Proof PDF · original proposition numbering →

Proof

Step 1 names the comparison. Steps 2–6 glue the levels and prove (a); step 4 shows why (Lev) asks for full faithfulness on \(D^b\). Steps 7–8 prove (b), step 9 proves (c), and steps 10–11 prove (d).

1. Purpose Compare three thick spans

   Every summand of a generator is a constituent of its perverse cohomology. The proof places \(\mathcal C_I^{\mathrm{Iwa}}\) between the spans of the two sets and identifies \(\mathcal C_I\) with the span of all simple objects:

   $$\operatorname{Thick}(\mathcal M_I)\overset{?}{\subseteq}\mathcal C_I^{\mathrm{Iwa}}\overset{?}{\subseteq}\operatorname{Thick}(\mathcal M_I^+)\subseteq\operatorname{Thick}(\mathcal S_I)\overset{?}{=}\mathcal C_I .$$
   Plan

   • Levels. Steps 2–6: the perverse hearts of the levels glue; simple objects and constituent subcategories do not depend on the level, and constituent subcategories are thick.
   • Generators. Steps 7–8: \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{G_w\}\), placed between the two spans without (DT).
   • Decomposition. Steps 9–11: (DT) closes the gap; it comes from a finite-type proper model, and must hold equivariantly.
2. Method Lemma · perverse hearts glue along the levels

   At each level the equivariant perverse heart has finite length, and the transitions preserve hearts, subquotients and simple objects:

   $$\pi^*[d]\,\mathrm{IC}(\bar o,\xi)\simeq\mathrm{IC}(\overline{o'},\pi^*\xi|_{o'}),\qquad\iota_*\mathrm{IC}(\bar o,\xi)=\mathrm{IC}(\iota\bar o,\xi).$$
   Proof

   • Premises. (Lev): \(H_\alpha\) connected linear, affine-space transitions \(\pi=\pi_{N,M',M}\) of locally constant relative dimension \(d\), closed transitions \(\iota:Y_{N,M}\hookrightarrow Y_{N+1,M}\); I1.
   • One level. \(H\) connected: \(\operatorname{Perv}_H(Y)\to\operatorname{Perv}(Y)\) is fully faithful with image closed under subquotients (BBD 4.2.5 and 4.2.6.2, for the action and projection \(H\times Y\to Y\)). \(\operatorname{Perv}(Y)\) has finite length (BBD 4.3.1); a simple equivariant object is \(\mathrm{IC}(\bar o,\xi)\), \(o\) an orbit by I1, \(\xi\) an irreducible representation of \(\pi_0(S)\) for \(o=H/S\); under I2, \(\xi=k\).
   • Smooth transitions. \(\pi^*[d]\) is t-exact and fully faithful on perverse sheaves, with image closed under subquotients (BBD 4.2.5–4.2.6.2). \(\pi^{-1}(o)\) is smooth and connected, so irreducible, with one dense orbit \(o'\) by I1; \(j_{!*}\) commutes with smooth base change and \(\pi^*\xi\) stays irreducible (the remark after BBD 4.2.6.1).
   • Closed transitions. \(\iota_*\) is t-exact (BBD 1.4.16) and fully faithful; a perverse subquotient of \(\iota_*A\) vanishes off \(\iota Y\), because \(j^*\) is t-exact.
   • Output. For every transition \(\tau\), the constituents of \(\bigoplus_i{}^pH^i(\tau F)\) are the \(\tau\)-images of those of \(\bigoplus_i{}^pH^i(F)\), up to shift. Run A2-2(a),(b); audit A7-1: holds, citations repaired.
3. Method Lemma · retracts stay at one level

   A retract in \(\mathcal C_I\) of an object of one level lies at that level. So \(\mathcal C_I\) is idempotent complete, and thick spans are computed level by level:

   $$X\in\mathcal C_I^{(\alpha)},\ \ R\ \text{a retract of}\ X\ \text{in}\ \mathcal C_I\ \Longrightarrow\ R\in\mathcal C_I^{(\alpha)}.$$
   Proof

   • Premises. (Lev): \(\pi_\alpha^*\) is fully faithful on \(D^b\). A triangulated category with a bounded t-structure is idempotent complete (Le–Chen, J. Algebra 310 (2007); cited).
   • Inference. The idempotent \(e\) of \(X\) defining \(R\) lies in \(\operatorname{End}(X)\) computed in \(\mathcal C_I^{(\alpha)}\) and splits there, \(X\simeq X_1\oplus X_2\); splittings of an idempotent are unique up to isomorphism, so \(R\simeq X_1\).
   • Output. Finitely many objects lie at a common level, which is closed under shifts, cofibres and retracts; \(\operatorname{Thick}\) is computed there. Run A2-1, step 2; audit A7-1, step 1.
4. Ruled out t-exact transitions alone

   Full faithfulness on hearts does not give full faithfulness on \(D^b\), so (Lev) must ask for the latter:

   $$\pi:\mathbb G_m\to\mathrm{pt}:\qquad\pi^*[1]\ \text{t-exact, fully faithful on}\ \operatorname{Perv},\qquad\operatorname{Hom}(\pi^*k,\pi^*k[1])=H^1(\mathbb G_m)\ne0=\operatorname{Hom}(k,k[1]).$$
   Proof

   • Hypotheses of BBD 4.2.5. \(\pi\) is smooth, surjective, with connected fibres, so \(\pi^*[1]\) is t-exact and fully faithful on perverse sheaves.
   • Failure on \(D^b\). \(\operatorname{Hom}(\pi^*k,\pi^*k[1])=H^1(\mathbb G_m,k)=k\), while \(\operatorname{Hom}(k,k[1])=0\) on a point.
   • Consequence. Along such transitions \(\bigcup_\alpha\mathcal C_I^{(\alpha)}\) is not a union of full subcategories, and a summand relation at one level could fail at the next. Version 2 of the criterion (A5-4) assumed only t-exactness; audit A7-4 adds the \(D^b\) clause, kept in (Lev).
5. Result One simple object per closure of a piece

   Simple objects and constituents do not depend on the level. Under I2 a simple object is determined by its support, the closure of a piece:

   $$\mathcal S_I\xrightarrow{\ \sim\ }\mathcal Z,\quad S\mapsto\operatorname{supp}S,\qquad\mathrm{IC}(\overline P)=\pi_{N,M}^*\mathrm{IC}(\bar o,k)\quad\big(P=\pi_{N,M}^{-1}(o)\big).$$
   Proof

   • Premises. Step 2; I2, or I2′.
   • Level-independence. A simple object stays simple, up to shift, at every finer level (step 2), so \(\mathcal S_I\) and \(\mathcal D_\Sigma\) do not depend on the level used.
   • Support. \(\operatorname{supp}\pi_\alpha^*\mathrm{IC}(\bar o,k)=\pi_\alpha^{-1}(\bar o)=\overline P\); the open piece \(P'\subseteq P\) of a finer level has \(\overline{P'}=\overline P\) and the same IC sheaf.
   • Injectivity. Two simple objects with the same support are compared at a common level, where both are \(\mathrm{IC}(\bar o,k)\): the open orbit of an irreducible orbit closure is unique.
   • Local systems. Under I2′, \(\mathcal S_I=\{\mathrm{IC}(Z,\xi)\}\); \(\pi_0\) of a stabilizer can grow under refinement, so new \(\xi\) on the same \(Z\) appear and are counted (audit A7-2, repair 2). Run A2-2(c); audit A7-1, step 4.
6. Method Constituent subcategories are thick spans

   For every \(\Sigma\subseteq\mathcal S_I\), \(\mathcal D_\Sigma\) is thick and equals \(\operatorname{Thick}(\Sigma)\); with \(\Sigma=\mathcal S_I\) this is (a):

   $$\mathcal D_\Sigma=\operatorname{Thick}(\Sigma),\qquad\mathcal C_I=\mathcal D_{\mathcal S_I}=\operatorname{Thick}(\mathcal S_I).$$
   Proof

   • Premises. Steps 2, 3 and 5.
   • Thick at one level. \(\mathcal P_\Sigma\) is a Serre subcategory; with \(q\) the exact quotient functor, \(h=q\circ{}^pH^0\) is homological and $$\mathcal D_\Sigma\cap\mathcal C_I^{(\alpha)}=\{F:\ h(F[n])=0\ \text{for all}\ n\},$$ a saturated triangulated subcategory (Stacks, Lemma 13.6.3 [05RD]).
   • Thick in \(\mathcal C_I\). The level-wise subcategories are compatible (step 5); their union is closed under shifts and cofibres (a common level) and under retracts (step 3).
   • Inclusion in \(\operatorname{Thick}(\Sigma)\). \(F\in\mathcal D_\Sigma\) has finitely many nonzero \({}^pH^i(F)\), each of finite length; the truncation triangles and Jordan–Hölder filtrations with (T1) put \(F\) in \(\operatorname{Thick}(\Sigma)\) (Stacks, Lemma 13.35.7 [0FX7]). The reverse inclusion holds because \(\mathcal D_\Sigma\) is thick and contains \(\Sigma\).
   • Output. Every constituent lies in \(\mathcal S_I\), so \(\mathcal D_{\mathcal S_I}=\mathcal C_I\). No weights are used: valid over \(\overline{\mathbb F}_q\) and \(\mathbb C\). Run A2-2(d); audit A7-1, step 5.
7. Method Lemma · generators \(G_w=\mathrm{IC}_w\ast_I\mathrm{IC}_0\)

   The Hecke category is spanned by the IC sheaves of Schubert varieties, so the Iwahori-generated part is spanned by the countably many \(G_w\):

   $$\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_w:w\in\widetilde W\},\qquad\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{G_w:w\in\widetilde W\}.$$
   Proof

   • Hecke side. An object of \(\mathcal H_{c,I}\) lives on finitely many cells \(IwI/I\simeq\mathbb A^{\ell(w)}\), on which \(I\) acts through a connected finite-dimensional quotient with connected stabilizers (Chen–Yi, Lemma 71(i)); steps 2 and 6 on this finite-type stack give \(\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_w\}\).
   • Exact image. \(-\ast_I\mathrm{IC}_0\) is exact; by (T2) it maps \(\operatorname{Thick}\{\mathrm{IC}_w\}\) into \(\operatorname{Thick}\{G_w\}\), and every \(G_w\) lies in \(\mathcal C_I\) by assumption.
   • The group. \(\widetilde W_G=\widetilde W_H\times\widetilde W_H\), \(\mathrm{IC}_{(w_1,w_2)}=\mathrm{IC}_{w_1}\boxtimes\mathrm{IC}_{w_2}\) (BBD 4.2.7), \((A\boxtimes B)\ast_IF=A\ast F\ast\mathrm{inv}^*B\), \(\mathrm{inv}^*\mathrm{IC}_w=\mathrm{IC}_{w^{-1}}\) and \(c=\mathrm{IC}_{w_0}[-\ell(w_0)]\): the generators are the \(\mathrm{IC}_{w_1}\ast c\ast\mathrm{IC}_{w_2}\).
   • Run. A2-3(a),(b); A3-5, step 0; audit A7-2: holds.
8. Result The sandwich, without (DT)

   Summands are reached, and only constituents are reached. This is (b), with no decomposition theorem:

   $$\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^+}=\operatorname{Thick}(\mathcal M_I^+).$$
   Proof

   • Summands are reached. If \(S\in\mathcal M_I\), then \(S[s]\) is a retract of \(G_w\in\mathcal C_I^{\mathrm{Iwa}}\), and thick subcategories contain retracts and shifts.
   • Only constituents. Every \(G_w\) lies in \(\mathcal D_{\mathcal M_I^+}\) by definition of \(\mathcal M_I^+\) (at any level, step 5); \(\mathcal D_{\mathcal M_I^+}\) is thick (step 6), and \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{G_w\}\) (step 7) is the smallest thick subcategory containing the \(G_w\).
   • Membership. If \(S\in\mathcal C_I^{\mathrm{Iwa}}\), take a level where \(S\) is a shifted simple perverse object: its only constituent is \(S\), so \(S\in\mathcal M_I^+\).
   • Run. A2-4(a),(b); audit A7-2, step 4: holds. Only the vanishing of nonnegative Jordan–Hölder multiplicities is used, never an alternating sum.
9. Result Under (DT), summands are constituents

   (DT) makes the perverse cohomology of each generator semisimple, so the two ends of the sandwich coincide. This is Proposition III:

   $$\mathcal M_I=\mathcal M_I^+,\qquad\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}(\mathcal M_I),\qquad\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I .$$
   Proof

   • Premises. (DT); steps 6 and 8.
   • Constituents are summands. \(G_w\simeq\bigoplus_bS_b[s_b]\) gives \({}^pH^i(G_w)=\bigoplus_{s_b=-i}S_b\), whose constituents are the \(S_b\), summands of \(G_w\) up to shift. So \(\mathcal M_I^+=\mathcal M_I\), and step 8 gives \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}(\mathcal M_I)\) and \(S\in\mathcal C_I^{\mathrm{Iwa}}\iff S\in\mathcal M_I\).
   • (⇐) If \(\mathcal M_I=\mathcal S_I\), then \(\mathcal C_I=\operatorname{Thick}(\mathcal S_I)=\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\) by (a) and (b). (DT) is not used.
   • (⇒) If \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\), every \(S\in\mathcal S_I\) lies in \(\mathcal C_I^{\mathrm{Iwa}}\), hence in \(\mathcal M_I^+=\mathcal M_I\).
   • Second proof. Version 2 (A5-4): \(\{\mathcal G\in\mathcal H_{c,I}:\mathcal G\ast_I\mathrm{IC}_0\in\operatorname{Thick}(\mathcal M_I)\}\) is thick and contains every \(\mathrm{IC}_w\) by (DT), hence is \(\mathcal H_{c,I}\) (step 7). Audit A7-4: holds once (Lev) has \(D^b\)-full faithfulness. Run A2-4(c)–(e), A3-5; audit A7-2.
10. Ruled out A non-equivariant splitting

    (DT) must hold in the equivariant category: a complex whose perverse cohomology is simple need not have its constituents as summands.

    $$\mathbb G_m\ \text{acting on a point}:\qquad E=\operatorname{Cone}(c:k\to k[2]),\qquad E\mapsto k[2]\oplus k[1]\ \text{after forgetting},\qquad k\notin\operatorname{Thick}(E).$$
    Proof

    • Forgetting. \(\operatorname{Hom}(k,k[n])=H^n(B\mathbb G_m)\), and \(H^*(B\mathbb G_m)=k[c]\), \(\deg c=2\). The forgetful functor kills \(c\), so \(E\) becomes \(k[2]\oplus k[1]\) and each \({}^pH^i(E)\) is simple.
    • A thick subcategory. \(\mathcal T=\{A:\sum_n\dim\operatorname{Hom}(k,A[n])\lt\infty\}\) is closed under shifts, cofibres (long exact sequences) and retracts.
    • Separation. Multiplication by \(c\) on \(k[c]\) is injective, and onto except in degree \(0\), so \(\sum_n\dim\operatorname{Hom}(k,E[n])=1\) and \(E\in\mathcal T\); \(\sum_n\dim H^n(B\mathbb G_m)=\infty\), so \(k\notin\mathcal T\supseteq\operatorname{Thick}(E)\).
    • Consequence. A splitting in \(D^b_c(Y_{N,M})\) (BBD 5.4.5, 6.2.5) does not make a constituent of \({}^pH^i(G_w)\) a summand in \(\mathcal C_I\); the groups \(H_\alpha\) contain \(T\) and have fixed points (\(0\in Y_{N,M}\) for \(\mathbb A^n\)). Audit A7-3: the objection to A5-7, step 3, and to A5-4 over \(\mathbb C\); A2-5 already used the theorem on stacks.
11. Method (DT) from a finite-type proper model

    Given \((\mathrm{FM}_w)\), the generator is a proper push-forward of a pure complex on a finite-type quotient stack, which splits on the stack, hence equivariantly. This is Proposition IV:

    $$G_w\simeq\pi_{N,M}^*\,p_!\,r^*\mathrm{IC}_w,\qquad G_w\simeq\bigoplus_i{}^pH^i(G_w)[-i]\ \text{in}\ \mathcal C_I,\qquad{}^pH^i(G_w)\ \text{semisimple}.$$
    Proof

    • Base change. The action map factors through the closed embedding \([g,y]\mapsto(gI,gy)\) of \(\overline{IwI}\times^IL^+X\) into \(\mathrm{Fl}_w\times Y_N\), then the projection, proper since \(\mathrm{Fl}_w\) is projective. By \((\mathrm{FM}_w)\) and proper base change, \(G_w\simeq\pi_{N,M}^*p_!r^*\mathrm{IC}_w\in\mathcal C_I^{(N,M)}\).
    • Purity. \(\mathrm{IC}_w=j_{!*}\overline{\mathbb Q}_\ell[\ell(w)]\) is pure of weight \(\ell(w)\) (BBD 5.3.2); \(r^*\), \(r\) smooth, and \(p_!=p_*\), \(p\) proper, preserve purity (Sun, Proposition 3.9).
    • Equivariant splitting. Sun, Theorem 1.2 (with 3.11 and 3.12), for the representable proper morphism \([H''\backslash Z]\to[H''\backslash Y_{N,M}]\) of finite-type stacks with affine stabilizers: \(p_*r^*\mathrm{IC}_w\simeq\bigoplus_i({}^pR^ip_*)[-i]\) with semisimple terms, on the stack. Inflation along the unipotent \(I_{N,M}/I''\) and \(\pi_{N,M}^*\) are fully faithful and t-exact up to shift ((Lev), step 2), so the splitting holds in \(\mathcal C_I\); a semisimple object of a finite-length heart is a finite sum of simple objects.
    • Over \(\mathbb C\). Replace the splitting by the equivariant decomposition theorem (Bernstein–Lunts, LNM 1578, §5; Sun, Theorem 3.15, stated without proof). Cited, not re-proved.
    • Where \((\mathrm{FM}_w)\) holds. \(\mathbb A^n\): \(E_w\) is a vector bundle over \(\mathrm{Fl}_{\le w}\), closed in \(\mathrm{Fl}_{\le w}\times Y_{N,M}\) (audit A7-4, step 5). The group: the twisted product \(\overline{Iw_1I}\times^I\overline{Iw_0I}\times^I\overline{Iw_2I}/I\to\mathrm{Fl}_H\) plays its role, with the same purity argument (A7-4, step 6; A8-1, R3). Run A2-5.
12. Conclusion Iwahori summand criterion

    Steps 6, 8, 9 and 11 prove (a)–(d). Under (Lev), I1, I2 (or I2′) and (DT), Iwahori generation says that every simple object of \(\mathcal C_I\) is a shifted summand of a generator:

    $$\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I,\qquad\text{and always}\quad\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}(\mathcal M_I^+).$$

    Proposition II (trace.html) reaches \(\mathcal M_I=\mathcal S_I\) from function-level generation over \(\mathbb F_q(\!(t)\!)\); Propositions III (rows.html), IV (gl2-t.html) and V (divergence.html) decide the summands row by row.

Remarks and literature

• No finiteness of orbit closures. The spherical run's hypothesis that each orbit closure is a finite union of orbits fails for \(\mathbb A^n\) at Iwahori level: \(\overline{O_{m,j}}=t^mL_j\) contains infinitely many \(I\)-orbits and \(\{0\}\), which is never a piece. Nothing above needs it. Finiteness enters only level by level (finite-length hearts, I1), thick spans use finite filtrations only, and the standard objects \(\Delta_P\) never enter (A3-5, checks).
• Two examples. \(\mathbb A^n\) under \(\mathrm{GL}_n\): the simple objects are the lattice sheaves \(\underline k_\Lambda\), \(\Lambda\in\{t^mL_j\}\) (A2-6(iv)). With \(\tau e_1=te_n\), \(\tau e_i=e_{i-1}\), a generator of \(\Omega\), $$\underline k_{\tau^k\mathcal O^n}=\Delta_{\tau^k}\ast_I\mathrm{IC}_0=\mathrm{IC}_{\tau^k}\ast_I\mathrm{IC}_0,$$ so \(\mathcal M_I=\mathcal S_I\) and \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) (A2-7; audits A8-8, A8-9). The group \(X=H\): with levels truncated in \(\Omega\), \(\mathcal S_I=\{\mathrm{IC}_z:z\in\widetilde W\}\) and \(\mathrm{IC}_\omega=\Delta_\omega\) for \(\omega\in\Omega\) (A2-8). Over \(\overline{\mathbb F}_q\) no \(\Delta_\omega\) is a constituent of a generator (A2-9; audit A8-5), so by (b) \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\): Proposition III(a).
• Relation to the Iwahori-generation pages. Their Proposition III proves \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) from I1–I3 by dévissage over the pieces of one level, with standard objects. Proposition I replaces I3 by \(\mathcal M_I=\mathcal S_I\) and uses no standard objects: for \(\mathbb A^n\) both apply; for the group I3 fails at the closed orbits and \(\mathcal M_I\ne\mathcal S_I\). Their Proposition I(a), through the retraction \(V\to q_*q^*V\) of the key lemma, makes \(p^*(V_\lambda\ast_K\mathrm{IC}_0)\) a retract of \(G_{w_\lambda}\) up to shift, \(w_\lambda\) the longest element of \(Wt^\lambda W\). So every spherical summand \(\mathrm{IC}_\nu\) of the spherical run's Proposition I, forgotten to \(I\), lies in \(\mathcal M_I\) (A2-3(c); audit A7-2: up to the shift \([-2d]\), with \(|W|\) invertible in the coefficients).