Iwahori generation · Proposition I

Generation

What the affine Hecke category generates from \(\mathrm{IC}_0\), in three descriptions.

1 · The question

Replace the arc group \(K=L^+G\) by an Iwahori subgroup \(I\subseteq K\). The basic object is the same \(\mathrm{IC}_0\) as at spherical level, the constant sheaf on \(L^+X\), regarded as an object of \(\mathcal C_I\); the Iwahori-generated part is the thick \(\mathcal H_{c,I}\)-submodule it generates:

$$\mathcal C_I:=\mathcal D_c(I\backslash LX),\qquad\mathcal H_{c,I}:=\mathcal D_c(I\backslash LG/I),\qquad\mathcal C_I^{\mathrm{Iwa}}:=\operatorname{Thick}_{\mathcal C_I}\{\mathcal G\ast_I\mathrm{IC}_0:\ \mathcal G\in\mathcal H_{c,I}\}.$$

Proposition I gives three descriptions of \(\mathcal C_I^{\mathrm{Iwa}}\): (a) through the bimodule \(\mathcal D_c(I\backslash\operatorname{Gr})=\mathcal D_c(I\backslash LG/K)\) acting on \(\mathrm{IC}_0\); (b) through the whole spherical category \(\mathcal C=\mathcal D_c(K\backslash LX)\); (c) as a relative tensor product over \(\mathcal H_{c,K}\). Its inputs are one key lemma (S1) and the two hypotheses H1, H2 of the two-definitions note; the concept graph merges this proof with the note's. Function level: Iwahori function pages.

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.

  • 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: \(\mathcal F\ast_K(\mathcal V\ast_KM)\simeq(\mathcal F\ast_K\mathcal V)\ast_KM\) and \(\mathcal G\ast_I(\mathcal F\ast_KM)\simeq(\mathcal G\ast_I\mathcal F)\ast_KM\).

    Comparison square. With \(r:=q\times\mathrm{id}\), both squares are Cartesian, \(m_I=m_K\circ r\), and \(\mathrm{pr}_2\circ r=p\circ\mathrm{pr}_2\):

    \(I\backslash LG/I\)\(I\backslash LG/K\) \(I\backslash LG\overset{I}{\times}LX\)\(I\backslash LG\overset{K}{\times}LX\)\(I\backslash LX\) \(I\backslash LX\)\(K\backslash LX\) \(q\) \(r\) \(m_K\) \(p\) \(\mathrm{pr}_1\) \(\mathrm{pr}_1\) \(\mathrm{pr}_2\) \(\mathrm{pr}_2\)

    \(m_I\) is the middle row read through \(r\). On the left, \(\mathcal G\mathbin{\widetilde\boxtimes}p^*M=\mathrm{pr}_1^*\mathcal G\otimes\mathrm{pr}_2^*M\); base change along the upper square turns \(\mathrm{pr}_1^*\mathcal G\) into \(\mathrm{pr}_1^*q_*\mathcal G\) after \(r_*\).

  • 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. \(\operatorname{Thick}_{\mathcal D}(S)\) is the smallest full subcategory of \(\mathcal D\) containing \(S\) and closed under shifts, cofibres and retracts. Two properties are used throughout:

    $$\text{(T1)}\ \ A\to B\to C\ \text{a cofibre sequence},\ A,C\in\operatorname{Thick}(S)\ \Longrightarrow\ B\in\operatorname{Thick}(S);\qquad\text{(T2)}\ \ \Phi\ \text{exact}\ \Longrightarrow\ \Phi\big(\operatorname{Thick}(S)\big)\subseteq\operatorname{Thick}(\Phi S).$$
  • Key lemma (retraction of \(\mathcal F\to q_*q^*\mathcal F\))

    \(\gamma:I\backslash\operatorname{Gr}\to\mathbb BG\) classifies the \(G\)-torsor induced from \(LG\to\operatorname{Gr}\), \(\pi:\mathbb BB\to\mathbb BG\), and \(I\backslash\operatorname{Fl}=I\backslash\operatorname{Gr}\times_{\mathbb BG}\mathbb BB\) with \(q\) the projection. Put \(\mathcal A:=\pi_*\pi^*\omega_{\mathbb BG}\), an algebra in \(\mathcal D(\mathbb BG)\) with unit \(u\) and trace \(\operatorname{tr}_{\mathcal A}:\mathcal A\to\omega_{\mathbb BG}\), \(\operatorname{tr}_{\mathcal A}\circ u=\dim\mathcal A=\chi(G/B)=|W|\). Under S1, for every \(\mathcal F\) on \(I\backslash\operatorname{Gr}\), the unit \(\eta_{\mathcal F}:\mathcal F\to q_*q^*\mathcal F\) has a natural retraction

    $$\beta_{\mathcal F}:q_*q^*\mathcal F\simeq\gamma^!\mathcal A\otimes^!\mathcal F\xrightarrow{\ \tfrac1{|W|}\gamma^!\operatorname{tr}_{\mathcal A}\otimes^!\mathrm{id}\ }\mathcal F,\qquad\beta_{\mathcal F}\circ\eta_{\mathcal F}=\mathrm{id}.$$

    The same square with \(K\backslash LX\) in place of \(I\backslash\operatorname{Gr}\) gives the retraction of \(M\to p_*p^*M\). Proof: key lemma page. For functions, \(q_*q^*f=[K:I]\,f\) and the retraction is division by \([K:I]\).

  • Relative tensor product and the functor \(F_c\) $$\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C:=\big(\operatorname{Ind}\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\operatorname{Ind}\mathcal H_{c,K}}\operatorname{Ind}\mathcal C\big)^\omega,$$

    the compact objects of the tensor product of the ind-completions (paper, Definition of the small relative tensor product). It is thickly generated by the pure tensors \(\operatorname{ins}(\mathcal F\boxtimes M)\) (paper, compact objects of a relative tensor product over a rigid monoidal category). The paper's functor is the restriction to compact objects of the continuous functor obtained from the \(LG\)-action by taking \(I\)-invariants:

    $$F_c:\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C\longrightarrow\mathcal C_I,\qquad\operatorname{ins}(\mathcal F\boxtimes M)\longmapsto\mathcal F\ast_KM,$$

    exact. Its essential image, when it is fully faithful, is closed under shifts, cofibres and retracts, hence thick.

3 · Hypotheses

S1 enters the key lemma (steps 3 and 5), H1 enters step 5, H2 enters steps 5 and 6. The labels are those of the note; click one to recall it.

4 · Proposition I

Proposition I · GenerationProved

Objects. All categories are coherent \(\mathcal D\)-modules, and every thick span below is taken inside \(\mathcal C_I\):

$$\mathcal C_I:=\mathcal D_c(I\backslash LX),\qquad\mathcal C:=\mathcal D_c(K\backslash LX),\qquad\mathcal H_{c,I}:=\mathcal D_c(I\backslash LG/I),\qquad\mathcal H_{c,K}:=\mathcal D_c(K\backslash LG/K),\qquad\mathcal D_c(I\backslash\operatorname{Gr}):=\mathcal D_c(I\backslash LG/K),$$ $$\mathcal C_I^{\mathrm{Iwa}}:=\operatorname{Thick}_{\mathcal C_I}\{\mathcal G\ast_I\mathrm{IC}_0:\ \mathcal G\in\mathcal H_{c,I}\},\qquad\text{the thick }\mathcal H_{c,I}\text{-submodule of }\mathcal C_I\text{ generated by }\mathrm{IC}_0 .$$

Statement. Assume S1, and let \(\mathcal F\) range over \(\mathcal D_c(I\backslash\operatorname{Gr})\) and \(M\) over \(\mathcal C\).

  • (a)\(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}_{\mathcal C_I}\{\mathcal F\ast_K\mathrm{IC}_0\}\). No further hypothesis; the right side is the paper's Definition 4.3, \(\mathcal D_c(I\backslash LX)^{\mathrm{Sat}}\).
  • (b)Given H2: \(\operatorname{Thick}_{\mathcal C_I}\{\mathcal F\ast_K\mathrm{IC}_0\}=\operatorname{Thick}_{\mathcal C_I}\{\mathcal F\ast_KM:\ M\in\mathcal C\}\) if and only if H1 holds.
  • (c)Under H1 and H2: \(F_c:\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C\xrightarrow{\ \sim\ }\mathcal C_I^{\mathrm{Iwa}}\), \(\operatorname{ins}(\mathcal F\boxtimes M)\mapsto\mathcal F\ast_KM\).

In one line:

$$\mathcal C_I^{\mathrm{Iwa}}\overset{(a)}{=}\operatorname{Thick}_{\mathcal C_I}\{\mathcal F\ast_K\mathrm{IC}_0\}\overset{(b)}{=}\operatorname{Thick}_{\mathcal C_I}\{\mathcal F\ast_KM:\ M\in\mathcal C\}\overset{(c)}{\simeq}\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C .$$

What \(\mathcal C\) is. \(\mathcal C=\mathcal D_c(K\backslash LX)\) is the spherical category: the \(K\)-equivariant objects on \(LX\), on which the spherical Hecke category \(\mathcal H_{c,K}\) acts. \(\mathcal C_I\) is its Iwahori-equivariant counterpart, and \(p^*:\mathcal C\to\mathcal C_I\) forgets from \(K\) to \(I\). So (b) says: given H2, the generators \(\mathcal F\ast_KM\) with \(M\) anywhere in \(\mathcal C\) can be replaced by those with \(M=\mathrm{IC}_0\) exactly when \(\mathcal C\) is itself generated by its Satake objects \(\mathcal V\ast_K\mathrm{IC}_0\) (H1). Under H1 the middle term is also \(\operatorname{Thick}_{\mathcal C_I}\{\mathcal G\ast_Ip^*M:\ \mathcal G\in\mathcal H_{c,I},\ M\in\mathcal C\}\), the thick \(\mathcal H_{c,I}\)-submodule generated by \(p^*\mathcal C\). Generators: \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}_{\mathcal C_I}\{\Delta^{\operatorname{Gr}}_v\ast_K\mathrm{IC}_0:v\in\widetilde W/W\}=\operatorname{Thick}_{\mathcal C_I}\{\Delta^{\operatorname{Fl}}_w\ast_I\mathrm{IC}_0:w\in\widetilde W\}\).

Classically. Over \(F=\mathbb F_q(\!(t)\!)\) replace sheaves by compactly supported functions: \(C_K=\mathcal S(X)^K\subseteq C_I=\mathcal S(X)^I\), \(\varphi_0=\mathbf 1_{X(\mathcal O)}\), the Hecke algebras \(\mathcal H_K\), \(\mathcal H_I\), and \(C_c(I\backslash G(F)/K)\) in place of \(\mathcal D_c(I\backslash\operatorname{Gr})\). Thick spans become linear spans, and a retract becomes division by \([K:I]\).

  • (a)\(\mathcal H_I\cdot\varphi_0=C_c(I\backslash G(F)/K)\cdot\varphi_0\): the Iwahori–Hecke span of the basic function is its span under the \((I,K)\)-bi-invariant functions.
  • (b)\(\mathcal H_I\cdot\varphi_0=\mathcal H_I\cdot C_K\) if and only if \(\mathcal H_K\cdot\varphi_0=C_K\): the \(K\)-invariant functions can be replaced by \(\varphi_0\) exactly under spherical generation.
  • (c)\(\mathcal H_I\cdot C_K\) is the image of \(C_c(I\backslash G(F)/K)\otimes_{\mathcal H_K}C_K\to C_I\); its cokernel \(Q_X=C_I/\mathcal H_I\cdot C_K\) measures the failure of fullness.

The dictionary between the two levels is in 5. No implication between the two levels is proved.

Table 1. · The terms of the chain, their names elsewhere, and what each equality needs.
TermElsewhereEquality
\(\mathcal C_I^{\mathrm{Iwa}}\)\(\mathcal S_H\) in the note; Devalapurkar's Conjecture 3.4.14the definition
\(\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\)\(\mathcal S_T\); the paper's Definition 4.3, \(\mathcal D_c(I\backslash LX)^{\mathrm{Sat}}\)(a): always, by the key lemma (S1)
\(\operatorname{Thick}\{\mathcal F\ast_KM:\ M\in\mathcal C\}\)—(b): if and only if H1, given H2
\(\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C\)\(\mathcal S_E\), the image of \(F_c\)(c): under H1 and H2
\(\mathcal C_I\)—false for the group (Proposition II); true under I1–I3 (Proposition III)

Map of the proof

Two forgetful maps carry the argument: \(q\), between the spaces of the two Hecke categories, and \(p\), between the spaces of the two equivariant categories. Both are pullbacks of \(\pi:\mathbb BB\to\mathbb BG\), so both are \(G/B\)-bundles and one key lemma serves both.

\(I\backslash LG/I=I\backslash\operatorname{Fl}\)\(I\backslash LG/K=I\backslash\operatorname{Gr}\) \(\mathbb BB\)\(\mathbb BG\) \(I\backslash LX\)\(K\backslash LX\) \(q\) \(\pi\) \(p\) \(\gamma'\) \(\gamma\) \(\gamma'_X\) \(\gamma_X\)

Both squares are Cartesian with fibre \(G/B\): \(q\) and \(p\) are proper and smooth, \(q^!=q^*[2d]\), \(p^!=p^*[2d]\).

The proof compares the two Hecke actions on the same \(\mathrm{IC}_0\) through \(q\), and the two equivariant categories through \(p\):

\(\mathcal H_{c,I}=\mathcal D_c(I\backslash LG/I)\)\(\mathcal D_c(I\backslash\operatorname{Gr})\) \(\mathcal C_I\)\(\mathcal C_I\)\(\mathcal C\) \(q_*\) \((-)\ast_I\mathrm{IC}_0\) \((-)\ast_K\mathrm{IC}_0\) \(p^*\)

The square commutes up to shift (step 1); the images of its two columns span the two sides of (a). \(q^*\) runs the top row backwards and \(\mathcal F\) is a retract of \(q_*q^*\mathcal F\): that gives (a) (steps 2–3). \(p_*\) runs \(p^*\) backwards and \(M\) is a retract of \(p_*p^*M\): that makes H1 necessary in (b) (step 5). (c) is formal from H2 (step 6).

Concept graph of this proof, merged with the two-definitions note →

Proof

Step 1 compares the two convolutions. Steps 2–3 prove (a), step 4 names generators, step 5 proves (b) and step 6 proves (c).

  1. Purpose Compare the two convolutions.

    $$\mathcal G\ast_Ip^*M\simeq(q_*\mathcal G)\ast_KM\qquad(\mathcal G\in\mathcal H_{c,I},\ M\in\mathcal C),\qquad\text{in particular}\quad\mathcal G\ast_I\mathrm{IC}_0\simeq(q_*\mathcal G)\ast_K\mathrm{IC}_0 .$$

    Proof
    • In the comparison square, \(r\) is the base change of \(q\), a \(G/B\)-bundle. Base change and the projection formula along it give \(r_*(\mathcal G\mathbin{\widetilde\boxtimes}p^!M)\simeq(q_*\mathcal G)\mathbin{\widetilde\boxtimes}M\) (note, step 5, with the comparison square drawn there). Then \(m_{I*}=m_{K*}r_*\).
    • \(p^!=p^*[2d]\), so the identity for \(p^*M\) holds up to the shift \([-2d]\). Thick spans are shift-stable, so shifts are suppressed from here on.
  2. Method The Iwahori span lies in the Iwahori–spherical span.

    $$\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0:\mathcal F\in\mathcal D_c(I\backslash\operatorname{Gr})\}.$$

    Proof
    • Each generator is \(\mathcal G\ast_I\mathrm{IC}_0\simeq(q_*\mathcal G)\ast_K\mathrm{IC}_0\) by step 1, and \(q_*\mathcal G\) is constructible because \(q\) is proper.
  3. Method The key lemma gives the other inclusion.

    $$\mathcal F\ast_K\mathrm{IC}_0\ \text{ is a retract of }\ (q^*\mathcal F)\ast_I\mathrm{IC}_0\in\mathcal C_I^{\mathrm{Iwa}};\qquad\text{hence (a).}$$

    Proof
    • By the key lemma S1, \(\mathcal F\) is a retract of \(q_*q^*\mathcal F\). The functor \(-\ast_K\mathrm{IC}_0\) is exact, so \(\mathcal F\ast_K\mathrm{IC}_0\) is a retract of \((q_*q^*\mathcal F)\ast_K\mathrm{IC}_0\simeq(q^*\mathcal F)\ast_I\mathrm{IC}_0\) (step 1 with \(\mathcal G=q^*\mathcal F\), constructible because \(q\) is smooth).
    • A thick span contains the retracts of its objects. With step 2 this is (a) (note, step 7).
  4. Result Generators.

    $$\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\Delta^{\operatorname{Gr}}_v\ast_K\mathrm{IC}_0:v\in \widetilde W/W\}=\operatorname{Thick}\{\Delta^{\operatorname{Fl}}_w\ast_I\mathrm{IC}_0:w\in \widetilde W\}.$$

    Proof
    • \(\mathcal D_c(I\backslash\operatorname{Gr})=\operatorname{Thick}\{\Delta^{\operatorname{Gr}}_v\}\): on one cell the equivariant sheaves are \(\operatorname{Thick}\{\underline k\}\) (affine space, connected stabilizer, truncation); each closure is a finite union of cells, and recollement with (T1), (T2) reduces to one cell fewer (the paper's Proposition on standard objects, stated there for \(I\)-orbits as well). Likewise \(\mathcal H_{c,I}=\operatorname{Thick}\{\Delta^{\operatorname{Fl}}_w\}\).
    • (T2) for the exact functors \(-\ast_K\mathrm{IC}_0\) and \(-\ast_I\mathrm{IC}_0\), together with (a).
  5. Method Spherical generation removes the basic object.

    $$\text{H1}\ \Longrightarrow\ \operatorname{Thick}\{\mathcal F\ast_KM:M\in\mathcal C\}=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}=\operatorname{Thick}\{\mathcal G\ast_Ip^*M:\mathcal G\in\mathcal H_{c,I},\ M\in\mathcal C\}.$$

    Proof
    • First equality, \(\supseteq\): take \(M=\mathrm{IC}_0\). \(\subseteq\): by H1, \(M\in\operatorname{Thick}\{\mathcal V\ast_K\mathrm{IC}_0:\mathcal V\in\mathcal H_{c,K}\}\); the functor \(\mathcal F\ast_K-\) is exact, so (T2) and associativity give \(\mathcal F\ast_KM\in\operatorname{Thick}\{(\mathcal F\ast_K\mathcal V)\ast_K\mathrm{IC}_0\}\), and \(\mathcal F\ast_K\mathcal V\in\mathcal D_c(I\backslash\operatorname{Gr})\) (right action, proper on bounded supports). This is (b) (note, step 3).
    • Converse, given H2. If \(\operatorname{Thick}\{\mathcal F\ast_KM\}=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\), then H1 holds: \(p^*M\) lies in the left side for every \(M\in\mathcal C\) (it is \(\delta\ast_KM\) up to shift, \(\delta\) the constant sheaf on \(I\backslash K/K\)); averaging, \(p_*(\mathcal F\ast_K\mathrm{IC}_0)\simeq(p'_*\mathcal F)\ast_K\mathrm{IC}_0\) carries the right side into \(\mathcal C^{\mathrm{Sat}}\); and the key lemma for \(p\) makes \(M\) a retract of \(p_*p^*M\). So \(M\in\mathcal C^{\mathrm{Sat}}\) (note, step 4).
    • Second equality: \(\mathcal G\ast_Ip^*M\simeq(q_*\mathcal G)\ast_KM\) (step 1) gives \(\supseteq\); \(\mathcal F\ast_KM\) is a retract of \((q^*\mathcal F)\ast_Ip^*M\) (step 3 with \(M\) in place of \(\mathrm{IC}_0\); the key lemma holds for every \(\mathcal F\)) gives \(\subseteq\). So \(\mathcal C_I^{\mathrm{Iwa}}\) is the thick \(\mathcal H_{c,I}\)-submodule of \(\mathcal C_I\) generated by \(p^*\mathcal C\).
  6. Conclusion The tensor product.

    $$\text{H1, H2}\ \Longrightarrow\ F_c:\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C\xrightarrow{\ \sim\ }\mathcal C_I^{\mathrm{Iwa}} .$$

    Proof
    • \(F_c\) is exact and, by H2, fully faithful out of an idempotent-complete stable category, so its essential image is thick. It contains \(F_c(\operatorname{ins}(\mathcal F\boxtimes M))=\mathcal F\ast_KM\), hence \(\operatorname{Thick}\{\mathcal F\ast_KM\}\).
    • The pure tensors generate the source thickly, so (T2) gives \(\operatorname{EssIm}(F_c)\subseteq\operatorname{Thick}\{\mathcal F\ast_KM\}\). Thus \(\operatorname{EssIm}(F_c)=\operatorname{Thick}\{\mathcal F\ast_KM:M\in\mathcal C\}=\mathcal C_I^{\mathrm{Iwa}}\) by (b), and \(F_c\) is an equivalence onto \(\mathcal C_I^{\mathrm{Iwa}}\). This is (c) (note, step 2).

5 · Dictionary

Function level over \(\mathbb F_q(\!(t)\!)\) against sheaf level over \(\mathbb C(\!(t)\!)\). The left column is the Iwahori function pages; the right column is Proposition I above and Propositions II–III.

Table 2. · The dictionary between the two levels.
Functions over \(\mathbb F_q((t))\)Sheaves over \(\mathbb C((t))\)
\(C_K\subseteq C_I\)\(p^*:\mathcal C\to\mathcal C_I\)
\(\mathcal H_K\), \(\mathcal H_I\), \(C_c(I\backslash G(F)/K)\)\(\mathcal H_{c,K}\), \(\mathcal H_{c,I}\), \(\mathcal D_c(I\backslash\operatorname{Gr})\)
\((f\ast\varphi)(x)=\sum_{g\in G(F)/J}f(g)\varphi(g^{-1}x)\)\(\mathcal F\ast_JM=m_{J*}(\mathcal F\mathbin{\widetilde\boxtimes}M)\)
\(\varphi_0=\mathbf 1_{X(\mathcal O)}\)\(\mathrm{IC}_0=\underline k_{L^+X}\)
\(\mathcal H_I\cdot\varphi_0=C_c(I\backslash G(F)/K)\cdot\varphi_0\) (Lemma A)\(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\), I(a)
\(q_*q^*f=[K:I]\,f\)\(\mathcal F\) a retract of \(q_*q^*\mathcal F\) (key lemma)
\(\mathcal H_I\cdot\varphi_0=\mathcal H_I\cdot C_K\) under spherical generation (Proposition B)\(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal G\ast_Ip^*M\}\), I(b)
image of \(C_c(I\backslash G(F)/K)\otimes_{\mathcal H_K}C_K\)\(\operatorname{EssIm}(F_c)\), I(c)
submodule generated; \(Q_X=C_I/\mathcal H_I\cdot C_K\)thick subcategory generated; \(\mathcal C_I/\mathcal C_I^{\mathrm{Iwa}}\)
sign character \(\varepsilon\), \(\mathcal H_{I_H}e_{K_H}\mathcal H_{I_H}\subsetneq\mathcal H_{I_H}\)affine Whittaker average \(\Phi\), \(\mathcal C_I^{\mathrm{Iwa}}\subsetneq\mathcal C_I\) (II)
Iwahori generation \(\mathcal H_I\cdot\varphi_0=C_I\)\(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) (III: sufficient conditions)
Remarkswithout H1 · what (b) does not say · sources