The group and three \(\GL_2\)-varieties
At the spherical level the basic function of a group is the unit; at the Iwahori level it is \([K_H:I_H]\) times a non-unit idempotent, and the sign character sees the difference. Three \(\GL_2\)-varieties with the same open orbit then show that the answer depends on the model, not only on the open orbit.
The group caseWhy does the unit stop being a unit?
\(X=H\), \(G=H\times H\) acting by \((g_1,g_2)\cdot x=g_1xg_2^{-1}\); \(K=K_H\times K_H\), \(I=I_H\times I_H\), and \(W\) the finite Weyl group of \(H\). Then \(\CK=\mathcal H_{K_H}\), \(\CI=\mathcal H_{I_H}\), and \(\HI=\mathcal H_{I_H}\otimes\mathcal H_{I_H}^{\mathrm{op}}\) acts by \((a\otimes b)\cdot\varphi=a\ast\varphi\ast b\): submodules are two-sided ideals.
| spherical | Iwahori | |
|---|---|---|
| module | \(\CK=\mathcal H_{K_H}\) | \(\CI=\mathcal H_{I_H}\) |
| \(\phiz=\one_{K_H}\) | the unit | \(\sum_{w\in W}T_w=[K_H:I_H]\,e_{K_H}\); the unit is \(T_e\) |
| generated part | everything | the ideal \(\mathcal H_{I_H}e_{K_H}\mathcal H_{I_H}\) |
| \(\QX\) | \(0\) | \(\mathcal H_{I_H}/\mathcal H_{I_H}e_{K_H}\mathcal H_{I_H}\ne0\) for \(W\ne1\) |
(group case; workspace “Proposition E”) Proved here
(SG) holds for every \(H\). For \(W\ne1\) the Iwahori-generated part is proper, so (IG) fails:
\[\HI\ast\phiz=\mathcal H_{I_H}\ast e_{K_H}\ast\mathcal H_{I_H}\ \subseteq\ \ker\varepsilon\ \not\ni T_e,\qquad\varepsilon:\mathcal H_{I_H}\to\C,\quad\varepsilon(T_w)=(-1)^{\ell(w)}\ \text{(the sign character)}.\]Proof
- Spherical. \(\one_{K_H}\) is the unit of \(\mathcal H_{K_H}\) (\(\vol K_H=1\)), and \(a\ast\one_{K_H}\ast b=a\ast b\) ranges over all of \(\mathcal H_{K_H}\). Equivalently, by idempotent ideals with \(e\) the unit, no nonzero module is killed, so nothing obstructs.
- Iwahori. \(K_H=\bigsqcup_{w\in W}I_HwI_H\) gives \(\one_{K_H}=\sum_WT_w\). \(\varepsilon\) is a character, because the Iwahori–Matsumoto presentation (\(T_s^2=(q-1)T_s+q\), the braid relations, \(T_\omega\) for \(\omega\) of length \(0\)) is respected by \(T_s\mapsto-1\), \(T_\omega\mapsto1\): at \(T_s=-1\), \(1=-(q-1)+q\). \(\varepsilon(\one_{K_H})=\sum_W(-1)^{\ell(w)}=0\), since \(w\mapsto ws\) pairs even with odd lengths when \(W\ne1\); and \(\varepsilon(T_e)=1\). So \(\ker\varepsilon\) is a two-sided ideal containing \(e_{K_H}\) and not \(T_e\). Citations (Iwahori–Matsumoto Thm. 3.5, Borel 3.5): the note's functions page, step 8; worked out on the tree: Iwahori–Matsumoto site.
- As a witness. \(\varepsilon\) is the \(\mathcal H_{I_H}\)-module \(\St_H^{I_H}\) [Borel 1976, 3.5, 5.7], and \(\St_H\boxtimes\widetilde{\St}_H\) is a quotient of \(\Sw(H)=C_c^\infty(H(F))\) through \(f\mapsto\pi(f)\), with \(I\)-invariants \(\St_H^{I_H}\otimes\widetilde{\St}_H^{I_H}\ne0\) and no \(K\)-invariants: the witness Proposition for \(G=H\times H\). The corner \(e_{K_H}\mathcal H_{I_H}e_{K_H}=\mathcal H_{K_H}\) [HKP, §4.1] is the spherical algebra inside the affine one, with unit \(e_{K_H}\ne T_e\).
The two levels differ by the modules on which \(\eK\) acts by zero, which exist at the Iwahori level and not at the spherical level.
Three \(\GL_2\)-varieties with one open orbitDoes (IG) depend only on the open \(G\)-orbit?
\(G=\GL_2\), \(I\) the upper triangular Iwahori subgroup. The three varieties share the open \(B\)-orbit; their function spaces differ.
| \(X\) | \(X(\cO)\) | (SG) | (IG) |
|---|---|---|---|
| \(\mathbb P^1=\GL_2/B\) (projective) | \(=X(F)\) | holds, \(\dim\CK=1\) | fails: quotient \(\St^I\) |
| \(\Aff^2\smallsetminus\{0\}\) (quasi-affine) | primitive vectors | holds | fails: radial integration onto \(\St\) |
| \(\Aff^2\) (affine) | \(\cO^2\) | holds | holds: lattice indicators |
(\(X=\mathbb P^1\): (IG) fails)
\(\HI\ast\phiz=\C\one=\CK\), while \(\CI\) is two-dimensional; \(Q_{X,\phiz}\simeq\St^I\).
Proof
\(X\) is proper, so \(X(\cO)=X(F)\) and \(\phiz=\one\) is \(G(F)\)-invariant: \(f\ast\one=\big(\int f\big)\one\), so \(\HI\ast\phiz=\C\one=\CK\). \(\CI=C^\infty(\mathbb P^1)^I=\C\one_{U_1}\oplus\C\one_{U_2}\), and the quotient by \(\C\one\) is \(\St^I\) (the Steinberg of \(\GL_2\)).
(\(X=\Aff^2\smallsetminus\{0\}=\GL_2/\mathrm{Mir}_2\): (IG) fails)
(SG) holds, and radial integration
\[\Rad:\Sw(X)\to C^\infty(\mathbb P^1(F)),\qquad(\Rad f)([v]):=\int_{F^\times}f(tv)\,d^\times t\quad(\vol\cO^\times=1),\]followed by \(C^\infty(\mathbb P^1(F))\to\St\), is a surjective \(G(F)\)-map that kills \(\phiz\). So \(\HI\ast\phiz\subsetneq\CI\).
Proof
- (SG). \(X(F)=F^2\smallsetminus0\), \(X(\cO)=\cO^2\smallsetminus\vpi\cO^2\), the primitive vectors. The \(K\)-orbits are the shells \(X_m=\vpi^mX(\cO)\), \(m\in\Z\), and \(\one_{\vpi^mK}\ast_K\phiz=\one_{X_m}\).
- \(\Rad\) is well defined. \(\{t:\ tv\in\supp f\}\) is compact in \(F^\times\), since \(\supp f\) is compact and misses \(0\); and \(\Rad f\) depends only on the line \([v]\).
- Equivariant and onto. \(\Rad\) commutes with \(G(F)\), which acts linearly on \(v\). For \(h\in C^\infty(\mathbb P^1(F))\), \(v\mapsto h([v])\one_{X(\cO)}(v)\) is a preimage: on each line, \(\{t:\ tv\in X(\cO)\}\) is one coset of \(\cO^\times\).
- It kills \(\phiz\) in \(\St\). \(\Rad\phiz=\one\), because \(tv\in X(\cO)\iff t\in\cO^\times\) for primitive \(v\); and \(\one\mapsto0\) in \(\St\). On \(I\)-invariants the composite is a nonzero \(\HI\)-map \(\CI\to\St^I\simeq\C\) killing \(\phiz\): the obstruction principle, part 1.
(\(X=\Aff^2\): (IG) holds)
With \(L_m:=\vpi^m\cO^2=\Lambda_{m,2}\) and \(L_m':=\vpi^m\cO e_1\oplus\vpi^{m+1}\cO e_2=\Lambda_{m,1}\), \(\one_{gK}\ast\phiz=(q+1)\one_{g\cO^2}\) for each of these \(I\)-stable lattices \(g\cO^2\), and the differences \(\one_{L_m}-\one_{L_m'}\), \(\one_{L_m'}-\one_{L_{m+1}}\) are the \(I\)-orbit indicators. Hence \(\HI\ast\phiz=\CI\).
This is the case \(n=2\) of the \(\Aff^n\) Proposition, proved there for every \(n\) (here \([K:I]=q+1\)).
\(\Aff^2\smallsetminus\{0\}\) and \(\Aff^2\) have the same open orbit. Adding the origin changes \(\Sw(X)\): \(\Rad\) diverges on \(\one_{\cO^2}\), since all small \(t\) contribute, and the Steinberg quotient of the punctured plane does not extend (why the origin matters). So \(\QX\) depends on the model and its basic function, not only on the open \(G\)-orbit. Of the three, only \(\Aff^2\) is affine; \(\mathbb P^1\) and the punctured plane lie outside the class of the Theorem.