Proposition I, with the two-definitions note

1 · The chain and the dictionary

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

Definitions

Thick spans: (T1) if two terms of a cofibre sequence lie in a thick subcategory, so does the third; (T2) an exact functor \(\Phi\) sends \(\operatorname{Thick}(S)\) into \(\operatorname{Thick}(\Phi S)\).

• 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; (T1) and (T2) as above.

• 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.

Hypotheses

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

• S1 · Invertible Euler characteristic.

  $$|W|\in k^\times .$$

  Automatic for \(k=\mathbb C\). Used through the key lemma, for \(q\) and for \(p\).

• H1 · Spherical generation.

  $$\mathcal C^{\mathrm{Sat}}:=\operatorname{Thick}\{\mathcal V\ast_K\mathrm{IC}_0:\mathcal V\in\mathcal H_{c,K}\}=\mathcal C .$$

  Discussion · where it is known

  The paper's Proposition "Satake equals full" proves H1 under its hypotheses (H1)–(H5), and its Corollary gives it for the group \(X=(H\times H)/H\) and for the symmetric varieties \(G/G^\theta\) with \(G^\theta\) and \(Z_{G^\theta}(S)\) connected and \(\varphi(\Lambda_T^+)=\Lambda_S^+\): \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(\mathrm{SL}_4/\mathrm{Sp}_4\), \(\mathrm{GL}_2/\mathrm{Sp}_2\). It is unknown for \(E_6/F_4\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\). Function shadow: \(\mathcal H_K\cdot\mathbf 1_0=C_c(K\backslash X(F))\), the note's F1.

• H2 · Full faithfulness.

  The functor \(F_c\) is fully faithful.

  Discussion

  The paper proves that the continuous functor \(\mathcal D(I\backslash\operatorname{Gr})\otimes_{\mathcal H_K}\mathcal D(K\backslash LX)\to\mathcal D(I\backslash LX)\) is fully faithful (Campbell–Dhillon and \(I\)-invariants, its §5), and \(F_c\) is its restriction to compact objects; the note's H2 also records that the pure tensors generate the source thickly.

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.

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 \(p\):

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).

Proof

Step 1 names the three descriptions. Steps 2–4 prove (a): the comparison of convolutions, the key lemma, the retract. Step 5 names generators. Steps 6–7 prove (b) and its converse, step 8 proves (c). Steps 9–10 record what the chain does not say.

1. Purpose Three descriptions of one subcategory

   Each term of the chain is a full subcategory of \(\mathcal C_I\); in the note's names,

   $$\mathcal S_E\overset{\text{H1, H2}}{=}\mathcal S_T\overset{\text{key lemma}}{=}\mathcal S_H,\qquad\mathcal S_H=\mathcal C_I^{\mathrm{Iwa}},\quad\mathcal S_T=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\},\quad\mathcal S_E=\operatorname{EssIm}(F_c).$$
   Proof

   • \(\mathcal C_I^{\mathrm{Iwa}}\) and \(\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\) are thick spans by definition; \(\operatorname{EssIm}(F_c)\) is the essential image of an exact functor, thick under H2 (step 6).
   • (a) is the note's claim (i); (b) and (c) are its (iii) and (iv), the goal of the handwritten note, which proves only the key lemma. The rest is reconstructed below.
2. Method Compare the two convolutions along \(q\)

   $$\mathcal G\ast_Ip^*M\simeq(q_*\mathcal G)\ast_KM\,[-2d]\qquad(\mathcal G\in\mathcal H_{c,I},\ M\in\mathcal C),\qquad\text{hence}\quad\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}.$$

   Proof

   • In the comparison square, \(r=q\times\mathrm{id}\) is the base change of the proper map \(q\), a \(G/B\)-bundle. Base change and the projection formula along \(r\) give \(r_*(\mathcal G\mathbin{\widetilde\boxtimes}p^!M)\simeq(q_*\mathcal G)\mathbin{\widetilde\boxtimes}M\); then \(m_{I*}=m_{K*}r_*\), and \(p^!=p^*[2d]\) accounts for the shift.
   • \(q_*\mathcal G\) is constructible because \(q\) is proper. Thick spans are shift-stable, so each generator \(\mathcal G\ast_I\mathrm{IC}_0\) of \(\mathcal C_I^{\mathrm{Iwa}}\) lies in \(\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\) (note, step 5).
3. Method Key lemma · \(\mathcal F\) is a retract of \(q_*q^*\mathcal F\)

   $$\mathcal F\xrightarrow{\ \eta_{\mathcal F}\ }q_*q^*\mathcal F\xrightarrow{\ \beta_{\mathcal F}\ }\mathcal F,\qquad\beta_{\mathcal F}\circ\eta_{\mathcal F}=\mathrm{id},\qquad\beta_{\mathcal F}=\tfrac1{|W|}\gamma^!\operatorname{tr}_{\mathcal A}\otimes^!\mathrm{id}\quad\text{(S1)}.$$

   Proof

   • \(q\) is the base change of \(\pi:\mathbb BB\to\mathbb BG\) along \(\gamma\), so \(q_*q^*\mathcal F\simeq\gamma^!\mathcal A\otimes^!\mathcal F\) with \(\mathcal A=\pi_*\pi^*\omega_{\mathbb BG}\) and \(\eta_{\mathcal F}=\gamma^!u\otimes^!\mathrm{id}\); \(\operatorname{tr}_{\mathcal A}\circ u=\chi(G/B)=|W|\), invertible by S1.
   • Only the square is used, so the same holds for \(p:I\backslash LX\to K\backslash LX\) (step 7). Full proof and graph: key lemma page of the note.
4. Result (a) · \(\mathcal F\ast_K\mathrm{IC}_0\) is a retract of an Iwahori translate

   $$\mathcal F\ast_K\mathrm{IC}_0\longrightarrow(q^*\mathcal F)\ast_I\mathrm{IC}_0[2d]\longrightarrow\mathcal F\ast_K\mathrm{IC}_0\ \text{ composes to the identity, so }\ \operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\subseteq\mathcal C_I^{\mathrm{Iwa}};\qquad\text{with step 2, (a).}$$

   Proof

   • Apply the exact functor \((-)\ast_K\mathrm{IC}_0\) to \(\beta_{\mathcal F}\circ\eta_{\mathcal F}=\mathrm{id}\); the middle term is \((q_*q^*\mathcal F)\ast_K\mathrm{IC}_0\simeq(q^*\mathcal F)\ast_I\mathrm{IC}_0[2d]\) by step 2 with \(\mathcal G=q^*\mathcal F\), constructible because \(q\) is smooth.
   • A thick span contains the retracts and shifts of its objects (note, step 7).
5. 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

   • \(\mathcal D_c(I\backslash\operatorname{Gr})=\operatorname{Thick}\{\Delta^{\operatorname{Gr}}_v\}\) and \(\mathcal H_{c,I}=\operatorname{Thick}\{\Delta^{\operatorname{Fl}}_w\}\): 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.
   • (T2) for the exact functors \((-)\ast_K\mathrm{IC}_0\) and \((-)\ast_I\mathrm{IC}_0\), together with (a).
6. Method H2, then H1, collapse the image to (a)

   $$\text{H2}\ \Longrightarrow\ \operatorname{EssIm}(F_c)=\operatorname{Thick}\{\mathcal F\ast_KM\};\qquad\text{H1}\ \Longrightarrow\ \operatorname{Thick}\{\mathcal F\ast_KM\}=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}=\operatorname{Thick}\{\mathcal G\ast_Ip^*M\}:\ \text{(b)}.$$

   Proof

   • Thick image. \(F_c\) is exact; a retract of \(F_c(x)\) is split by an idempotent, which by full faithfulness comes from an idempotent of \(x\), split in the idempotent-complete source. With H2(b) and (T2), \(\operatorname{EssIm}(F_c)=\operatorname{Thick}\{\mathcal F\ast_KM\}\) (note, step 2).
   • Collapse. \(\supseteq\): take \(M=\mathrm{IC}_0\). \(\subseteq\): by H1, \(M\in\operatorname{Thick}\{\mathcal V\ast_K\mathrm{IC}_0\}\); \(\mathcal F\ast_K(-)\) is exact, so (T2) and associativity give \(\mathcal F\ast_KM\in\operatorname{Thick}\{(\mathcal F\ast_K\mathcal V)\ast_K\mathrm{IC}_0\}\), with \(\mathcal F\ast_K\mathcal V\in\mathcal D_c(I\backslash\operatorname{Gr})\) (note, step 3). For the restricted functor \(F_c^{\mathrm{Sat}}\) the collapse needs no H1.
   • Through \(\mathcal H_{c,I}\). Step 2 gives \(\operatorname{Thick}\{\mathcal G\ast_Ip^*M\}\subseteq\operatorname{Thick}\{\mathcal F\ast_KM\}\); step 4 with \(M\) in place of \(\mathrm{IC}_0\) gives the reverse. So \(\mathcal C_I^{\mathrm{Iwa}}\) is the thick \(\mathcal H_{c,I}\)-submodule of \(\mathcal C_I\) generated by \(p^*\mathcal C\).
7. Result H1 is necessary for (b)

   $$\text{H2, S1 and }\operatorname{Thick}\{\mathcal F\ast_KM\}=\operatorname{Thick}\{\mathcal F\ast_K\mathrm{IC}_0\}\ \Longrightarrow\ \mathcal C^{\mathrm{Sat}}=\mathcal C .$$

   Proof

   • \(p^*M\simeq\delta\ast_KM\) up to shift, \(\delta\) the constant sheaf on \(I\backslash K/K\): so \(p^*M\) lies in the left side for every \(M\in\mathcal C\).
   • Averaging: \(p_*(\mathcal F\ast_K\mathrm{IC}_0)\simeq(p'_*\mathcal F)\ast_K\mathrm{IC}_0\) with \(p':I\backslash\operatorname{Gr}\to K\backslash\operatorname{Gr}\), so \(p_*\) carries the right side into \(\mathcal C^{\mathrm{Sat}}\) by (T2).
   • The key lemma for \(p\) makes \(M\) a retract of \(p_*p^*M\in\mathcal C^{\mathrm{Sat}}\); so \(M\in\mathcal C^{\mathrm{Sat}}\) (note, step 4).
8. Conclusion (c) · the tensor product; the three agree

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

   Proof

   • By step 6, \(\operatorname{EssIm}(F_c)=\operatorname{Thick}\{\mathcal F\ast_KM\}=\mathcal C_I^{\mathrm{Iwa}}\) by (b); \(F_c\) is fully faithful by H2, hence an equivalence onto \(\mathcal C_I^{\mathrm{Iwa}}\).
   • Without H1, steps 2–4 and 6 give \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal F\ast_KM:M\in\mathcal C^{\mathrm{Sat}}\}\simeq\mathcal D_c(I\backslash\operatorname{Gr})\otimes_{\mathcal H_{c,K}}\mathcal C^{\mathrm{Sat}}\), the paper's theorem on \(F_c^{\mathrm{Sat}}\); H1 removes the superscript.
9. Result Beyond (b) · the group: \(\mathcal C_I^{\mathrm{Iwa}}\) is a proper ideal

   $$X=H\ \text{under}\ H\times H,\ W\ne1:\qquad\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal A\ast c\ast\mathcal B\}\subseteq\ker\Phi,\qquad\Phi(\Delta_e)\ne0,\qquad\mathcal C_I^{\mathrm{Iwa}}\subsetneq\mathcal C_I .$$

   Proof

   • \(\Phi\) is right Whittaker averaging; it is exact, left \(\mathcal H_{c,I}\)-linear, and kills every left \(K\)-equivariant object such as \(c\ast\mathcal B\). This is Proposition II; the function shadow is the sign character of \(\mathcal H_I\).
10. Purpose Beyond (b) · when is \(\mathcal C_I^{\mathrm{Iwa}}\) everything?

    $$\mathcal C_I^{\mathrm{Iwa}}\overset{?}{=}\mathcal C_I:\qquad\text{false for the group (II),}\quad\text{I1–I3}\Rightarrow\text{true (III),}\quad\text{true for }\mathbb A^n\text{ under }\mathrm{GL}_n\text{ and }\mathbb G_m\text{ (IV)}.$$

    Proof

    • Proposition III works one finite level at a time with the generators of step 5; IV checks its conditions on \(\mathbb A^n\) and on the group; the status by family tab lists every family. The function-level shadow is the Iwahori function pages.