III · Sufficient conditions

1 · The question

For which $X$ is $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$? Proposition II shows that it fails for the group. Proposition III gives a sufficient condition, modelled on the paper's proof of spherical generation but checked on one finite level $(N,M)$ of $LX$ at a time, so that an orbit closure may contain infinitely many $I$-orbits:

$$\underbrace{\text{I1 finite levels},\qquad\text{I2 connected stabilizers},\qquad\text{I3 triangularity on pieces}}_{\text{at every level }(N,M)}\ \Longrightarrow\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I .$$

Proposition IV checks the conditions on $\mathbb A^n$ under $\mathrm{GL}_n$, where they hold although every orbit closure is infinite, on the group, where I3 fails, and on $\mathbb G_m$. Function level: Iwahori pages, Corollary D, and 𝔸ⁿ pages; no implication between the two levels is proved.

2 · Definitions

Iwahori orbits on $LX$

$\mathcal V$ is the set of $I$-orbits $O_v\subseteq LX$, ordered by $u\le v\iff O_u\subseteq\overline{O_v}$. Chen–Yi parametrize $\mathcal V$ for symmetric $X$, prove that each $\overline{O_v}$ is placid and a finite union of orbits, and present $LX=\varinjlim_v\overline{O_v}$ as an $I$-placid ind-scheme (the paper's ). For $\mathbb A^n$ each closure is a lattice and contains infinitely many orbits (𝔸ⁿ, orbits lemma).
Levels, pieces and their standard objects

A level presentation is
$$LX=\varinjlim_N Y_N,\qquad Y_N=\varprojlim_M Y_{N,M},\qquad\pi_{N,M}:Y_N\to Y_{N,M},$$
with $Y_N$ closed, $I$-stable and placid (for instance an orbit closure), $Y_{N,M}$ of finite type, $\pi_{N,M}$ smooth, affine and surjective, and $I$ acting on $Y_{N,M}$ through a finite-dimensional quotient $I/I_{N,M}$. Then
$$\begin{gathered}\text{piece of level }(N,M):\quad P=\pi_{N,M}^{-1}(o),\qquad o\subseteq Y_{N,M}\text{ an }I/I_{N,M}\text{-orbit},\\[6pt]\mathcal P_{N,M}:=\{\text{pieces of level }(N,M)\},\qquad P'\le P\iff P'\subseteq\overline P,\qquad\Delta_P:=j_{P!}\,\underline k_P,\\[6pt]\mathcal C_I^{(N,M)}:=\{\pi_{N,M}^*\bar{\mathcal F}\ \text{extended by zero}:\ \bar{\mathcal F}\in\mathcal D_c(I/I_{N,M}\backslash Y_{N,M})\}\subseteq\mathcal C_I .\end{gathered}$$
Pieces of one level partition $Y_N$; each is $I$-stable, locally closed and a union of $I$-orbits, and $\overline P=\pi_{N,M}^{-1}(\overline o)$, so $\partial P:=\overline P\setminus P$ is a union of lower pieces of the same level. An orbit pulled back from a finite level is a piece for $M\gg0$, with $\Delta_P=i_{v!}\underline k$; an orbit of infinite codimension, such as $\{0\}\subseteq L\mathbb A^n$, is never a piece and has no standard object in $\mathcal C_I$.

Example A · $\mathbb A^n$ under $\mathrm{GL}_n$ (IIa). Lattices give the levels, and the finite-dimensional quotient is a congruence quotient of $I$:
$$Y_N=t^{-N}\mathcal O^n,\qquad Y_{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n\simeq\mathbb A^{(N+M)n},\qquad I_{N,M}=1+t^{N+M}M_n(\mathcal O).$$
The $I/I_{N,M}$-orbits on $Y_{N,M}$ are the images of the orbits $O_{m,j}$ with $-N\le m<M$ and the point $0$; so
$$\mathcal P_{N,M}=\{O_{m,j}:-N\le m<M,\ 1\le j\le n\}\sqcup\{T_M\},\qquad T_M:=\pi_{N,M}^{-1}(0)=t^M\mathcal O^n=\{0\}\sqcup\bigsqcup_{m\ge M,\,j}O_{m,j},\qquad\Delta_{T_M}=\underline k_{T_M}.$$
The closed piece $T_M$ contains infinitely many orbits; raising $M$ splits off $n$ more of them.

Example B · the group $X=H$ under $H\times H$ (IV.2). Bruhat closures give the levels, and the quotient is by the right action of a congruence subgroup $I_M=\ker(I\to H(\mathcal O/t^M))$:
$$Y_N=\bigcup_{\ell(w)\le N}\overline{IwI},\qquad Y_{N,M}=Y_N/I_M,\qquad I_{N,M}=I_{M'}\ \text{ with }M'\ge M\text{ such that }w^{-1}I_{M'}w\subseteq I_M\text{ for all }\ell(w)\le N .$$
Each $IvI$ is right $I_M$-stable, so it is the preimage of its image, and the pieces are the orbits at every level:
$$\mathcal P_{N,M}=\{IvI:\ v\le w,\ \ell(w)\le N\},\qquad\Delta_P=\Delta_v=j_{v!}\,\underline k_{IvI},\qquad|\mathcal P_{N,M}|=\#\{v\in\widetilde W:v\le w\text{ for some }\ell(w)\le N\}<\infty .$$
Every piece is an orbit, so here a level presentation is the Bruhat stratification.
The affine Lusztig–Vogan module and the orbit graph

Chen–Yi's module $M$ is the free $\mathbb Z[q^{\pm1/2}]$-module on the pairs $(\xi,v)$, $\xi$ an $I$-equivariant local system on $O_v$, with the action of the affine Hecke algebra $\mathcal H$ of $G$ given by their formula (29): $T_s$ acts on $e_{\xi,v}$ by a combination of $e_{\xi,v}$ and of $e_{\xi',v'}$ for the orbits $v'$ of $P_s\cdot O_v$, according to the type of $v$ for $s$ (the affine form of the Lusztig–Vogan and Mars–Springer tables). The class of $\mathcal F\in\mathcal C_I$ is $[\mathcal F]=\sum_{v,\xi,i}(-1)^i\operatorname{rk}H^i(i_v^*\mathcal F)_\xi\,e_{\xi,v}$ at $q=1$, and $[\Delta_s\ast\mathcal F]=T_s\cdot[\mathcal F]$. The orbit graph has vertex set $\mathcal V$ and an edge $v\to v'$ for each simple affine $s$ and each orbit $v'\ne v$ in $P_s\cdot O_v$ on which $T_se_v$ has a nonzero coefficient. It is the tool for checking I3 on a symmetric $X$.

3 · Hypotheses

Click a label to recall it.

I1 · Finite levels.

$LX$ has a level presentation such that
$$\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)},\qquad|\mathcal P_{N,M}|<\infty\ \text{ for all }N,M .$$
Discussion
The first part says that every coherent $I$-equivariant object is supported on some $Y_N$ and pulled back from a finite level; it is the reading, for coherent equivariant objects, of the paper's , $\mathcal D^*(Y_N)=\varinjlim_M\mathcal D(Y_{N,M})$ along $*$-pullbacks, and is assumed, not proved, here. The second part replaces “every orbit closure is a finite union of $I$-orbits” (Chen–Yi for symmetric $X$; the Iwahori form of the paper's finiteness hypothesis on orbit closures): when that holds, the pieces of $Y_N=\overline{O_v}$ at large level are its orbits. Function analogue: compact support and local constancy.
I2 · Connected stabilizers at each level.

Every stabilizer of $I/I_{N,M}$ on $Y_{N,M}$ is connected.

Discussion
On one finite-level orbit $o=H/S$, $S$ connected, the equivariant constructible objects are $\operatorname{Thick}\{\underline k_o\}$: their cohomology sheaves are representations of $\pi_0S=1$. With disconnected stabilizers replace $\Delta_P$ by the $\Delta_{P,\xi}$ for all local systems $\xi$, in I3 as well.
I3 · Triangularity on pieces.

For every level $(N,M)$ and every $P\in\mathcal P_{N,M}$ there are $\mathcal G\in\mathcal H_{c,I}$ and $Y\in\{\mathrm{IC}_0\}\cup\{\Delta_{P'}:P'\in\mathcal P_{N,M},\ P'<P\}$ such that $A:=\mathcal G\ast_IY\in\mathcal C_I^{(N,M)}$ and
$$\operatorname{supp}A\subseteq\overline P,\qquad A|_P\simeq\bigoplus_{s=1}^{r}\underline k_P[d_s],\qquad0<r<\infty .$$
Discussion
The Iwahori form of the paper's triangularity hypotheses, with pieces in place of orbits: a Hecke translate of something already generated reaches $P$ with nonzero constant restriction and does not leak above $P$. For a closed piece only $Y=\mathrm{IC}_0$ is available. For $\mathbb A^n$ a single lattice translate of $\mathrm{IC}_0$ works for each piece (IIa); for the group it fails at the closed orbits (IV.2).

4 · Proposition III

Proposition III · Sufficient conditionsProved

Assume I1, I2 and I3. Then

$$\boxed{\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I .\ }$$

Checked on three cases, with every hypothesis verified: Proposition IV.

Concept graph and literature reader of this proof →

Proof

Step 1 generates one level by standards; step 2 reaches each standard by induction inside the level; step 3 concludes.

Method Dévissage at one level.
$$Z\subseteq Y_N\ \text{a closed union of pieces of level }(N,M)\ \Longrightarrow\ \{\mathcal F\in\mathcal C_I^{(N,M)}:\operatorname{supp}\mathcal F\subseteq Z\}=\operatorname{Thick}\{\Delta_P:\ P\in\mathcal P_{N,M},\ P\subseteq Z\}.$$
Proof
- $\pi_{N,M}^*$ is exact and commutes with $j_!$ and $i_*$ along preimages of locally closed subsets (smooth base change), and $\pi_{N,M}^*j_{o!}\underline k_o=\Delta_P$ for $P=\pi_{N,M}^{-1}(o)$. So it suffices to prove the statement on $Y_{N,M}$, which has finitely many $I/I_{N,M}$-orbits (I1).
- Induction on the number of orbits in $\bar Z:=\pi_{N,M}(Z)$. Choose an orbit $o$ open in $\bar Z$, $\bar Z'=\bar Z\setminus o$ closed. For $\bar{\mathcal F}$ supported on $\bar Z$: $j_!j^*\bar{\mathcal F}\to\bar{\mathcal F}\to i_*i^*\bar{\mathcal F}$; $j^*\bar{\mathcal F}\in\operatorname{Thick}\{\underline k_o\}$ by I2, so $j_!j^*\bar{\mathcal F}\in\operatorname{Thick}\{j_{o!}\underline k_o\}$, and $i_*i^*\bar{\mathcal F}$ is supported on $\bar Z'$. Conclude by (T1).
Method Induction inside one level: every standard is Iwahori-generated.
$$\Delta_{P'}\in\mathcal C_I^{\mathrm{Iwa}}\ \text{for all }P'<P\text{ in }\mathcal P_{N,M}\ \Longrightarrow\ \Delta_P\in\mathcal C_I^{\mathrm{Iwa}} .$$
Proof
- $\mathcal P_{N,M}$ is finite (I1), so the induction is well founded even when orbit closures are infinite. Take $\mathcal G$, $Y$, $A$ from I3. $Y\in\mathcal C_I^{\mathrm{Iwa}}$ and $\mathcal C_I^{\mathrm{Iwa}}$ is an $\mathcal H_{c,I}$-submodule (associativity and (T2)), so $A\in\mathcal C_I^{\mathrm{Iwa}}$.
- With $j:P\hookrightarrow\overline P$, $i:\partial P\hookrightarrow\overline P$: in $j_!j^*A\to A\to i_*i^*A$ the third term lies in $\mathcal C_I^{(N,M)}$ and is supported on $\partial P$, a closed union of pieces $P'<P$; by step 1 and induction it lies in $\mathcal C_I^{\mathrm{Iwa}}$. By (T1), $j_!j^*A\in\mathcal C_I^{\mathrm{Iwa}}$.
- By I3, $j_!j^*A\simeq\bigoplus_s\Delta_P[d_s]$ with at least one summand, so $\Delta_P$ is a retract of an object of $\mathcal C_I^{\mathrm{Iwa}}$, hence in it. (The paper's argument for Proposition 4.30, with $\mathcal H_{c,I}$ and pieces.)
Conclusion The claim.
$$\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)}=\bigcup_{N,M}\operatorname{Thick}\{\Delta_P:P\in\mathcal P_{N,M}\}\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal C_I .$$
Proof
- I1, then step 1 with $Z=Y_N$, then step 2 for every piece.

Literature and remarkstemplate · orbits · not claimed

Template. Proposition III is the paper's proof of spherical generation (its orbit hypotheses and Proposition 4.30), with the pieces of a level in place of orbits; the Spherical generation workspace records that proof. Working level by level removes the paper's finiteness hypothesis on orbit closures, which fails for $\mathbb A^n$.
Orbits. Chen–Yi, Singularities of orbit closures in loop spaces of symmetric varieties, arXiv:2310.20006v4: the $I_0$-orbits on $LX$ and the closed ones (Lemmas 23, 31), placidity of closures (§5), dimensions (Proposition 63), the affine Hecke module $M$ and formula (29) (Theorem 1, Proposition 74, Lemma 75), the action (21) on $D(I_0\backslash LX)$. Finite-field model: Mars–Springer, Definition 3.2.7 and table 4.3.1 (located on the note's functions page).
Not claimed. The finite-level presentation in I1 is assumed, not derived from the paper's , so every case read in $\mathcal D_c(I\backslash LX)$ is conditional on it. This page proves no implication between the sheaf statement and its function-level shadow (thick spans contain retracts, functions do not see them); functions ⇒ sheaves over $\mathbb F_q(\!(t)\!)$ is Proposition II of the Iwahori run. I3 for a symmetric $X$ other than a torus is open (status by family).