Verdict · 2026-10-18

Iwahori generation at two levels

partial

Iwahori generation can fail for sheaves: $\mathcal C_I^{\mathrm{Iwa}}\ne\mathcal C_I$ for the group and for $\mathrm{GL}_2/T$, while $\mathbb A^n$ and $\mathbb G_m$ generate. The Iwahori summand criterion and functions ⇒ sheaves hold, so over $\mathbb F_q(\!(t)\!)$ a divergence can only be “sheaves generate, functions do not”. Every decided row of Table 1 agrees at the two levels; the other symmetric rows are open at sheaf level.

Terms · every symbol used below

Groups and Hecke objects. $G\supseteq B\supseteq T$ split, $K=L^+G\supseteq I$ the Iwahori, $\widetilde W=X_*(T)\rtimes W$, $\Omega\subseteq\widetilde W$ the elements of length zero; $\mathcal H_{c,I}=\operatorname{Shv}_c(I\backslash LG/I)$, with $\mathrm{IC}_w$, $\Delta_w$ the intersection complex and the standard object of the cell $IwI/I$.
Categories. $\mathcal C_K=\operatorname{Shv}_c(K\backslash LX)$ (the spherical run's category); $\mathcal C_I=\operatorname{Shv}_c(I\backslash LX)=\bigcup_{N,M}\mathcal C_I^{(N,M)}$, objects pulled back from a finite level $Y_{N,M}$ of a level presentation; $\mathrm{IC}_0$ the constant sheaf on $L^+X$; $\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal G\ast_I\mathrm{IC}_0:\mathcal G\in\mathcal H_{c,I}\}=\operatorname{Thick}\{\mathrm{IC}_w\ast_I\mathrm{IC}_0\}$.
Simple objects. $\mathcal S_I$: the IC sheaves of closures of pieces, with irreducible equivariant local systems; $\mathcal M_I\subseteq\mathcal S_I$ those that are shifted direct summands of some $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; $\mathcal M_I^+$ those that are perverse constituents of one.
Functions. $F$ a non-archimedean local field with residue field $\mathbb F_q$; $\mathcal H_I=C_c(I\backslash G(F)/I)$, $\operatorname{vol}I=1$, $T_w=\mathbf 1_{IwI}$; $C_I=C_c^\infty(X(F))^I$, $\varphi_0=\mathbf 1_{X(\mathcal O)}$. (IG): $\mathcal H_I\ast\varphi_0=C_I$. (SG): $\mathcal H_K\ast\varphi_0=C_K$.
Hypotheses. (Lev) the level axioms; I1 finitely many pieces per level; I2 connected stabilizers; (DT) the equivariant decomposition of every $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; (Dict) the trace dictionary over $\mathbb F_q$. Full statements on the pages of Propositions I and II.
Fields. Sheaf statements are over $\overline{\mathbb F}_q$ ($\ell$-adic) or $\mathbb C$ (Betti, equivalently the pages' $\mathcal D$-modules); each proposition says which.

1 · The answer, variety by variety

$$\text{sheaves:}\ \ \mathcal C_I^{\mathrm{Iwa}}\overset?=\mathcal C_I,\qquad\qquad\text{functions:}\ \ \text{(IG)}\ \ \mathcal H_I\ast\varphi_0\overset?=C_I .$$

Table 2. Generation at the two levels.
$X$	sheaves	functions	same?	where
$H$ under $H\times H$, $W\ne1$	fail: $\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}$	fail, every $F$	yes	V
$\mathbb A^n$ under $\mathrm{GL}_n$	generate	generate, every $F$	yes	VI
$\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m$	generate	generate, every $F$	yes	VI · cor.
$\mathrm{GL}_2/T$	fail at $\underline k_{A_\pm}$, given I1	fail, every $F$, codimension $2$	yes	Proof
$E_6/F_4$	open	fail over $\mathbb F_q(\!(t)\!)$ and $p$-adic $F$, $p$ odd	?	—
$\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$, $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$, $n\ge2$	open	fail, $\operatorname{char}F=0$	?	—
$\mathrm{GL}_{2p}/\mathrm{GL}_p^2$, $\mathrm{GL}_3/\mathrm{GL}_1\times\mathrm{GL}_2$	open	fail, $\operatorname{char}F=0$	?	—
$\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$, $q>p$, $n\ge4$; $\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$, $n\ge4$	open	open	?	—

The group's sheaf failure is proved here over $\overline{\mathbb F}_q$; over $\mathbb C$ it follows from a cited comparison, and the pages' Proposition II proves it there directly. $\mathrm{Spin}_4/\mathrm{Spin}_3$ is the group $\mathrm{SL}_2$, and $\mathrm{Spin}_6/\mathrm{Spin}_5$ is $\mathrm{SL}_4/\mathrm{Sp}_4$. In every open row with a function failure, by Proposition II a divergence would be "sheaves generate, functions do not", and only over $\mathbb F_q(\!(t)\!)$ does Proposition II connect the levels.

2 · Key propositions

Colour = proof status. Proved every step proved here or a cited published result. Conditional proved assuming a named statement that is not yet proved. Not proved open. Whether a level “holds” or “fails” is in the words, not the colour.

I Finite-level spans
Over $\mathbb C$ or $\overline{\mathbb F}_q$: assume (Lev), I1, I2 (or count local systems at every level), and $G_w\in\mathcal C_I$.
$$\mathcal C_I=\operatorname{Thick}(\mathcal S_I).$$ $$\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}(\mathcal M_I^+).$$
No decomposition theorem is needed.
Proof →
II Function generation implies sheaf generation

Let $X$ be a smooth affine variety under a split connected reductive group $G$ over $\mathbb F_q$, $q$ odd. Put $F=\mathbb F_q(\!(t)\!)$, $\mathcal O=\mathbb F_q[\![t]\!]$, $k=\overline{\mathbb F}_q$, and $\ell\ne\operatorname{char}k$. Let $I$ be the Iwahori subgroup and $\widetilde W$ the extended affine Weyl group.

For the finite-level presentation $LX=\varinjlim_N\varprojlim_MY_{N,M}$, with projections $\pi_{N,M}$ and congruence subgroups $I_{N,M}$, the category of $I$-equivariant constructible $\ell$-adic sheaves is
$$\mathcal C_I=\bigcup_{N,M}\pi_{N,M}^*D^b_c\big([Y_{N,M}/(I/I_{N,M})],\overline{\mathbb Q}_\ell\big).$$
Quotients are over $k$, and level objects are extended by zero to $LX$. With $\mathrm{IC}_w$ the perverse-normalized Schubert intersection complex and $\mathrm{IC}_0$ the unshifted constant sheaf on $L^+X$, define
$$\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_w\ast_I\mathrm{IC}_0:w\in\widetilde W\}.$$
Thus $\mathcal C_I^{\mathrm{Iwa}}$ is closed under shifts, cones and direct summands. For $\overline{\mathbb Q}_\ell$-valued functions, with $\operatorname{vol}I=1$, put
$$\mathcal H_I=C_c(I\backslash G(F)/I),\qquad C_I=C_c^\infty(X(F))^I,\qquad\varphi_0=\mathbf1_{X(\mathcal O)}.$$
Assume (Lev), I1, I2, (Dict) and (DT): finite levels, connected stabilizers, the trace dictionary and equivariant geometric decomposition.
$$\boxed{\mathcal H_I\ast\varphi_0=C_I\quad\Longrightarrow\quad\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I.}$$
Only the first Frobenius is used: $n=1$. The sheaf conclusion is over $k$.¹
Input: Sun’s Theorem 1.2 · precise statement

Cited result, restated. Let $f_0:\mathscr X_0\to\mathscr Y_0$ be proper with finite diagonal. Suppose both are finite-type Artin stacks over $\mathbb F_q$, and all their geometric stabilizers are affine. Fix $\iota:\overline{\mathbb Q}_\ell\hookrightarrow\mathbb C$ and take an $\iota$-pure constructible complex $K_0$ on $\mathscr X_0$.

After base change to $k=\overline{\mathbb F}_q$, write $f:\mathscr X\to\mathscr Y$ and $K=K_0|_{\mathscr X}$. Then
$$Rf_*K\simeq\bigoplus_{i\in\mathbb Z}{}^p\!\mathcal H^i(Rf_*K)[-i],$$
with semisimple perverse cohomology. The conclusion also holds for a semisimple perverse $K_0$ without a single-weight assumption.

[Sun, Theorem 1.2, arXiv v3] (Duke Math. J. 161 (2012), 2297–2310). Statement checked 18 October 2026. Application to (DT) →
Statement and proof →
III · Affine space under GLₙ
$$G_{\tau^k}=S_k[k],\qquad\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I=\operatorname{Thick}\{G_{\tau^k}:k\in\mathbb Z\}.$$
The length-zero kernels already give every simple object; IIa supplies the general finite-level classification.
Statement and proof →

3 · What remains open

Sheaves for the symmetric rows. For $\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$, $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$, $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$, $\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$ ($n\ge3$) and $E_6/F_4$, Proposition I reduces the question to whether every IC sheaf of an $I$-orbit closure, with its local systems, is a summand of some $\mathrm{IC}_w\ast_I\mathrm{IC}_0$: a computation in Chen–Yi's affine Hecke module. It also needs the level axioms for Chen–Yi's presentation.
The level presentation I1. In the strict form (finite quotients of $I$ acting on finite-type $Y_{N,M}$), I1 is proved for the group, $\mathbb A^n$ and $\mathbb G_m$, and assumed for $\mathrm{GL}_2/T$ with $N\ge1$, as on the pages for Proposition IV.1.
Over $\mathbb C$. Proposition II and the Proposition V are proved over $\overline{\mathbb F}_q$; the Betti versions need a spreading-out uniform in $w$, or the comparison of BBD 6.1.
Function failures in characteristic 0. The symplectic and $\mathrm{GL}_{2p}/\mathrm{GL}_p^2$ failures rest on theorems over $p$-adic fields; Proposition II needs $\mathbb F_q(\!(t)\!)$.
The group's summand set beyond type $A_1$: the lowest two-sided cell, cited from Lusztig and Shi, not re-proved.

Problem

Question. For the varieties $X$ of the Iwahori-generation pages (Table 1), where do Iwahori generation for sheaves and for functions differ?

$$\text{sheaves:}\quad \mathcal C_I^{\mathrm{Iwa}}:=\operatorname{Thick}_{\mathcal C_I}\{\mathcal G\ast_I\mathrm{IC}_0:\mathcal G\in\mathcal H_{c,I}\}\overset{?}{=}\mathcal C_I:=\operatorname{Shv}_c(I\backslash LX),$$ $$\text{functions:}\quad \text{(IG)}\qquad \mathcal H_I\ast\varphi_0\overset{?}{=}C_I:=C_c^\infty(X(F))^I,\qquad \varphi_0=\mathbf 1_{X(\mathcal O)}.$$

A divergence is a variety $X$ for which exactly one of the two equalities holds. First step: the Iwahori analogues of Propositions I and II of the spherical run, and from them a second proof of the first two rows of Table 1.

Setting and conventions

$k=\mathbb C$ (Betti constructible sheaves; the Iwahori-generation pages write coherent $\mathcal D$-modules, the same category by Riemann–Hilbert at each finite level) or $k=\overline{\mathbb F}_q$, $p$ odd, with $\overline{\mathbb Q}_\ell$ coefficients. $G\supseteq B\supseteq T$ split connected reductive, $\widetilde W=X_*(T)\rtimes W$, $K=L^+G$, $I=\mathrm{ev}^{-1}(B)$, $\operatorname{Fl}=LG/I$; $\mathrm{IC}_w$ the intersection complex of the Schubert cell $IwI/I$, $\Delta_w$ its standard object.
$X$ a smooth affine spherical $G$-variety of Table 1, $LX=X(k(\!(t)\!))\supseteq L^+X=X(k[\![t]\!])$. A level presentation $LX=\varinjlim_NY_N$, $Y_N=\varprojlim_MY_{N,M}$, with $\pi_{N,M}:Y_N\to Y_{N,M}$ smooth affine surjective and $I$ acting on $Y_{N,M}$ through $I/I_{N,M}$; a piece is $P=\pi_{N,M}^{-1}(o)$, $o$ an orbit.
$\mathcal C_K=\operatorname{Shv}_c(K\backslash LX)$ (the spherical run's category), $\mathcal C_I=\operatorname{Shv}_c(I\backslash LX)=\bigcup_{N,M}\mathcal C_I^{(N,M)}$, objects pulled back from a finite level; $\mathcal H_{c,I}=\operatorname{Shv}_c(I\backslash LG/I)$ acting by $\mathcal G\ast_IM=m_!(\mathcal G\mathbin{\widetilde\boxtimes}M)$; $\mathrm{IC}_0$ the constant sheaf on $L^+X$. Thick = closed under isomorphisms, shifts, cofibres and retracts.
Function level: $F$ a non-archimedean local field with residue field $\mathbb F_q$ (for traces $F=\mathbb F_q(\!(t)\!)$), $K=G(\mathcal O)\supseteq I$, $\mathcal H_I=C_c(I\backslash G(F)/I)$ with $\operatorname{vol}I=1$, $T_w=\mathbf 1_{IwI}$.

Sources

References

iwahori-generation/index.html · Table 1; Propositions I–IV
summary.html · every variety at both levels, every citation
hypothesis.html · Proposition III: levels, pieces, I1–I3
group.html · Proposition II: the group, by the affine Whittaker functor
case-affine.html · Proposition IV.1: $\mathbb A^n$ (retired; now IIa)
classical.html · function level: Proposition B, Corollary D, E (the group)
classical-affine.html · F3: $\mathbb A^n$ at function level
satake-two-levels/verdict.html · the spherical run: Propositions I and II

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