1 · The goal
Let \(G\supseteq B\) be connected reductive over \(k=\mathbb C\), \(K=L^+G\supseteq I\) the arc group and an Iwahori subgroup, and \(X\) a smooth affine spherical \(G\)-variety; categories are coherent \(\mathcal D\)-modules. The basic object \(\mathrm{IC}_0\), the constant sheaf on \(L^+X\subseteq LX\), is \(K\)-equivariant, hence \(I\)-equivariant. Two equivariant categories and two Hecke categories act:
$$\mathcal C:=\mathcal D_c(K\backslash LX),\qquad\mathcal C_I:=\mathcal D_c(I\backslash LX),\qquad\mathcal H_{c,K}:=\mathcal D_c(K\backslash LG/K),\qquad\mathcal H_{c,I}:=\mathcal D_c(I\backslash LG/I),$$
$$\mathcal C_I^{\mathrm{Iwa}}:=\operatorname{Thick}_{\mathcal C_I}\{\mathcal G\ast_I\mathrm{IC}_0:\ \mathcal G\in\mathcal H_{c,I}\}\ \subseteq\ \mathcal C_I,\qquad\text{the thick }\mathcal H_{c,I}\text{-submodule of }\mathcal C_I\text{ generated by }\mathrm{IC}_0 .$$
At spherical level the Satake objects \(\mathcal V\ast_K\mathrm{IC}_0\) generate \(\mathcal C\) for the group and for symmetric spaces satisfying two root-datum conditions (the Spherical generation workspace). The Iwahori level asks two questions:
$$\textbf{Generation:}\ \ \text{what is }\mathcal C_I^{\mathrm{Iwa}}\,?\qquad\qquad\textbf{Fullness:}\ \ \boxed{\ \mathcal C_I^{\mathrm{Iwa}}\overset{?}{=}\mathcal C_I\ }$$
Proposition I answers the first: \(\mathcal C_I^{\mathrm{Iwa}}\) is the paper's Definition 4.3, it is the Iwahori span of the whole spherical category \(\mathcal C\) exactly when spherical generation holds, and then it is a relative tensor product. The second has no uniform answer: it fails for the group (Proposition II) and holds under three conditions checked one finite level at a time (Proposition III), which Proposition IV tests on \(\mathbb A^n\), the group and \(\mathbb G_m\).
2 · Definitions
Setting
\(k=\mathbb C\). \(G\supseteq B\supseteq T\) connected reductive with a Borel and a maximal torus, \(W=N_G(T)/T\), \(\widetilde W=X_*(T)\rtimes W\). \(X\) is a smooth affine spherical \(G\)-variety satisfying the paper's standing assumption; \(LX=X(F)\) and \(L^+X=X(\mathcal O)\), closed in \(LX\) since \(X\) is affine, is the \(K\)-orbit of the base point.
$$K=L^+G\xrightarrow{\ \mathrm{ev}\ }G,\qquad I=\mathrm{ev}^{-1}(B),\qquad\operatorname{Gr}=LG/K,\qquad\operatorname{Fl}=LG/I,\qquad d=\dim G/B .$$
\(I\)-orbits on \(\operatorname{Gr}\) and \(\operatorname{Fl}\) are the Iwahori–Schubert cells \(It^vK/K\) (\(v\in \widetilde W/W\)) and \(IwI/I\) (\(w\in \widetilde W\)): affine spaces, connected stabilizers, finitely many cells in each closure.
Example. The group case is \(X=H\) under \(G=H\times H\), \((h_1,h_2)\cdot x=h_1xh_2^{-1}\); then \(K=K_H\times K_H\), \(I=I_H\times I_H\), \(I\backslash LX=I_H\backslash LH/I_H\), and \(\Omega\subseteq\widetilde W\), the elements of length zero, indexes the connected components of \(\operatorname{Fl}\).
Categories, forgetful maps, convolutions
$$\mathcal C=\mathcal D_c(K\backslash LX),\quad\mathcal C_I=\mathcal D_c(I\backslash LX),\quad\mathcal H_{c,K}=\mathcal D_c(K\backslash\operatorname{Gr}),\quad\mathcal H_{c,I}=\mathcal D_c(I\backslash LG/I),\quad\mathcal D_c(I\backslash\operatorname{Gr})=\mathcal D_c(I\backslash LG/K).$$
Small, idempotent-complete, stable; \(\mathcal H_{c,K}\) and \(\mathcal H_{c,I}\) monoidal under convolution. The forgetful maps are proper and smooth with fibre \(K/I=G/B\), so \(p^!=p^*[2d]\), \(q^!=q^*[2d]\), and \(p_*\), \(q_*\) preserve constructibility:
$$p:I\backslash LX\longrightarrow K\backslash LX,\qquad q:I\backslash LG/I\longrightarrow I\backslash LG/K .$$
For \(J\in\{K,I\}\) an object on \(I\backslash LG/J\) acts on a \(J\)-equivariant object through the twisted product and the action map \(m_J:I\backslash LG\times^JLX\to I\backslash LX\), proper on bounded supports:
$$\mathcal F\ast_JM:=m_{J*}\big(\mathcal F\mathbin{\widetilde\boxtimes}M\big):\qquad\ast_K:\mathcal D_c(I\backslash\operatorname{Gr})\times\mathcal C\to\mathcal C_I,\qquad\ast_I:\mathcal H_{c,I}\times\mathcal C_I\to\mathcal C_I .$$
The same formula gives the left action of \(\mathcal H_{c,I}\) and the right action of \(\mathcal H_{c,K}\) on \(\mathcal D_c(I\backslash\operatorname{Gr})\). All instances are exact in each variable and associative. The comparison square relating \(\ast_I\) and \(\ast_K\) is drawn on the Proposition I page.
Basic object, standards, thick span
\(\mathrm{IC}_0:=i_!\underline k_{K\backslash L^+X}=i_*\underline k_{K\backslash L^+X}\in\mathcal C\) for the closed embedding \(i\); on \(I\backslash LX\) its image \(p^*\mathrm{IC}_0\) is written \(\mathrm{IC}_0\) again. \(\Delta^{\operatorname{Gr}}_v\in\mathcal D_c(I\backslash\operatorname{Gr})\) and \(\Delta^{\operatorname{Fl}}_w\in\mathcal H_{c,I}\) are the constant sheaves on the cells, extended by zero; \(\Delta_e\) is the unit of \(\mathcal H_{c,I}\). \(\operatorname{Thick}_{\mathcal D}(S)\) is the smallest full subcategory of \(\mathcal D\) containing \(S\) and closed under shifts, cofibres and retracts. Two properties are used throughout:
$$\text{(T1)}\ \ A\to B\to C\ \text{a cofibre sequence},\ A,C\in\operatorname{Thick}(S)\ \Longrightarrow\ B\in\operatorname{Thick}(S);\qquad\text{(T2)}\ \ \Phi\ \text{exact}\ \Longrightarrow\ \Phi\big(\operatorname{Thick}(S)\big)\subseteq\operatorname{Thick}(\Phi S).$$
Thick versus two other spans →
Spherical generation
At spherical level the Satake part is the thick \(\mathcal H_{c,K}\)-span of the basic object, and spherical generation (H1) is the statement that it is everything:
$$\mathcal C^{\mathrm{Sat}}:=\operatorname{Thick}\{\mathcal V\ast_K\mathrm{IC}_0:\mathcal V\in\mathcal H_{c,K}\}\subseteq\mathcal C,\qquad\text{H1}:\ \ \mathcal C^{\mathrm{Sat}}=\mathcal C .$$
The paper proves H1 under its orbit hypotheses (Proposition 4.30), for the group and for the symmetric varieties with \(Z_{G^\theta}(S)\) connected and \(\varphi(\Lambda_T^+)=\Lambda_S^+\); the Spherical generation workspace records that proof.
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 \(\partial P:=\overline P\setminus P\) is a union of lower pieces of the same level. For \(\mathbb A^n\) the levels are lattices and a closed piece contains infinitely many orbits; for the group the pieces are the Bruhat cells: examples on the Proposition III page.
Function level
Over \(F=\mathbb F_q((t))\), compactly supported functions replace sheaves: \(C_K=\mathcal S(X)^K\subseteq C_I=\mathcal S(X)^I\), the Hecke algebras \(\mathcal H_K\), \(\mathcal H_I\), and the basic function \(\varphi_0=\mathbf 1_{X(\mathcal O)}\). Spherical and Iwahori generation read
$$\mathcal H_K\cdot\varphi_0=C_K,\qquad\mathcal H_I\cdot\varphi_0=C_I,$$
and the obstruction module \(Q_X=C_I/\mathcal H_I\cdot C_K\) measures the failure of the second. The full dictionary is on the Proposition I page.
3 · The four propositions
One statement per page. A label such as H1 or I3, a dotted symbol, or a name such as Proposition III opens a copy of what it refers to; each page defines the symbols and hypotheses it uses and links a paper-style PDF of its statement and proof. The concept graph and the literature reader of each proposition are listed in section 5.
I · Generation
Proved under S1, H1, H2
Three descriptions of \(\mathcal C_I^{\mathrm{Iwa}}\), with \(\mathcal F\in\mathcal D_c(I\backslash\operatorname{Gr})=\mathcal D_c(I\backslash LG/K)\), \(M\in\mathcal C=\mathcal D_c(K\backslash LX)\), and every thick span inside \(\mathcal C_I\):
- (a)\(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\), the paper's Definition 4.3; needs only S1.
- (b)\(\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}=\operatorname{Thick}\{\mathcal F\ast_KM:\ M\in\mathcal C\}\) if and only if H1, given H2.
- (c)\(\mathcal C_I^{\mathrm{Iwa}}\simeq\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C\) under H1 and H2.
Hypotheses, statement and proof →
II · Group case
Proved
\(X=H\) under \(G=H\times H\) with \(W\ne1\). Then \(\mathcal C_I=\mathcal H_{c,I_H}\), the affine Hecke category of \(H\), and the Iwahori-generated part is a proper thick two-sided ideal: the unit is missing.
$$\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal A\ast\mathrm{IC}_0\ast\mathcal B:\ \mathcal A,\mathcal B\in\mathcal H_{c,I_H}\}.$$
Detected by an affine Whittaker functor \(\Phi:\mathcal C_I\to\mathrm{Vect}^{\Omega}\) that kills the ideal and not \(\Delta_e\); the only input is \(H_c^*(\mathbb A^1,\exp)=0\).
Statement and proof →
III · Sufficient conditions
Proved
Three conditions, checked on one finite level \((N,M)\) of \(LX\) at a time so that an orbit closure may contain infinitely many \(I\)-orbits, force fullness:
$$\begin{gathered}\text{I1 finite levels},\qquad\text{I2 connected stabilizers},\\ \text{I3 triangularity on pieces, at every level }(N,M)\\[2pt]\Longrightarrow\qquad\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I .\end{gathered}$$
The proof is a dévissage at one level with (T1) and (T2), modelled on the spherical proof.
Hypotheses, statement and proof →
IV · Case studies
Proved
The conditions of Proposition III on two \(X\); \(\mathbb A^n\) is Proposition IIa:
- IV.2The group: I3 fails at every closed piece \(I\omega I\), consistently with Proposition II.
- IV.3\(\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\): I1–I3 hold, so \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\).
Statements, side by side →
4 · Status by family
For the symmetric families, deciding I3 needs the orbit graph of Chen–Yi's module (Proposition III, definitions): \(L^+X\) splits into several \(I\)-orbits, and the function-level failures suggest that no Hecke translate of \(\mathrm{IC}_0\) is a nonzero constant sheaf on a closed one.
The function column is the classical site, for compactly supported functions: the Iwahori function pages (criterion, group case, three \(\mathrm{GL}_2\) examples, symmetric varieties), F1 · Tools and the Steinberg sweep, F2 · Family by family and \(\mathbb A^n\) under \(\mathrm{GL}_n\) (formerly F3). No implication between the two levels is proved. The Summary tab lists every variety at both levels with its witness or reason, and every cited theorem with its hypotheses, verified or assumed.
5 · Concept graphs and literature
Each proposition has a concept-role graph page: A, the key ideas and the justified arrows between them; B, the selected idea's statement, scope and proof; C, the draft; D, the literature reader, with up to three ranked and highlighted extracts for each idea. The reader is where the literature enters this workspace.
Related workspaces: the Spherical generation workspace proves H1; the Iwahori–Matsumoto site treats the Iwahori–Hecke algebra on the tree of \(\mathrm{SL}_2\).