Two definitions of the Iwahori Satake category

1 · The goal

$G$ is a connected reductive group over an algebraically closed field $k$ of characteristic zero, $K=L^+G$ its arc group, $I\subseteq K$ the Iwahori subgroup of a Borel $B$, and $X$ a smooth affine spherical $G$-variety as in the paper's Definition 4.2. One full subcategory of $\mathcal D_c(I\backslash LX)$, the Iwahori Satake category, has three descriptions. Each starts from the same basic object $\mathrm{IC}_0$ and differs only in which Hecke-type category is convolved with it:

$$\mathcal S_T:=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0:\ \mathcal F\in\mathcal D_c(I\backslash LG/K)\},\qquad\mathcal S_H:=\operatorname{Thick}\{\mathcal G\ast_I\mathrm{IC}_0:\ \mathcal G\in\mathcal D_c(I\backslash LG/I)\},$$ $$\mathcal S_E:=\operatorname{EssIm}\big(F_c:\mathcal D_c(I\backslash LG/K)\otimes_{\mathcal H_{c,K}}\mathcal D_c(K\backslash LX)\longrightarrow\mathcal D_c(I\backslash LX)\big).$$

$\mathcal S_T$ is the paper's Definition 4.3. $\mathcal S_H$ is the right-hand side of the handwritten note and the category of Devalapurkar's Conjecture 3.4.14. $\mathcal S_E$ is the working definition of the original paper, with the spherical factor there restricted to its Satake part. The goal in one line, with H1 the spherical Satake generation $\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}=\mathcal D_c(K\backslash LX)$ and H2 the paper's full faithfulness of $F_c$:

$$\boxed{\ \mathcal S_T=\mathcal S_H\ \text{ always;}\qquad\text{given H2,}\ \ \mathcal S_E=\mathcal S_T\iff\text{H1}.\ }$$

Original paper · essential image

$$\mathcal S_E=\operatorname{EssIm}(F_c)$$

contains $\mathcal S_T$; equal exactly when H1 holds

=if and only if H1, given H2

Definition 4.3 · thick span

$$\mathcal S_T=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}$$

$\mathcal F\in\mathcal D_c(I\backslash LG/K)$

=always: key lemma, $\mathcal F$ is a retract of $q_*q^*\mathcal F$

Note and Conjecture 3.4.14 · affine Hecke span

$$\mathcal S_H=\operatorname{Thick}\{\mathcal G\ast_I\mathrm{IC}_0\}$$

$\mathcal G\in\mathcal D_c(I\backslash LG/I)=\mathcal H_{c,I}$

The note assumes H1 and states $\mathcal S_E\simeq\mathcal S_H$; it proves only the key lemma. The Proposition page reconstructs the whole chain and the functions page checks the same chain for literal functions on $X(F)$.

2 · What $\mathrm{IC}_0$ is, and which convolutions exist

The basic object lives on $K\backslash LX$. $L^+X=X(\mathcal O)$ is closed in $LX=X(F)$ because $X$ is affine, and it is the $K$-orbit of the base point. $\mathrm{IC}_0$ is the constant $\mathcal D$-module on that closed orbit, extended by zero:

$$\mathrm{IC}_0:=i_!\,\underline k_{K\backslash L^+X}=i_*\,\underline k_{K\backslash L^+X}\in\mathcal D_c(K\backslash LX),\qquad i:K\backslash L^+X\hookrightarrow K\backslash LX\ \text{closed},\qquad\text{function shadow: }\mathbf 1_{X(\mathcal O)}.$$

So $\mathrm{IC}_0$ is $K$-equivariant, not merely $I$-equivariant. The forgetful functor along $p:I\backslash LX\to K\backslash LX$ sends it to an $I$-equivariant object, written $\mathrm{IC}_0$ again (a $K$-invariant function is $I$-invariant):

$$\mathrm{Oblv}=p^*:\mathcal D_c(K\backslash LX)\longrightarrow\mathcal D_c(I\backslash LX),\qquad p^!=p^*[2d],\qquad d=\dim G/B.$$

$\mathcal S_T$ and $\mathcal S_E$ use $\mathrm{IC}_0$ on $K\backslash LX$; $\mathcal S_H$ uses its image on $I\backslash LX$.

One convolution, two instances. For $J\in\{K,I\}$, objects on $I\backslash LG/J$ act on $J$-equivariant objects on $LX$ through the correspondence

$I\backslash LG\times LX$ $I\backslash LG/J\times J\backslash LX$ $I\backslash LG\overset{J}{\times}LX$ $I\backslash LX$ $\mathrm{pr}$ $\pi_J$ $\tilde p_J$ $m_J$

$\pi_J$ quotients by $J$ acting as $(g,x)\mapsto(gj^{-1},jx)$; $m_J[g,x]=gx$; $\tilde p_J[g,x]=(gJ,Jx)$. $\tilde p_J$ is not a map of $\mathcal D$-module categories by pullback alone: the twisted product is defined by descent along $\pi_J$.

$$\mathcal F\ast_J\mathcal M:=m_{J*}\big(\mathcal F\,\widetilde\boxtimes\,\mathcal M\big),\qquad\pi_J^*\big(\mathcal F\,\widetilde\boxtimes\,\mathcal M\big)=\mathrm{pr}^*(\mathcal F\boxtimes\mathcal M),\qquad(f\ast_J\psi)(x)=\sum_{g\in G(F)/J}f(g)\,\psi(g^{-1}x)\ \ \text{for functions}.$$

Table 1. · The convolutions in play, all instances of the diagram above.
$J$	Left factor	Right factor	Result	Used for
$K$	$\mathcal D_c(I\backslash LG/K)$	$\mathcal D_c(K\backslash LX)$	$\mathcal D_c(I\backslash LX)$	$\mathcal S_T$, $\mathcal S_E$ (the functor $F_c$)
$I$	$\mathcal D_c(I\backslash LG/I)=\mathcal H_{c,I}$	$\mathcal D_c(I\backslash LX)$	$\mathcal D_c(I\backslash LX)$	$\mathcal S_H$ (the affine Hecke action)
$I$	$\mathcal H_{c,I}$	$\mathcal D_c(I\backslash LG/K)$	$\mathcal D_c(I\backslash LG/K)$	left action on $I\backslash\mathrm{Gr}$; $\mathcal S_T$ is an $\mathcal H_{c,I}$-submodule
$K$	$\mathcal D_c(I\backslash LG/K)$	$\mathcal H_{c,K}$	$\mathcal D_c(I\backslash LG/K)$	right action; the tensor product over $\mathcal H_{c,K}$

What cannot be done, and what relates the two spans. A function on $I\backslash G(F)/K$ cannot be convolved on the right with a function on $I\backslash G(F)/I$: the group in the middle must agree. $\mathcal S_T$ and $\mathcal S_H$ never need this. They convolve two different Hecke-type categories with one and the same $\mathrm{IC}_0$, using its $K$-equivariance for $\mathcal S_T$ and only its $I$-equivariance for $\mathcal S_H$. The forgetful map $q:I\backslash LG/I\to I\backslash LG/K$, proper with fibre $K/I=G/B$, carries one to the other:

$$\mathcal G\ast_I\mathrm{IC}_0\simeq(q_*\mathcal G)\ast_K\mathrm{IC}_0,\qquad(q^*\mathcal F)\ast_I\mathrm{IC}_0\simeq(q_*q^*\mathcal F)\ast_K\mathrm{IC}_0\simeq\big(\mathcal F\otimes^!\gamma^!\mathcal A\big)\ast_K\mathrm{IC}_0 .$$

For functions $q_*q^*f=[K:I]\,f$, so the second identity reads $(q^*f)\ast_I\mathbf 1_{X(\mathcal O)}=[K:I]\cdot f\ast_K\mathbf 1_{X(\mathcal O)}$ and the two spans coincide at once. For sheaves the number $[K:I]$ becomes the cohomology $\mathcal A$ of $G/B$, and $\mathcal F$ is only a direct summand of $q_*q^*\mathcal F$; producing that summand is the note's key lemma.

3 · Is $\mathcal S_T=\mathcal S_H$ true, and why compare with $\mathcal S_H$?

$\mathcal S_T=\mathcal S_H$ holds, for every $X$ as above, in characteristic zero. $\supseteq$: $\mathcal G\ast_I\mathrm{IC}_0\simeq(q_*\mathcal G)\ast_K\mathrm{IC}_0$ with $q_*\mathcal G$ coherent because $q$ is proper. $\subseteq$: $\mathcal F\ast_K\mathrm{IC}_0$ is a direct summand of $(q^*\mathcal F)\ast_I\mathrm{IC}_0\in\mathcal S_H$ by the key lemma, and a thick span contains the direct summands of its objects. For functions both inclusions are one-line identities, and the spans agree with $\mathcal H_I\cdot\mathbf 1_{X(\mathcal O)}$, the cyclic $\mathcal H_I$-module generated by the basic function (functions page).
The note's strategy. The note writes the goal with $\mathcal S_E$ on the left and $\mathcal S_H$ on the right, then proves only the key lemma. The key lemma is exactly the non-formal half of $\mathcal S_T=\mathcal S_H$. The passage from $\mathcal S_E$ to $\mathcal S_T$ is the note's standing assumption “Sat = everything” (H1) together with the paper's §5 (H2). Reading the note as “$\mathcal S_E=\mathcal S_T$ under H1–H2, then $\mathcal S_T=\mathcal S_H$ by the key lemma” accounts for every line of it.
Why $\mathcal S_H$. Devalapurkar, Conjecture 3.4.14 defines the Iwahori Satake category as the full subcategory of $\mathrm{Shv}^c_I(X(\mathbb C(\!(t)\!)))$ generated by $\mathrm{IC}_0=i_!\mathbb Q$ under the action of the affine Hecke category $\mathrm{Shv}^{c,\mathrm{Sat}}_{I\times I}(G(\mathbb C(\!(t)\!)))$ by convolution: that is $\mathcal S_H$. The paper's Definition 4.3 is $\mathcal S_T$, because $\mathcal S_T$ is what its functor $F_c^{\mathrm{Sat}}$ produces (Theorem 5.1), and the paper presents its main theorem as a variant of Conjecture 3.4.14. The Proposition identifies the two. A second reason: $\mathcal S_H$ is visibly the thick $\mathcal H_{c,I}$-submodule generated by $\mathrm{IC}_0$, so Definition 4.3 is stable under the affine Hecke action that the main theorem is equivariant for.
Where H1 enters, sharply. Given H2, $\mathcal S_T\subseteq\mathcal S_E$ always, and $\mathcal S_E\subseteq\mathcal S_T$ forces H1: $p^*\mathcal M\in\mathcal S_E$ for every $\mathcal M\in\mathcal D_c(K\backslash LX)$, averaging over $G/B$ carries $\mathcal S_T$ into $\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}$, and the key lemma for $p:I\backslash LX\to K\backslash LX$ exhibits $\mathcal M$ as a direct summand of $p_*p^*\mathcal M$. The note's assumption is therefore necessary, not only sufficient; the same holds for functions, where it reads $\mathcal H_K\cdot\mathbf 1_{X(\mathcal O)}=C_c(X(F))^K$.
Caveats. Characteristic zero enters only through $|W|\in k^\times$, in both uses of the key lemma. Devalapurkar's Hecke factor is the Satake part of the affine Hecke category of constructible sheaves, while the paper's $\mathcal H_{c,I}$ is all coherent $\mathcal D$-modules on $I\backslash LG/I$; the Proposition is proved for $\mathcal H_{c,I}$, the span by the Satake part sits inside $\mathcal S_H$, and equality of the two spans is not checked here.

4 · Definitions

Setting

$k$ is algebraically closed of characteristic zero; $G\supseteq B\supseteq T$ a connected reductive group, a Borel and a maximal torus; $W=N_G(T)/T$. Evaluation at $t=0$ defines the arc group, the Iwahori subgroup and the two homogeneous ind-schemes:
$$K=L^+G\xrightarrow{\ \mathrm{ev}\ }G,\qquad I=\mathrm{ev}^{-1}(B),\qquad\mathrm{Gr}=LG/K,\qquad\mathrm{Fl}=LG/I,\qquad d=\dim G/B.$$
$X$ is a smooth affine spherical $G$-variety with connected $B$-stabilizers on its open $B$-orbit (paper, Definition 4.2); $LX$ is its loop space and $L^+X$ the closed orbit of the base point.
Categories

$\mathcal D(-)$ is the DG category of $\mathcal D$-modules on a prestack and $\mathcal D_c(-)$ its small subcategory of coherent objects (paper, Definition 3.27). The Hecke categories and the module categories of the Proposition are
$$\mathcal H_{c,K}=\mathcal D_c(K\backslash\mathrm{Gr}),\qquad\mathcal H_{c,I}=\mathcal D_c(I\backslash LG/I),\qquad\mathcal D_c(I\backslash\mathrm{Gr})=\mathcal D_c(I\backslash LG/K),\qquad\mathcal D_c(K\backslash LX),\ \ \mathcal D_c(I\backslash LX).$$
The three subcategories
$$\mathcal S_E:=\operatorname{EssIm}\big(F_c:\mathcal D_c(I\backslash\mathrm{Gr})\otimes_{\mathcal H_{c,K}}\mathcal D_c(K\backslash LX)\to\mathcal D_c(I\backslash LX)\big),$$ $$\mathcal S_T:=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0:\mathcal F\in\mathcal D_c(I\backslash\mathrm{Gr})\},\qquad\mathcal S_H:=\operatorname{Thick}\{\mathcal G\ast_I\mathrm{IC}_0:\mathcal G\in\mathcal H_{c,I}\}.$$
The full dictionary to the two versions of the paper, to the note and to Conjecture 3.4.14 is on the Proposition page.

5 · The pages

I · Two definitions agree

S_T = S_H proved; S_E = S_T ⟺ H1 under H2

Unconditionally $\mathcal S_T=\mathcal S_H$. Under H2, $\mathcal S_E=\mathcal S_T$ holds if and only if H1 does; so under H1 and H2,

$$\mathcal S_E=\mathcal S_T=\mathcal S_H.$$

Restricting the spherical factor to its Satake subcategory, as the original paper did, gives $\mathcal S_T$ under H2 alone.

Hypotheses, statement, proof and graph →

Key lemma · the retraction

Proved in characteristic zero

For any $Y\to\mathbb BG$ and $q:Y\times_{\mathbb BG}\mathbb BB\to Y$, every $\mathcal F\in\mathcal D(Y)$ is a direct summand of $q_*q^*\mathcal F$:

$$\beta_{\mathcal F}\circ\eta_{\mathcal F}=\mathrm{id}_{\mathcal F},\qquad\beta_{\mathcal F}=\tfrac{1}{|W|}\,\gamma^!\operatorname{tr}_{\mathcal A}\otimes^!\mathrm{id}_{\mathcal F}.$$

Used for $I\backslash\mathrm{Fl}\to I\backslash\mathrm{Gr}$ (the note) and for $I\backslash LX\to K\backslash LX$ (necessity of H1).

Statement, proof and graph →

Functions · the classical statement

Elementary, with references

For compactly supported functions on $X(F)$, $F$ a local field: the two spans are $\mathcal H_I\cdot\mathbf 1_{X(\mathcal O)}$, the image of the tensor product equals it exactly when $\mathcal H_K\cdot\mathbf 1_{X(\mathcal O)}=C_c(X(F))^K$, and in the group case all three are the two-sided ideal $\mathcal H_Ie_K\mathcal H_I$, which is proper.

Statement, proof, example and graph →

Table 2. · Status of each statement and what it depends on.
Statement	Status	Depends on
$\mathcal S_T=\mathcal S_H$	proved here	key lemma; convolution compatibility; $\|W\|\in k^\times$
$\operatorname{EssIm}(F_c\|_{\otimes\,\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}})=\mathcal S_T$	conditional	H2 (paper §5: full faithfulness; GR Cor. 8.7.4 and Neeman Lemma 2.2 for the generators)
$\mathcal S_E=\mathcal S_T\iff$ H1	conditional on H2	H2; key lemma for $p$; averaging over $G/B$
$\mathcal S_E=\mathcal S_T=\mathcal S_H$ (the note's goal)	conditional	H1 (paper §4.4, Proposition 4.30 under its five orbit hypotheses) and H2
Key lemma	proved here	base change, projection formula, $\|W\|\in k^\times$
Function-level statement	proved here	Fubini on $G(F)/I\to G(F)/K$; $[K:I]$ invertible; classical Hecke algebra facts

One concept graph for this Proposition and Proposition I of the Iwahori generation workspace, in that workspace's notation: concept graph. Sources: the handwritten note and the paper snapshots in panel C of each page; literature with located highlights in panel D. H1 is proved for symmetric spaces in the Spherical generation workspace. The Iwahori generation workspace uses this Proposition as its I′ and asks when $\mathcal S_H$ is all of $\mathcal D_c(I\backslash LX)$: false for the group (II′), with sufficient conditions (III′), the case $\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}$ (IV′), and a function-level page with the Steinberg representation and explicit $\mathrm{GL}_2$ examples.

\(J\)	Left factor	Right factor	Result	Used for
\(K\)	\(\mathcal D_c(I\backslash LG/K)\)	\(\mathcal D_c(K\backslash LX)\)	\(\mathcal D_c(I\backslash LX)\)	\(\mathcal S_T\), \(\mathcal S_E\) (the functor \(F_c\))
\(I\)	\(\mathcal D_c(I\backslash LG/I)=\mathcal H_{c,I}\)	\(\mathcal D_c(I\backslash LX)\)	\(\mathcal D_c(I\backslash LX)\)	\(\mathcal S_H\) (the affine Hecke action)
\(I\)	\(\mathcal H_{c,I}\)	\(\mathcal D_c(I\backslash LG/K)\)	\(\mathcal D_c(I\backslash LG/K)\)	left action on \(I\backslash\mathrm{Gr}\); \(\mathcal S_T\) is an \(\mathcal H_{c,I}\)-submodule
\(K\)	\(\mathcal D_c(I\backslash LG/K)\)	\(\mathcal H_{c,K}\)	\(\mathcal D_c(I\backslash LG/K)\)	right action; the tensor product over \(\mathcal H_{c,K}\)