This copy is kept unchanged for the links of earlier runs; its \(E_6/F_4\) row (“open”) is superseded: (SG) fails there, so (IG) fails.

Conventions: \(X=G/H\), left actions, left \(\mathcal H_I\)-modules.

1 · Setting

\(F\) a non-archimedean local field, \(\mathcal O\), \(\varpi\), \(\mathbb F_q\); \(\operatorname{char}F=0\) where a cited theorem needs it (5). \(G\supseteq B\) split connected reductive over \(\mathcal O\), \(X\) a spherical \(G\)-variety over \(\mathcal O\) (the paper takes \(X\) smooth affine; 4 notes where an example leaves that class). Coefficients \(\mathbb C\).

$$K=G(\mathcal O)\xrightarrow{\ \mathrm{ev}\ }G(\mathbb F_q),\qquad I:=\mathrm{ev}^{-1}(B(\mathbb F_q)),\qquad\mathcal S(X):=C_c^\infty(X(F)),\qquad C_K:=\mathcal S(X)^K\ \subseteq\ C_I:=\mathcal S(X)^I .$$

Hecke algebras, the Iwahori–spherical bimodule, the basic function and the Iwahori–Matsumoto basis:

$$\mathcal H_I:=C_c(I\backslash G(F)/I),\qquad\mathcal H_K:=C_c(K\backslash G(F)/K),\qquad C_c(I\backslash G(F)/K),\qquad\varphi_0:=\mathbf 1_{X(\mathcal O)}\in C_K,\qquad T_w:=\mathbf 1_{IwI}\ (w\in\widetilde W),\qquad\mathbf 1_K=\sum_{w\in W}T_w .$$

Convolution. Haar measure on \(G(F)\) with \(\operatorname{vol}(I)=1\). For \(f\in\mathcal H_I\) and \(\varphi\in C_I\),

$$(f\ast\varphi)(x):=\int_{G(F)}f(g)\,\varphi(g^{-1}x)\,dg=\sum_{g\in G(F)/I}f(g)\,\varphi(g^{-1}x).$$

The same formula with \(\operatorname{vol}(K)=1\), summed over \(G(F)/K\), is written \(\ast_K\); an unadorned \(\ast\) is the Iwahori one, \(\ast_I\). The two measures differ by the factor \([K:I]\): \(f\ast_Ig=[K:I]\,f\ast_Kg\) whenever \(f\) is right \(K\)-invariant and \(g\) left \(K\)-invariant, since the \([K:I]\) cosets \(hI\) inside one \(hK\) contribute equal terms.

Well defined; \(C_I\) is a unital left \(\mathcal H_I\)-module
  • The summand depends only on \(gI\), and finitely many cosets contribute. \((f\ast\varphi)(ix)=(f\ast\varphi)(x)\) for \(i\in I\) by the substitution \(g\mapsto i^{-1}g\). \(\operatorname{supp}(f\ast\varphi)\subseteq\operatorname{supp}f\cdot\operatorname{supp}\varphi\).
  • The same formula makes \(\mathcal H_I\) an associative algebra with unit \(T_e=\mathbf 1_I\), and \((f_1\ast f_2)\ast\varphi=f_1\ast(f_2\ast\varphi)\). With \(\operatorname{vol}(K)=1\) instead, \(C_K\) is a unital \(\mathcal H_K\)-module with unit \(\mathbf 1_K\). Hecke algebras of compact open subgroups and their modules: Cartier, §1.3–1.4; the Iwahori–Hecke algebra and its basis: Haines–Kottwitz–Prasad, §1.2–1.3.
  • For \(f\in\mathcal H_I\) the operator \(\varphi\mapsto f\ast\varphi\) on \(\mathcal S(X)\) is \(\pi(f)=\int f(g)\pi(g)\,dg\) for the \(G(F)\)-representation \((\pi(g)\varphi)(x)=\varphi(g^{-1}x)\); so every \(G(F)\)-map \(\mathcal S(X)\to V\) restricts to an \(\mathcal H_I\)-map \(C_I\to V^I\).

Lemma (the spherical idempotent). Put

$$e_K:=[K:I]^{-1}\mathbf 1_K=[K:I]^{-1}\sum_{w\in W}T_w\in\mathcal H_I,\qquad[K:I]=|G(\mathbb F_q)/B(\mathbb F_q)|=\sum_{w\in W}q^{\ell(w)} .$$
  1. \(e_K\ast e_K=e_K\). For \(f\in\mathcal H_I\), \(f\ast e_K\) is the average of \(f\) over the coset \(xK\); for \(\varphi\in C_I\), \(e_K\ast\varphi\) is the average of \(\varphi\) over the \(K\)-translates of \(x\): $$(f\ast e_K)(x)=\frac1{[K:I]}\sum_{k\in K/I}f(xk),\qquad(e_K\ast\varphi)(x)=\frac1{[K:I]}\sum_{k\in K/I}\varphi(k^{-1}x).$$
  2. \(\mathcal H_I\ast e_K=C_c(I\backslash G(F)/K)\) and \(e_K\ast C_I=C_K\). Both maps are projections: \(f\ast e_K=f\) when \(f\) is right \(K\)-invariant, \(e_K\ast\varphi=\varphi\) when \(\varphi\) is \(K\)-invariant.
  3. \(e_K\mathcal H_Ie_K:=\{e_K\ast f\ast e_K:\ f\in\mathcal H_I\}\) is the space of bi-\(K\)-invariant functions in \(\mathcal H_I\). It is closed under \(\ast\), with unit \(e_K\), and $$\Phi:(\mathcal H_K,\ast_K)\xrightarrow{\ \sim\ }(e_K\mathcal H_Ie_K,\ast),\qquad\Phi(f)=[K:I]^{-1}f,\qquad\Phi(\mathbf 1_K)=e_K,$$ is an isomorphism of algebras. For \(f\in\mathcal H_K\) and \(\varphi\in C_K\), \(\Phi(f)\ast\varphi=f\ast_K\varphi\); hence \((e_K\mathcal H_Ie_K)\cdot\varphi=\mathcal H_K\cdot\varphi\).
Proof
  • (1) Averaging. In \((f\ast e_K)(x)=\sum_{g\in G(F)/I}f(g)\,e_K(g^{-1}x)\) only the cosets \(gI\subseteq xK\) contribute, each with weight \([K:I]^{-1}\), and they are the \(xkI\) for \(k\in K/I\). Likewise \((e_K\ast\varphi)(x)=[K:I]^{-1}\sum_{g\in K/I}\varphi(g^{-1}x)\). The average over \(xK\) is right \(K\)-invariant in \(x\) and equals \(f(x)\) when \(f\) is already right \(K\)-invariant; the same on the left for \(\varphi\). For \(f=e_K\), which is right \(K\)-invariant, this reads \(e_K\ast e_K=e_K\).
  • (2) Images. By (1), \(f\ast e_K\) is right \(K\)-invariant and a right \(K\)-invariant \(f\) equals \(f\ast e_K\); so \(\mathcal H_I\ast e_K\) is the right \(K\)-invariant part of \(\mathcal H_I\), which is \(C_c(I\backslash G(F)/K)\). In the same way \(e_K\ast C_I\) is the \(K\)-invariant part of \(C_I\), which is \(C_K\).
  • (3) The corner. \(e_K\ast f\ast e_K\) is bi-\(K\)-invariant by (1), and a bi-\(K\)-invariant \(f\) equals \(e_K\ast f\ast e_K\); so \(e_K\mathcal H_Ie_K\) is the space of bi-\(K\)-invariant functions, which as a vector space is \(C_c(K\backslash G(F)/K)=\mathcal H_K\). It is closed under \(\ast\) because \(e_K\) is idempotent, \((e_K\ast f\ast e_K)\ast(e_K\ast g\ast e_K)=e_K\ast(f\ast e_K\ast g)\ast e_K\), and \(e_K\) is a two-sided unit for it. The two products differ only through the Haar measures: for bi-\(K\)-invariant \(f,g\) the \([K:I]\) cosets \(hI\) inside one \(hK\) contribute equal terms to \(\sum_{h\in G(F)/I}f(h)g(h^{-1}x)\), so \(f\ast g=[K:I]\,f\ast_Kg\). Hence \(\Phi(f)\ast\Phi(g)=[K:I]^{-2}f\ast g=[K:I]^{-1}f\ast_Kg=\Phi(f\ast_Kg)\) and \(\Phi(\mathbf 1_K)=e_K\). The same count gives \(f\ast\varphi=[K:I]\,f\ast_K\varphi\) for \(f\in\mathcal H_K\), \(\varphi\in C_K\), that is \(\Phi(f)\ast\varphi=f\ast_K\varphi\).
  • References. \(K=\bigsqcup_{w\in W}IwI\) with \(|IwI/I|=q^{\ell(w)}\), hence \(\mathbf 1_K=\sum_WT_w\) and \([K:I]=\sum_Wq^{\ell(w)}\): Iwahori–Matsumoto, Proposition 2.4, Theorem 2.5 and Proposition 3.2; in Cartier's survey, formula (22). The idempotent \(e_K\) and the corner \(\mathcal H_K=e_K\mathcal H_Ie_K\): Haines–Kottwitz–Prasad, Iwahori–Hecke algebras, §4.1. The same facts, located line by line: the note's functions page, steps 4–5 and 7.

2 · Generation at the two levels: the criterion

Three statements about the same function \(\varphi_0\):

$$\text{spherical generation}\quad\mathcal H_K\cdot\varphi_0=C_K;\qquad\text{Iwahori-generated part}\quad\mathcal H_I\cdot\varphi_0\subseteq C_I;\qquad\text{Iwahori generation}\quad\mathcal H_I\cdot\varphi_0=C_I .$$

Lemma A (what \(\varphi_0\) generates). \(\mathcal H_I\cdot\varphi_0=(\mathcal H_I\ast e_K)\cdot\varphi_0=C_c(I\backslash G(F)/K)\cdot\varphi_0\): only the Iwahori–spherical translates of \(\varphi_0\) are needed.

Proof
  • \(f\ast\varphi_0=f\ast e_K\ast\varphi_0\) since \(\varphi_0\) is \(K\)-invariant; then (2) of the Lemma in 1.

Proposition B (criterion).

$$\boxed{\ \mathcal H_I\cdot\varphi_0=C_I\iff\big(\mathcal H_K\cdot\varphi_0=C_K\ \text{ and }\ \mathcal H_I\cdot C_K=C_I\big)\ }$$

Moreover \(\mathcal H_I\cdot\varphi_0=\mathcal H_I\cdot C_K\) if and only if spherical generation holds, and then both equal \(C_c(I\backslash G(F)/K)\cdot C_K\), the image of \(C_c(I\backslash G(F)/K)\otimes_{\mathcal H_K}C_K\to C_I\). In particular Iwahori generation forces spherical generation.

Proof
  • (⇒) Apply \(e_K\): \(C_K=e_K\ast C_I=e_K\ast\mathcal H_I\cdot\varphi_0=(e_K\mathcal H_Ie_K)\cdot\varphi_0=\mathcal H_K\cdot\varphi_0\), using \(\varphi_0=e_K\ast\varphi_0\). And \(\varphi_0\in C_K\) gives \(C_I=\mathcal H_I\cdot\varphi_0\subseteq\mathcal H_I\cdot C_K\).
  • (⇐) \(\mathcal H_I\cdot C_K=\mathcal H_I\cdot(\mathcal H_K\cdot\varphi_0)=\mathcal H_I\cdot(e_K\mathcal H_Ie_K)\cdot\varphi_0\subseteq\mathcal H_I\cdot\varphi_0\). The same inclusion, with \(\varphi_0\in C_K\), gives the equality under spherical generation; and \(f\ast\varphi=(f\ast e_K)\ast\varphi\) for \(\varphi\in C_K\) gives the last form. Conversely, if \(\mathcal H_I\cdot\varphi_0=\mathcal H_I\cdot C_K\), apply \(e_K\): \(\mathcal H_K\cdot\varphi_0=(e_K\mathcal H_Ie_K)\cdot\varphi_0=e_K\ast\mathcal H_I\cdot\varphi_0=e_K\ast\mathcal H_I\cdot C_K\supseteq e_K\ast C_K=C_K\).

The obstruction module. Put

$$Q_X:=C_I\big/\mathcal H_I\cdot C_K,\qquad e_K\ast Q_X=0\quad(\text{because }e_K\ast C_I=C_K\subseteq\mathcal H_I\cdot C_K).$$

Under spherical generation, Iwahori generation holds iff \(Q_X=0\). In general two modules are in play (the pasted rank-one note's distinction):

$$Q_{X,\varphi_0}:=C_I\big/\mathcal H_I\cdot\varphi_0,\qquad0\to\mathcal H_I\cdot C_K\big/\mathcal H_I\cdot\varphi_0\to Q_{X,\varphi_0}\to Q_X\to0 ;$$

\(Q_{X,\varphi_0}=0\) is Iwahori generation by the chosen basic function, while \(Q_X\ne0\) says that no \(K\)-invariant function whatever generates \(C_I\). The next lemma says that any nonzero \(\mathcal H_I\)-module killed by \(e_K\) and reached from \(C_I\) obstructs, and that \(Q_X\) is the universal one. For split symmetric \(X\), \(C_I\) is finitely generated over \(\mathcal H_I\) (Shtotland, arXiv:2406.16070v3, Theorem 5.9), so \(Q_X\ne0\) is then equivalent to the existence of a witness representation as in Corollary D (F1, Tool 1).

Lemma C (obstruction principle). Let \(f:C_I\to Q\) be a nonzero \(\mathcal H_I\)-linear map.

  1. If \(f(\varphi_0)=0\), then \(\mathcal H_I\cdot\varphi_0\subseteq\ker f\subsetneq C_I\).
  2. If \(e_K\ast Q=0\), then \(f\) kills \(\mathcal H_I\cdot C_K\), so \(f(\varphi_0)=0\) and (1) applies; \(f\) factors through \(Q_X\).
  3. Conversely, if spherical generation holds and \(\mathcal H_I\cdot\varphi_0\ne C_I\), then \(C_I\to Q_X\) is such a map. For a unital algebra \(A\) and an idempotent \(e\): \(AeA=A\) iff \(eL\ne0\) for every nonzero \(A\)-module \(L\).
Proof
  • (1) \(f(a\ast\varphi_0)=a\ast f(\varphi_0)=0\). (2) \(f(C_K)=f(e_K\ast C_I)=e_K\ast f(C_I)\subseteq e_K\ast Q=0\), then \(\mathcal H_I\)-linearity. (3) Proposition B. The algebra statement: if \(eL=0\) then \(AeA\cdot L=0\), so \(AeA=A\) forces \(L=0\); conversely \(L=A/AeA\) has \(eL=0\).

Corollary D (Steinberg-type obstruction). Let \(\pi\) be an irreducible smooth representation of \(G(F)\) with

$$\pi^I\ne0,\qquad\pi^K=0,\qquad\operatorname{Hom}_{G(F)}\big(\mathcal S(X),\pi\big)\ne0 .$$

Then \(\mathcal H_I\cdot\varphi_0\subsetneq C_I\). For \(X=G/H\) with \(H\) reductive, \(\operatorname{Hom}_{G(F)}(\mathcal S(X),\pi)\ne0\) iff the contragredient \(\tilde\pi\) carries a nonzero \(H(F)\)-invariant functional (\(\tilde\pi\) is \(H\)-distinguished).

Proof
  • A nonzero \(G(F)\)-map \(f:\mathcal S(X)\to\pi\) is onto (\(\pi\) irreducible). Invariants under the compact open \(I\) are exact (\(e_I\) projects), so \(f(C_I)=f(e_I\mathcal S(X))=e_I\pi=\pi^I\ne0\), and \(f|_{C_I}\) is \(\mathcal H_I\)-linear (1). \(f(\varphi_0)\in\pi^K=0\). Lemma C(1).
  • Frobenius reciprocity: the smooth dual of \(\mathcal S(G/H)\) is \(C^\infty(G/H)\) (a distribution invariant under a compact open \(J\) is a function on the discrete set \(J\backslash G/H\)), and \(\operatorname{Hom}_G(\mathcal S(X),\pi)=\operatorname{Hom}_G(\tilde\pi,C^\infty(G/H))=\operatorname{Hom}_H(\tilde\pi,\mathbb C)\) for admissible \(\pi\), \(H\) unimodular. Borel's dictionary (Theorem 4.10, Corollary 4.11): \(\pi\mapsto\pi^I\) is an equivalence between representations generated by \(I\)-fixed vectors and \(\mathcal H_I\)-modules, and \(\pi^K=e_K\pi^I\); so the irreducible \(\mathcal H_I\)-modules killed by \(e_K\) are exactly the \(\pi^I\) above. If \(Q_X\) has an irreducible quotient (for instance when \(C_I\) is finitely generated over the Noetherian algebra \(\mathcal H_I\)), the obstruction is of this shape, with \(\mathcal S(X)\) replaced by the subrepresentation it generates from \(C_I\).

3 · The group case

\(X=H\), \(G=H\times H\) acting by \((g_1,g_2)\cdot x=g_1xg_2^{-1}\); \(K=K_H\times K_H\), \(I=I_H\times I_H\), \(W\) the finite Weyl group of \(H\). Then \(C_K=\mathcal H_{K_H}\), \(C_I=\mathcal H_{I_H}\), and \(\mathcal H_I=\mathcal H_{I_H}\otimes\mathcal H_{I_H}^{\mathrm{op}}\) acts by \((a\otimes b)\cdot\varphi=a\ast\varphi\ast b\): submodules are two-sided ideals. The basic function is the unit at one level and a non-unit idempotent at the other:

Table 1. and spherical level and Iwahori level
spherical levelIwahori level
module\(C_K=\mathcal H_{K_H}\)\(C_I=\mathcal H_{I_H}\)
\(\varphi_0=\mathbf 1_{K_H}\)the unit\(\sum_{w\in W}T_w=[K_H:I_H]\,e_{K_H}\), unit is \(T_e\)
generated parteverythingthe ideal \(\mathcal H_{I_H}\,e_{K_H}\,\mathcal H_{I_H}\)
\(Q_X\)\(0\)\(\mathcal H_{I_H}/\mathcal H_{I_H}e_{K_H}\mathcal H_{I_H}\ne0\) for \(W\ne1\)

Proposition E (group case). Spherical generation holds for every \(H\). For \(W\ne1\) the Iwahori-generated part is proper:

$$\mathcal H_I\cdot\varphi_0=\mathcal H_{I_H}\ast e_{K_H}\ast\mathcal H_{I_H}\ \subseteq\ \ker\varepsilon\ \not\ni T_e,\qquad\varepsilon:\mathcal H_{I_H}\to\mathbb C,\ \ \varepsilon(T_w)=(-1)^{\ell(w)}\ \text{ (sign character)}.$$
Proof
  • Spherical. \(\mathbf 1_{K_H}\) is the unit of \(\mathcal H_{K_H}\) (\(\operatorname{vol}K_H=1\)), and \(a\ast\mathbf 1_{K_H}\ast b=a\ast b\) ranges over all of \(\mathcal H_{K_H}\). By Lemma C(3) the unit kills no module, so nothing obstructs.
  • Iwahori. \(K_H=\bigsqcup_{w\in W}I_HwI_H\) gives \(\mathbf 1_{K_H}=\sum_WT_w\); \(\varepsilon\) is a character because the Iwahori–Matsumoto presentation (\(T_s^2=(q-1)T_s+q\), braid relations, \(T_\omega\) for \(\omega\in\Omega\)) is respected by \(T_s\mapsto-1\), \(T_\omega\mapsto1\); \(\varepsilon(\mathbf 1_{K_H})=\sum_W(-1)^{\ell(w)}=0\) since \(w\mapsto ws\) pairs even with odd lengths when \(W\ne1\); \(\varepsilon(T_e)=1\). \(\ker\varepsilon\) is a two-sided ideal containing \(e_{K_H}\). Citations (Iwahori–Matsumoto Theorem 3.5, Borel 3.5): the note's functions page, step 8; details on the tree: Iwahori–Matsumoto site.
  • In the language of 2. \(\varepsilon\) is the \(\mathcal H_{I_H}\)-module \(\mathrm{St}_H^{I_H}\) (Borel 3.5, 5.7), and \(\mathrm{St}_H\boxtimes\widetilde{\mathrm{St}}_H\) is a quotient of \(\mathcal S(H)=C_c^\infty(H(F))\) through \(f\mapsto\pi(f)\), with \(I\)-invariants \(\mathrm{St}_H^{I_H}\otimes\widetilde{\mathrm{St}}_H^{I_H}\ne0\) and no \(K\)-invariants: Corollary D for \(G=H\times H\). The corner \(e_{K_H}\mathcal H_{I_H}e_{K_H}=\mathcal H_{K_H}\) (Haines–Kottwitz–Prasad §4.1) is the spherical algebra sitting inside the affine one with unit \(e_{K_H}\ne T_e\).

The group argument is therefore not an argument against spherical generation: the two statements differ by the modules on which \(e_K\) acts by zero, which exist at Iwahori level and not at spherical level.

4 · Three \(\mathrm{GL}_2\) examples

\(G=\mathrm{GL}_2\), \(I\) the upper-triangular Iwahori. Three varieties sharing the open \(B\)-orbit, with different function spaces:

Table 2. · The three examples; the third is the only one in the paper's affine class.
\(X\)type\(X(\mathcal O)\)\(\mathcal H_K\cdot\varphi_0=C_K\)\(\mathcal H_I\cdot\varphi_0=C_I\)
\(\mathbb P^1=\mathrm{GL}_2/B\)projective\(=X(F)\)holds, \(\dim C_K=1\)fails: \(C_I/\mathcal H_I\cdot\varphi_0\simeq\mathrm{St}^I\)
\(\mathbb A^2\setminus\{0\}=\mathrm{GL}_2/P_1\)quasi-affineprimitive vectorsholdsfails: radial integration onto \(\mathrm{St}\)
\(\mathbb A^2\)affine\(\mathcal O^2\)holdsholds: lattice indicators
A · \(X=\mathbb P^1\): fails
  • \(X\) is proper, so \(X(\mathcal O)=X(F)\) and \(\varphi_0=\mathbf 1\) is \(G(F)\)-invariant: \(f\ast\mathbf 1=(\int f)\,\mathbf 1\), so \(\mathcal H_I\cdot\varphi_0=\mathbb C\mathbf 1=C_K\) while \(C_I=V^I\) is two-dimensional (F1, 4). The quotient is \(\mathrm{St}^I\).
B · \(X=\mathbb A^2\setminus\{0\}\): fails
  • \(X(F)=F^2\setminus0\), \(X(\mathcal O)=\mathcal O^2\setminus\varpi\mathcal O^2\) (primitive vectors), \(\varphi_0=\mathbf 1_{X(\mathcal O)}\). The \(K\)-orbits are the shells \(X_m=\varpi^mX(\mathcal O)\), \(m\in\mathbb Z\), and \(\mathbf 1_{\varpi^mK}\ast\varphi_0=\mathbf 1_{X_m}\) (\(\operatorname{vol}K=1\)): spherical generation holds.
  • Radial integration, with \(\operatorname{vol}(\mathcal O^\times)=1\) for \(d^\times t\): $$\mathcal R:\mathcal S(X)\to C^\infty(\mathbb P^1(F)),\qquad(\mathcal Rf)([v]):=\int_{F^\times}f(tv)\,d^\times t .$$ Well defined: \(\{t:tv\in\operatorname{supp}f\}\) is compact in \(F^\times\) since \(\operatorname{supp}f\) is compact and misses \(0\); \(G(F)\)-equivariant; surjective (\(h\mapsto h([v])\mathbf 1_{X(\mathcal O)}(v)\) is a section); and \(\mathcal R\varphi_0=\mathbf 1\) because \(tv\in X(\mathcal O)\iff t\in\mathcal O^\times\) for primitive \(v\).
  • \(\mathcal S(X)\xrightarrow{\mathcal R}C^\infty(\mathbb P^1(F))\twoheadrightarrow\mathrm{St}\) is a surjective \(G(F)\)-map killing \(\varphi_0\); its \(I\)-invariants \(C_I\twoheadrightarrow\mathrm{St}^I\simeq\mathbb C\) kill \(\varphi_0\). Lemma C(1).
C · \(X=\mathbb A^2\): holds
  • \(\varphi_0=\mathbf 1_{\mathcal O^2}\). Lattices \(L_m:=\varpi^m\mathcal O^2\) and \(L_m':=\varpi^m\mathcal Oe_1\oplus\varpi^{m+1}\mathcal Oe_2\) are \(I\)-stable and of the form \(g\mathcal O^2\). For an \(I\)-stable \(L=g\mathcal O^2\): \(\mathbf 1_{IgK}\ast\varphi_0=\sum_{h\in IgK/K}\operatorname{vol}(K)\mathbf 1_{h\mathcal O^2}=(q+1)\mathbf 1_L\), as \(IgK/K\) is the single point \(gK\). So \(\mathbf 1_{L_m},\mathbf 1_{L_m'}\in\mathcal H_I\cdot\varphi_0\).
  • The \(I\)-orbits on \(F^2\setminus0\) are \(L_m'\setminus L_{m+1}\) and \(L_m\setminus L_m'\) (reduce \(\varpi^{-m}v\) modulo \(\varpi\): the two \(B(\mathbb F_q)\)-orbits on \(\mathbb F_q^2\setminus0\), fibres transitive as on F1, 4). Their indicators \(\mathbf 1_{L_m'}-\mathbf 1_{L_{m+1}}\) and \(\mathbf 1_{L_m}-\mathbf 1_{L_m'}\) lie in \(\mathcal H_I\cdot\varphi_0\).
  • An \(I\)-invariant \(\varphi\in\mathcal S(X)\) is constant on some \(L_N\) and vanishes outside some \(L_{-N}\), hence is a finite combination of \(\mathbf 1_{L_N}\) and orbit indicators: \(\mathcal H_I\cdot\varphi_0=C_I\). Spherical generation: \(\mathbf 1_{L_m}=\mathbf 1_{\varpi^mK}\ast\varphi_0\) span \(C_K\).

B and C have the same open orbit; adding the origin changes \(\mathcal S(X)\), and \(\mathcal R\) diverges on \(\mathbf 1_{\mathcal O^2}\) (all small \(t\) contribute). \(Q_X\) depends on the embedding and the basic function, not only on the open \(G\)-orbit. C is a positive example inside the paper's class; the same proof for \(\mathbb A^n\), with pictures in valuation coordinates, is on F3.

5 · Symmetric varieties

Proposition F (\(X=\mathrm{GL}_2/T\), \(T\) the diagonal torus; the smallest \(\mathrm{GL}_p\times\mathrm{GL}_q\) case). Spherical generation holds and \(\mathcal H_I\cdot\varphi_0\subsetneq C_I\): \(\mathrm{St}\) is \(T\)-distinguished.

Proof
  • \(X(F)=\mathrm{GL}_2(F)/T(F)\) (Hilbert 90) and \(X(\mathcal O)=K/T(\mathcal O)\) (\(\operatorname{Pic}\mathcal O=0\)). Spherical generation: the paper's family \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) (Spherical generation workspace).
  • \(\lambda([f]):=f([1:0])-f([0:1])\) is well defined on \(\mathrm{St}=C^\infty(\mathbb P^1)/\mathbb C\mathbf 1\), nonzero, and \(T\)-invariant since \(T\) fixes both points. \(\mathrm{St}\) is self-contragredient (unique irreducible quotient of \(C^\infty(\mathbb P^1)\) and unique irreducible subrepresentation of its dual). Corollary D with \(H=T\): \(\operatorname{Hom}_G(\mathcal S(X),\mathrm{St})\simeq\operatorname{Hom}_T(\mathrm{St},\mathbb C)\ni\lambda\ne0\), and \(\mathrm{St}^I\ne0=\mathrm{St}^K\) (F1, 4).

Proposition G (\(X_n=\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), nondegenerate alternating forms). \(n=1\): \(\mathcal H_I\cdot\varphi_0=C_I\). \(n\ge2\), \(\operatorname{char}F=0\): \(\mathcal H_I\cdot\varphi_0\subsetneq C_I\), witnessed by the Speh-type representation

$$\pi_n:=U(\mathrm{St}_2,2)\times\mathbf 1_{\mathrm{GL}_2}\times\dots\times\mathbf 1_{\mathrm{GL}_2}\quad(n-2\text{ trivial factors}),\qquad\pi_n^I\ne0,\quad\pi_n^K=0,\quad\pi_n\ \mathrm{Sp}_{2n}\text{-distinguished.}$$
Definitions (unramified principal series, Speh representations, symplectic distinction)
  • \(\chi_1\times\dots\times\chi_m\): normalized parabolic induction of unramified characters to \(\mathrm{GL}_m(F)\); \(\nu:=|\det|\). An irreducible \(\pi\) has \(\pi^I\ne0\) iff it is a subquotient of such a series (Casselman, Propositions 2.6–2.7; Borel, 4.9 (∗)); \(\dim(\chi_1\times\dots\times\chi_m)^K=1\) by \(G=PK\), so exactly one subquotient has a \(K\)-fixed vector. \(\mathrm{St}_2\subseteq\nu^{1/2}\times\nu^{-1/2}\).
  • Tadić's \(U(\delta,m)\), \(\delta\) square-integrable on \(\mathrm{GL}_a\): the unique irreducible quotient of \(\nu^{(m-1)/2}\delta\times\dots\times\nu^{-(m-1)/2}\delta\), irreducible unitary (Theorem D, Theorem A.8, \(\operatorname{char}F=0\)); \(U(\mathbf 1_{\mathrm{GL}_1},m)=\mathbf 1_{\mathrm{GL}_m}\); \(U(\mathrm{St}_2,2)\) is a subquotient of \(\nu^1\times\nu^0\times\nu^0\times\nu^{-1}\).
  • \(\pi\) is \(\mathrm{Sp}_{2n}\)-distinguished if \(\operatorname{Hom}_{\mathrm{Sp}_{2n}(F)}(\pi,\mathbb C)\ne0\); Heumos–Rallis, Theorem 2.4.2: the space is at most one-dimensional. Offen–Sayag, J. Number Theory (2007), Theorem 1: every \(\sigma_1\times\dots\times\sigma_t\times\tau_{t+1}\times\dots\times\tau_s\) with \(\sigma_i=U(\delta_i,2m_i)\) and \(\tau_i=\nu^{\alpha_i}U(\delta_i,2m_i)\times\nu^{-\alpha_i}U(\delta_i,2m_i)\) is distinguished; IMRN (2008), Theorem 8 and Corollary 2, with J. Funct. Anal. (2008), Corollary 1: these are all the distinguished irreducible unitary representations.
Proof
  • \(n=1\). \(\mathrm{GL}_2\) acts on \(X_1=\mathbb G_m\) through \(\det\), and \(\det I=\det K=\mathcal O^\times\): \(I\backslash X_1(F)=K\backslash X_1(F)=F^\times/\mathcal O^\times\simeq\mathbb Z\), \(C_I=C_K=C_c(\mathbb Z)\), and \(T_{t^\lambda}\ast\mathbf 1_{\mathcal O^\times}=[It^\lambda I:I]\,\mathbf 1_{\varpi^{\langle\det,\lambda\rangle}\mathcal O^\times}\) span it.
  • Iwahori-spherical. Induction in stages makes \(\pi_n\) a subquotient of an unramified principal series of \(\mathrm{GL}_{2n}\), so \(\pi_n^I\ne0\); it is irreducible unitary since products of elements of Tadić's set \(B\) are (Theorem D(i)).
  • No spherical vector. \(\pi_n\) is a subquotient of \(J_n:=\nu^{1/2}\mathrm{St}_2\times\nu^{-1/2}\mathrm{St}_2\times\mathbf 1_{\mathrm{GL}_2}^{\times(n-2)}=\operatorname{Ind}_P^G\sigma\). By \(G=PK\), \((\operatorname{Ind}_P^G\sigma)^K\simeq\sigma^{M(\mathcal O)}=0\) because \(\mathrm{St}_2^{\mathrm{GL}_2(\mathcal O)}=0\) (F1, 4). \(K\)-invariants are exact, so \(\pi_n^K=0\).
  • Distinguished. Offen–Sayag, Theorem 1, with \(\sigma_1=U(\mathrm{St}_2,2)\) and \(\sigma_2=\dots=\sigma_{n-1}=U(\mathbf 1_{\mathrm{GL}_1},2)=\mathbf 1_{\mathrm{GL}_2}\) (for \(n=2\) also Heumos–Rallis, Theorem 11.1). \(\pi_n\) unitary gives \(\tilde\pi_n\simeq\bar\pi_n\), and conjugation carries an invariant functional on \(\pi_n\) to one on \(\bar\pi_n\): \(\tilde\pi_n\) is distinguished. Corollary D.
Table 3. · Function level over the paper's families: spherical generation (from the paper) and Iwahori generation.
\(X\)\(\mathcal H_K\cdot\varphi_0=C_K\)\(\mathcal H_I\cdot\varphi_0=C_I\)witness
group \(H\), \(W\ne1\)holdsfails\(\mathrm{St}_H\) (3)
\(\mathrm{GL}_2/\mathrm{Sp}_2=\mathbb G_m\)holdsholds—
\(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge2\)paper: \(\mathrm{SL}_4/\mathrm{Sp}_4\); \(n\ge3\) openfails (\(\operatorname{char}F=0\))\(U(\mathrm{St}_2,2)\times\mathbf 1_{\mathrm{GL}_2}^{n-2}\)
\(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge2\)paper: \(n=2\); \(n\ge3\) openfails (\(\operatorname{char}F=0\))Speh restricted to \(\mathrm{SL}_{2n}\) (F2, Theorem C)
\(\mathrm{GL}_2/(\mathrm{GL}_1\times\mathrm{GL}_1)\)holdsfails\(\mathrm{St}_2\), \(T\)-distinguished
\(\mathrm{SL}_2/T\)failsfailsparity and square classes; sign summand (F2, D)
\(\mathrm{GL}_{2p}/(\mathrm{GL}_p\times\mathrm{GL}_p)\), \(\mathrm{GL}_3/(\mathrm{GL}_1\times\mathrm{GL}_2)\)holdsfails\(\mathrm{St}_{2p}\) (Shalika model), \(\mathrm{St}_2\times\mathbf 1\) (F2, B)
\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(q>p\), \(n\ge4\)holdsopencandidate \(\mathrm{St}_{2p}\times\mathbf 1_{\mathrm{GL}_{q-p}}\) (F2, B)
\(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\)holds\(n=2,3\) fail; \(n\ge4\) opengroup \(\mathrm{SL}_2\); \(\mathrm{SL}_4/\mathrm{Sp}_4\) (F2, 2)
\(E_6/F_4\)openopenSteinberg sweep silent (F2, 2)
\(\mathbb A^n\) under \(\mathrm{GL}_n\) (not homogeneous)holdsholdslattice chain (4; F3)
\(M_n\) under \(\mathrm{GL}_n\times\mathrm{GL}_n\), \(n\ge2\) (not homogeneous)holdsfails\(\mathrm{St}_n\boxtimes\mathrm{St}_n\) via Godement–Jacquet (F2, E)

Pattern: wherever a Steinberg-, induced-Steinberg- or Speh-type representation is \(X\)-distinguished, the Iwahori-generated part is proper; for quasi-split split symmetric \(X\) the twisted Steinberg always is (F1, Theorem A). The positive cases are the torus-like \(n=1\), the non-homogeneous \(\mathbb A^n\), and the non-split \(\mathrm{PGL}_2/T_u\).

Literature and remarksspherical generation · Iwahori modules · Steinberg distinction · representations · the two pasted notes