Hypotheses

• H1 · Spherical Satake generation.

  The spherical Satake objects thickly generate the coherent spherical category: $$\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}=\operatorname{Thick}\{\mathcal V\ast\mathrm{IC}_0:\mathcal V\in\mathcal H_{c,K}\}=\mathcal D_c(K\backslash LX).$$

  Discussion · where H1 is proved

  The paper proves it in §4.4 (Proposition 4.30) from its five hypotheses on the orbit stratification (numbered in §4.2 of the paper), and verifies those for the group case and for symmetric spaces satisfying two root-datum conditions. The Spherical generation workspace records that proof. By derived Satake, the span may be taken over the \(V_\lambda\) alone (paper, Remark 4.4(3)).

• H2 · The Iwahori convolution functor.

  1. \(F_c\) is exact and fully faithful.
  2. Its source is the thick span of the pure tensors: $$\mathcal D_c(I\backslash\mathrm{Gr})\otimes_{\mathcal H_{c,K}}\mathcal D_c(K\backslash LX)=\operatorname{Thick}\{\operatorname{ins}(\mathcal F\boxtimes\mathcal M):\mathcal F\in\mathcal D_c(I\backslash\mathrm{Gr}),\ \mathcal M\in\mathcal D_c(K\backslash LX)\}.$$

  Discussion · where H2 comes from

  (a) is the paper's §5: the large functor (5.1) is fully faithful by Campbell–Dhillon, Theorem 3.1.5 and Lemma 3.30, and §5.3–5.5 show that the embedding \(\iota\) of the small tensor product is fully faithful (Lemma 5.4, the Künneth formula and the monad computation). (b) combines Gaitsgory–Rozenblyum, Corollary 8.7.4 (compact generation by pure tensors, with rigidity of \(\mathcal H_K^{\mathrm{ren}}\) as in the paper's Lemma 5.3) and Neeman, Lemma 2.2 (compact objects form the thick span of the generators).

Proof

Step 1 names the three subcategories. Steps 2–4 prove (ii) and (iii): H2 makes the image a thick span of convolutions, H1 collapses the generators, and averaging over \(G/B\) shows that H1 is also necessary. Steps 5–7 prove (i): the easy inclusion, the key lemma, and the retract. Step 8 concludes.

1. Purpose Three descriptions of one subcategory

   Each description is a full subcategory of \(\mathcal D_c(I\backslash LX)\), and (iv) is the conjunction of two equalities with different inputs:

   $$\mathcal S_E\overset{\text{H1, H2}}{=}\mathcal S_T\overset{\text{key lemma}}{=}\mathcal S_H.$$
   Proof

   • \(\mathcal S_T\) and \(\mathcal S_H\) are thick spans by definition. \(\mathcal S_E\) is the essential image of an exact functor; H2(a) will make it thick (step 2).
   • The note proves only the key lemma; the rest of this page is the reconstruction of the chain from it.
2. Assumption H2 · the image is a thick span of convolutions

   Under H2,

   $$\mathcal S_E=\operatorname{Thick}\{\mathcal F\ast\mathcal M:\mathcal F\in\mathcal D_c(I\backslash\mathrm{Gr}),\ \mathcal M\in\mathcal D_c(K\backslash LX)\},$$

   and the same with \(\mathcal M\) restricted to \(\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}\) for the restricted functor.

   Proof

   • The essential image is thick. \(F_c\) is exact, so \(\mathcal S_E\) is closed under shifts and cofibers. Let \(Y\) be a retract of \(F_c(x)\): \(Y\xrightarrow{i}F_c(x)\xrightarrow{r}Y\), \(ri=\mathrm{id}\). The idempotent \(e=ir\) of \(F_c(x)\) is \(F_c(e')\) for an endomorphism \(e'\) of \(x\) by full faithfulness, and \(e'\) is idempotent in the homotopy category. The source is idempotent complete, so by HA, Lemma 1.2.4.6 \(e'\) splits, \(x\simeq x'\oplus x''\) with \(e'\) the projector to \(x'\); then \(Y\simeq F_c(x')\).
   • Generators. By H2(b) the source is \(\operatorname{Thick}\{\operatorname{ins}(\mathcal F\boxtimes\mathcal M)\}\); (T2) gives \(\mathcal S_E\subseteq\operatorname{Thick}\{\mathcal F\ast\mathcal M\}\), and the reverse inclusion holds because \(\mathcal S_E\) is thick and contains each \(\mathcal F\ast\mathcal M\).
   • For the restricted functor, replace \(\mathcal D_c(K\backslash LX)\) by its thick subcategory \(\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}\); H2(b) restricts because the restricted tensor product is the thick span of the pure tensors with \(\mathcal M\) in the subcategory.
3. Method Collapse the generators to Definition 4.3

   If every \(\mathcal M\) in the second factor lies in \(\operatorname{Thick}\{\mathcal V\ast\mathrm{IC}_0:\mathcal V\in\mathcal H_{c,K}\}\), then

   $$\operatorname{Thick}\{\mathcal F\ast\mathcal M\}=\operatorname{Thick}\{\mathcal F\ast\mathrm{IC}_0:\mathcal F\in\mathcal D_c(I\backslash\mathrm{Gr})\}=\mathcal S_T.$$

   For the restricted functor this is automatic, proving (ii); for \(F_c\) itself it is H1, proving \(\mathcal S_E=\mathcal S_T\) under H1 and H2.

   Proof

   • \(\supseteq\): \(\mathcal F\ast\mathrm{IC}_0\) is one of the generators, since \(\mathrm{IC}_0\in\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}\).
   • \(\subseteq\): fix \(\mathcal F\). The functor \(\mathcal F\ast(-):\mathcal D_c(K\backslash LX)\to\mathcal D_c(I\backslash LX)\) is exact, so by (T2) $$\mathcal F\ast\mathcal M\in\operatorname{Thick}\{\mathcal F\ast(\mathcal V\ast\mathrm{IC}_0)\}=\operatorname{Thick}\{(\mathcal F\ast\mathcal V)\ast\mathrm{IC}_0\}\subseteq\mathcal S_T,$$ using associativity and \(\mathcal F\ast\mathcal V\in\mathcal D_c(I\backslash\mathrm{Gr})\).
4. Result H1 is necessary and sufficient

   Under H2 and S1, \(\mathcal S_T\subseteq\mathcal S_E\), and \(\mathcal S_E\subseteq\mathcal S_T\) implies H1:

   $$p^*\mathcal M\in\mathcal S_E\ \ (\forall\mathcal M),\qquad p_*(\mathcal S_T)\subseteq\mathcal D_c(K\backslash LX)^{\mathrm{Sat}},\qquad\mathcal M\ \text{a retract of}\ p_*p^*\mathcal M .$$

   With step 3 this is (iii). The original paper's restricted definition sidesteps H1; the note's unrestricted statement needs it.

   Proof

   • \(\mathcal S_T\subseteq\mathcal S_E\) is the inclusion \(\supseteq\) of step 3 applied to \(F_c\).
   • Spherical objects lie in \(\mathcal S_E\). Let \(\delta\in\mathcal D_c(I\backslash LG/K)\) be the constant \(\mathcal D\)-module on the closed substack \(I\backslash K/K=\mathbb BI\) (the base point of \(\mathrm{Gr}\)), extended by zero. \(\delta\,\widetilde\boxtimes\,\mathcal M\) is supported on \(I\backslash K\times^KLX\simeq I\backslash LX\), where \(m_K\) is the identity and \(\mathrm{pr}_2\) is \(p\), so \(\delta\ast_K\mathcal M\simeq p^*\mathcal M\) up to a shift. Thick spans contain shifts, so \(p^*\mathcal M\in\mathcal S_E\) for every \(\mathcal M\in\mathcal D_c(K\backslash LX)\). (For functions: \(\mathbf 1_K\ast_K\varphi=\varphi\).)
   • Averaging over \(G/B\). \(p\) is proper with fibre \(K/I=G/B\), so \(p_*=p_!:\mathcal D_c(I\backslash LX)\to\mathcal D_c(K\backslash LX)\) is exact and preserves coherence. With \(p':I\backslash LG/K\to K\backslash LG/K\), base change and the projection formula exactly as in step 5 give \(p_*(\mathcal F\ast_K\mathrm{IC}_0)\simeq(p'_*\mathcal F)\ast_K\mathrm{IC}_0\) with \(p'_*\mathcal F\in\mathcal H_{c,K}\). By (T2), \(p_*(\mathcal S_T)\subseteq\operatorname{Thick}\{\mathcal V\ast\mathrm{IC}_0\}=\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}\).
   • Key lemma for \(p\). \(I=K\times_GB\) gives \(I\backslash LX=K\backslash LX\times_{\mathbb BG}\mathbb BB\), so \(p\) is a base change of \(\pi:\mathbb BB\to\mathbb BG\) and the key lemma applies: \(\mathcal M\) is a direct summand of \(p_*p^*\mathcal M\). If \(\mathcal S_E\subseteq\mathcal S_T\), then \(p^*\mathcal M\in\mathcal S_T\), hence \(p_*p^*\mathcal M\in\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}\), and the thick span contains its direct summand \(\mathcal M\). That is H1.
   • The same argument for functions: averaging over \(K/I\) shows \(C_c(I\backslash G(F)/K)\ast_K\mathbf 1_{X(\mathcal O)}\cap C_c(X(F))^K=\mathcal H_K\ast_K\mathbf 1_{X(\mathcal O)}\) (functions page).
5. Method Iwahori translates are spherical convolutions

   With \(q:I\backslash LG/I\to I\backslash LG/K\) the forgetful map, for \(\mathcal G\in\mathcal H_{c,I}\) and \(\mathcal M\in\mathcal D_c(K\backslash LX)\),

   $$\mathcal G\ast_Ip^!\mathcal M\simeq(q_*\mathcal G)\ast_K\mathcal M,\qquad q_*\mathcal G\in\mathcal D_c(I\backslash\mathrm{Gr}),\qquad\text{hence}\quad\mathcal S_H\subseteq\mathcal S_T.$$
   Proof

   • This is the comparison square. Since \(I\backslash LX\simeq\mathbb BI\times_{\mathbb BK}(K\backslash LX)\), the two convolution spaces are fibre products over \(\mathbb BK\): $$I\backslash LG\times^ILX\simeq(I\backslash LG/I)\times_{\mathbb BK}(K\backslash LX)\xrightarrow{\ r=q\times\mathrm{id}\ }(I\backslash LG/K)\times_{\mathbb BK}(K\backslash LX)\simeq I\backslash LG\times^KLX,$$ and \(m_I=m_K\circ r\). On the left, \(\mathcal G\,\widetilde\boxtimes\,p^!\mathcal M=\mathrm{pr}_1^!\mathcal G\otimes^!\mathrm{pr}_2^!\mathcal M\) with \(\mathrm{pr}_2\) to \(K\backslash LX\).
   • \(r\) is the base change of the proper map \(q\) along \(\mathrm{pr}_1\). The projection formula for \(r_*=r_!\) and base change give $$r_*(\mathrm{pr}_1^!\mathcal G\otimes^!\mathrm{pr}_2^!\mathcal M)\simeq r_*\mathrm{pr}_1^!\mathcal G\otimes^!\mathrm{pr}_2^!\mathcal M\simeq\mathrm{pr}_1^!q_*\mathcal G\otimes^!\mathrm{pr}_2^!\mathcal M=(q_*\mathcal G)\,\widetilde\boxtimes\,\mathcal M.$$ Applying \(m_{K*}\) gives the displayed isomorphism.
   • \(q\) is proper with fibre \(G/B\), so \(q_*=q_!\) preserves coherence. Since \(p^!\mathrm{IC}_0=p^*\mathrm{IC}_0[2d]\) and thick spans contain shifts, every \(\mathcal G\ast_I\mathrm{IC}_0\) lies in \(\mathcal S_T\).
6. Method Key lemma · \(\mathcal F\) is a retract of \(q_*q^*\mathcal F\)

   For \(q:I\backslash\mathrm{Fl}\to I\backslash\mathrm{Gr}\) and every \(\mathcal F\in\mathcal D_c(I\backslash\mathrm{Gr})\), the unit of \(q^*\dashv q_*\) has a natural retraction:

   $$\mathcal F\xrightarrow{\ \eta_{\mathcal F}\ }q_*q^*\mathcal F\xrightarrow{\ \beta_{\mathcal F}\ }\mathcal F,\qquad\beta_{\mathcal F}\circ\eta_{\mathcal F}=\mathrm{id}_{\mathcal F},\qquad q^*\mathcal F\in\mathcal H_{c,I}.$$
   Proof

   • \(q\) is the base change of \(\pi:\mathbb BB\to\mathbb BG\) along the classifying map \(\gamma:I\backslash\mathrm{Gr}\to\mathbb BG\); base change and the projection formula give \(q_*q^*\mathcal F\simeq\gamma^!\mathcal A\otimes^!\mathcal F\) with \(\mathcal A=\pi_*\pi^*\omega_{\mathbb BG}\), and \(\eta_{\mathcal F}=\gamma^!u\otimes^!\mathrm{id}\) for the unit \(u:\omega_{\mathbb BG}\to\mathcal A\).
   • \(\mathcal A\) is a commutative algebra with unit \(u\), dualizable, and the trace of multiplication satisfies \(\operatorname{tr}_{\mathcal A}\circ u=\chi(G/B)=|W|\), invertible in \(k\). Set \(\beta_{\mathcal F}=\tfrac1{|W|}\gamma^!\operatorname{tr}_{\mathcal A}\otimes^!\mathrm{id}\).
   • \(q^*\mathcal F\) is coherent because \(q\) is smooth. The argument uses only the square, so it holds for every base change of \(\pi\); step 4 uses it for \(p:I\backslash LX\to K\backslash LX\). Full proof and graph →
7. Result Extract \(\mathcal F\ast\mathrm{IC}_0\) as a retract

   Convolving the retraction with \(\mathrm{IC}_0\) and rewriting by step 5,

   $$\mathcal F\ast\mathrm{IC}_0\longrightarrow(q^*\mathcal F)\ast_I\mathrm{IC}_0[-2d]\longrightarrow\mathcal F\ast\mathrm{IC}_0\quad\text{composes to the identity, so}\quad\mathcal F\ast\mathrm{IC}_0\in\mathcal S_H,\quad\mathcal S_T\subseteq\mathcal S_H.$$
   Proof

   • \((-)\ast_K\mathrm{IC}_0\) is a functor, so \((\beta_{\mathcal F}\ast\mathrm{id})\circ(\eta_{\mathcal F}\ast\mathrm{id})=\mathrm{id}\); the middle term is \((q_*q^*\mathcal F)\ast_K\mathrm{IC}_0\simeq(q^*\mathcal F)\ast_Ip^!\mathrm{IC}_0[-2d]\) by step 5 with \(\mathcal G=q^*\mathcal F\).
   • \(\mathcal S_H\) contains \((q^*\mathcal F)\ast_I\mathrm{IC}_0\), hence its shifts and direct summands (Stacks 05RB). With step 5, \(\mathcal S_T=\mathcal S_H\), which is (i).
8. Conclusion The three agree

   (i) is step 7; (ii) is step 3 for the restricted functor; (iii) is step 4; (iv) combines step 3 under H1 with (i):

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

   • In the note's words: assuming \(\mathcal D_c(K\backslash LX)^{\mathrm{Sat}}\simeq\mathcal D_c(K\backslash LX)\), \(\operatorname{EssIm}(\mathcal D_c(I\backslash\mathrm{Gr})\otimes_{\mathcal H_{c,K}}\mathcal D_c(K\backslash LX))\simeq\langle\mathcal H_{c,I}\ast\mathrm{IC}_0\rangle\).