Proposition I as a paper · statement and complete proof (PDF) →

Definitions

• Symmetric space, orbits and labels

  \(G\) is connected reductive over an algebraically closed field, \(\theta\) an involution with \(G^\theta\) connected, \(X=G/G^\theta\), \(x_0=eG^\theta\). \(S\) is a maximal \(\theta\)-split torus inside a \(\theta\)-stable maximal torus \(T\); the dominant cocharacters are ordered by dominance:

  $$\Lambda_T^+\subseteq X_*(T),\qquad\Lambda_S^+=X_*(S)\cap\Lambda_T^+,\qquad\nu\le\mu\iff\mu-\nu\in\textstyle\sum_{\check\alpha\in\check\Phi^+}\mathbb Z_{\ge0}\check\alpha.$$

  The arc group acts on the loop space with one orbit per label, and translating the base point by a Hecke label lands in the orbit of its leading term (Chen–Yi, Proposition 17):

  $$LX=\bigsqcup_{\mu\in\Lambda_S^+}\mathcal O_\mu,\qquad\mathcal O_\mu=L^+G\cdot x_\mu,\qquad\varphi:\Lambda_T^+\to\Lambda_S^+,\ \lambda\mapsto\lambda-\theta(\lambda),\qquad t^\lambda x_0=x_{\varphi(\lambda)}.$$

  A label \(\mu\) is covered if \(\mu\in\varphi(\Lambda_T^+)\). \(\delta(\mu)\) is the dimension function of the orbits (Chen–Yi, Proposition 41).

• The category and the Satake objects

  \(\mathcal C_K\) is the idempotent-complete stable category of bounded \(L^+G\)-equivariant constructible complexes on \(LX\) with coefficients in a field \(k\), each supported on finitely many orbit closures: complex coefficients in the Betti setting of the pages, \(\overline{\mathbb Q}_\ell\) over \(\overline{\mathbb F}_q\) (\(p\ne2\)) in Chen–Yi's.

  $$\mathcal C_K=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX),\qquad a_\mu:\mathcal O_\mu\hookrightarrow LX,\qquad i_\mu:\overline{\mathcal O_\mu}\setminus\mathcal O_\mu\hookrightarrow LX,\qquad\Delta_\mu=a_{\mu!}\underline k_{\mathcal O_\mu}.$$

  The Satake objects convolve the Satake sheaves \(V_\lambda\) with \(\mathrm{IC}_0=\Delta_0\), the constant sheaf on \(L^+X\); \(\operatorname{Thick}\) is the smallest full subcategory containing \(0\) and closed under isomorphisms, shifts, cofibres and retracts:

  $$\mathcal C_K^{\mathrm{Sat}}=\operatorname{Thick}_{\mathcal C_K}\{V_\lambda\ast\mathrm{IC}_0:\lambda\in\Lambda_T^+\}.$$
• Intersection complexes of orbits

  For each label, the intersection complex of the orbit closure with constant coefficients, normalized to be perverse:

  $$\mathrm{IC}_\nu=\mathrm{IC}(\overline{\mathcal O_\nu},k),\qquad\operatorname{supp}\mathrm{IC}_\nu=\overline{\mathcal O_\nu},\qquad a_\nu^*\mathrm{IC}_\nu\simeq\underline k_{\mathcal O_\nu}[\delta(\nu)].$$

  Its stalk at \(x_\mu\) vanishes unless \(\mu\le\nu\) (closure relation, Chen–Yi Theorem 18), and for \(\mu\lt\nu\) it sits in degrees \(\lt-\delta(\mu)\) (support condition).

• The summand set

  The orbit labels whose intersection complex occurs, up to shift, as a direct summand of a Satake object:

  $$\mathcal M=\{\mu\in\Lambda_S^+:\ \mathrm{IC}_\mu[s]\ \text{is a direct summand of}\ V_\lambda\ast\mathrm{IC}_0\ \text{for some}\ \lambda\in\Lambda_T^+,\ s\in\mathbb Z\}.$$

  In an idempotent-complete stable category a retract is a direct summand, so "retract" and "summand" define the same set.

• Constituent subcategories

  For \(\Sigma\subseteq\Lambda_S^+\), \(\mathcal P_\Sigma\) is the Serre subcategory of the perverse heart \(\mathcal P\) whose objects have all Jordan–Hölder constituents among \(\mathrm{IC}_\nu\), \(\nu\in\Sigma\), and

  $$\mathcal D_\Sigma=\{M\in\mathcal C_K:\ {}^pH^i(M)\in\mathcal P_\Sigma\ \text{for all}\ i\in\mathbb Z\}.$$
• Defect of a stratum

  For a proper \(H\)-equivariant morphism \(m:Y\to Z\) of varieties of finite type, \(Y\) smooth irreducible of dimension \(D\), and \(Z=\bigsqcup_\nu\mathcal O_\nu\) finitely many \(H\)-orbits with base points \(x_\nu\):

  $$f_\nu=\dim m^{-1}(x_\nu),\qquad c_\nu=D-\dim\mathcal O_\nu,\qquad\operatorname{def}(\nu)=2f_\nu-c_\nu,\qquad r=\max_\nu\operatorname{def}(\nu).$$

  \(r\) is the defect of semismallness; \(m\) is semismall when \(r=0\).

Hypotheses H1–H3, PH and DT

H1–H3 are the hypotheses of Proposition I of the spherical workspace, copied here.

• H1 · Finite orbit filtration.

  1. Each orbit closure is the disjoint union of the finitely many orbits below it: $$\overline{\mathcal O_\mu}=\bigsqcup_{\nu\le\mu}\mathcal O_\nu,\qquad\#\{\nu\in\Lambda_S^+:\nu\le\mu\}\lt\infty.$$
  2. Every object of \(\mathcal C_K\) is supported on a finite union of orbit closures: $$\forall M\in\mathcal C_K\ \ \exists J\subseteq\Lambda_S^+\ \text{finite}:\qquad\operatorname{supp}(M)\subseteq\bigcup_{\mu\in J}\overline{\mathcal O_\mu}.$$
• H2 · Equivariant recollement.

  Let \(D\subseteq Z\) be closed finite unions of orbits and \(\mathcal O_\mu\subseteq Z\) an orbit, with embeddings $$i:D\hookrightarrow Z,\qquad r_Z:Z\hookrightarrow LX,\qquad a_{\mu,Z}:\mathcal O_\mu\hookrightarrow Z.$$

  1. Restriction \(f^*\) and extension by zero \(f_!\) along these embeddings and their composites are defined and exact. For a closed embedding \(f\), the extension \(f_!=f_*\) is right adjoint to \(f^*\).
  2. If \(\mathcal O_\mu\) is open in \(Z\) with complement \(D\), then \(j=a_{\mu,Z}\) and \(i\) give every \(M\in\mathcal C_Z\) a localization triangle: $$j_!j^*M\longrightarrow M\longrightarrow i_*i^*M\xrightarrow{+1}.$$
  3. Restriction and extension by zero respect composition: \((fg)^*\simeq g^*f^*\) and \((fg)_!\simeq f_!g_!\).

  Kuijper (2026), Definition 2.3.4 and Lemma 4.3.3 supply the abstract recollement mechanism; preservation of these constructible subcategories is part of H2.

  Recollement needs only that \(Z\) and \(D\) be cut out at a finite jet level (\(Z\setminus D\) quasi-compact); finiteness of the orbit union is used for the inductions of H1. For symmetric \(X\), closed finite unions of orbits are cut out at a finite level (Chen–Yi Thm. 36). Reader question →^[T]

• H3 · Constant coefficients.

  Each orbit category has a bounded t-structure whose heart consists of the finite direct sums of the constant sheaf: $$\mathcal C_\mu^\heartsuit\simeq\operatorname{Vect}_k^{\mathrm{fd}},\qquad V\longmapsto\underline k_{\mathcal O_\mu}\otimes_kV.$$ For the ordinary t-structure the heart is \(\operatorname{Rep}^{\mathrm{fd}}_k\big(\pi_0(\operatorname{Stab}_{L^+G}(x_\mu))\big)\) (draft Construction 3.34), so connected stabilizers give H3.
• PH · Perverse heart.

  \(\mathcal C_K\) carries a bounded t-structure whose heart \(\mathcal P\) has finite length, whose simple objects are the \(\mathrm{IC}_\nu\), \(\nu\in\Lambda_S^+\), pairwise non-isomorphic, and whose cohomology functors are cohomological: $${}^pH^i:\mathcal C_K\longrightarrow\mathcal P,\qquad\operatorname{Simple}(\mathcal P)=\{\mathrm{IC}_\nu:\nu\in\Lambda_S^+\}.$$

  Where it holds

  Over \(\overline{\mathbb F}_q\): the δ-normalized perverse t-structure of Chen–Yi A.7, with finite length on finite-type models (BBD 4.3.1); under H3 the equivariant local systems on orbits are constant, so the simple objects are the \(\mathrm{IC}_\nu\). This is part of D1 of the run.

• DT · Decomposition.

  For each \(\lambda\in\Lambda_T^+\) the Satake object lies in \(\mathcal C_K\) and is a finite direct sum of shifted intersection complexes: $$V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{b=1}^{n_\lambda}\mathrm{IC}_{\nu_b}[s_b],\qquad\nu_b\in\Lambda_S^+,\ s_b\in\mathbb Z.$$

  Where it holds

  Over \(\overline{\mathbb F}_q\), \(p\ne2\), with \(Z_{G^\theta}(S)\) connected: the decomposition theorem on finite-type equivariant models (Sun, Theorem 1.2; BBD 5.4.5–5.4.6) as in Chen–Yi Proposition 70 and the proof of Theorem 93. This is D2 of the run; over \(\mathbb C\) it is the workspace's open item 1 (D5).

Proof

Step 1 names the summand set. Steps 2–4 bound \(\mathcal C_K^{\mathrm{Sat}}\) from above and below and prove (a); steps 5–6 prove (b). Steps 7–8 place covered labels and H5 inside the criterion; the ruled-out route explains why restriction data cannot go further, and the maximal-defect lemma with the GL₄/Sp₄ example detects summands by fibre dimension instead.

1. Purpose Name the IC summands

   Thick subcategories contain retracts and shifts, so every \(\mathrm{IC}_\nu\) with \(\nu\in\mathcal M\) already lies in \(\mathcal C_K^{\mathrm{Sat}}\). The proof compares the two subcategories:

   $$\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\in\mathcal M\}\ \overset{?}{=}\ \mathcal C_K^{\mathrm{Sat}}\ \overset{?}{=}\ \mathcal C_K.$$
   Plan

   • From above. Steps 3–4 build a thick subcategory that contains every Satake object and no \(\mathrm{IC}_\mu\) with \(\mu\notin\mathcal M\).
   • From below. Step 2 writes each generator as a sum of these ICs; steps 5–6 trade ICs for standard objects.
2. Result Span: summands generate

   The Satake subcategory is the thick span of the intersection complexes it contains as summands:

   $$\mathcal C_K^{\mathrm{Sat}}=\operatorname{Thick}_{\mathcal C_K}\{\mathrm{IC}_\nu:\nu\in\mathcal M\}.$$
   Proof

   • Premises. DT; the definition of \(\mathcal M\).
   • ⊇. For \(\nu\in\mathcal M\), \(\mathrm{IC}_\nu[s]\) is a retract of some \(V_\lambda\ast\mathrm{IC}_0\in\mathcal C_K^{\mathrm{Sat}}\), so \(\mathrm{IC}_\nu\in\mathcal C_K^{\mathrm{Sat}}\).
   • ⊆. By DT each generator is \(\bigoplus_b\mathrm{IC}_{\nu_b}[s_b]\); every summand has \(\nu_b\in\mathcal M\) by definition, and a finite direct sum is an iterated cofibre of split triangles. Minimality of \(\mathcal C_K^{\mathrm{Sat}}\) gives equality.
3. Method Constituent subcategories are thick

   For every \(\Sigma\subseteq\Lambda_S^+\) the constituent subcategory is thick:

   $$\mathcal D_\Sigma=\{M\in\mathcal C_K:{}^pH^i(M)\in\mathcal P_\Sigma\ \forall i\}\ \ \text{is a thick subcategory of}\ \mathcal C_K.$$
   Proof

   • Premises. PH: finite length (so Jordan–Hölder constituents exist) and cohomological \({}^pH^0\).
   • Inference. \(\mathcal P_\Sigma\) is closed under subobjects, quotients and extensions, so the Serre quotient \(q:\mathcal P\to\mathcal P/\mathcal P_\Sigma\) is exact and \(h=q\circ{}^pH^0\) is cohomological. Since \({}^pH^i(M)={}^pH^0(M[i])\), $$\mathcal D_\Sigma=\{M\in\mathcal C_K:\ h(M[i])=0\ \text{for all}\ i\}.$$
   • Output. Stacks, Lemma 13.6.3 [05RD]: a saturated triangulated subcategory, i.e. closed under shifts, cofibres and direct summands.
4. Result Only summands are reached

   An intersection complex lies in the Satake subcategory only if it is a summand of a Satake object:

   $$\mathrm{IC}_\mu\in\mathcal C_K^{\mathrm{Sat}}\iff\mu\in\mathcal M,\qquad\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\ \Longrightarrow\ \mathcal M=\Lambda_S^+.$$
   Proof

   • Premises. DT, PH, step 3 with \(\Sigma=\mathcal M\).
   • Satake objects. By DT, \({}^pH^i(V_\lambda\ast\mathrm{IC}_0)=\bigoplus_{s_b=-i}\mathrm{IC}_{\nu_b}\in\mathcal P_{\mathcal M}\), so every generator lies in the thick subcategory \(\mathcal D_{\mathcal M}\), and \(\mathcal C_K^{\mathrm{Sat}}\subseteq\mathcal D_{\mathcal M}\).
   • Missing labels. If \(\mu\notin\mathcal M\), \({}^pH^0(\mathrm{IC}_\mu)=\mathrm{IC}_\mu\) is simple and not isomorphic to \(\mathrm{IC}_\nu\) for \(\nu\in\mathcal M\), so \(\mathrm{IC}_\mu\notin\mathcal D_{\mathcal M}\supseteq\mathcal C_K^{\mathrm{Sat}}\).
   • Output. With step 2 this is (a), and the direction ⇒ of (b).
5. Method ICs and standards span alike

   Below every label, intersection complexes and standard objects generate the same thick subcategory:

   $$\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\le\mu\}=\operatorname{Thick}\{\Delta_\nu:\nu\le\mu\}\qquad(\mu\in\Lambda_S^+).$$
   Proof

   • Premises. H1–H3; the normalization \(a_\mu^*\mathrm{IC}_\mu\simeq\underline k_{\mathcal O_\mu}[\delta(\mu)]\).
   • Induction. On \(h(\mu)=\#\{\nu\le\mu\}\), finite by H1(a). \(\mathrm{IC}_\mu\) lives on \(\overline{\mathcal O_\mu}\), in which \(\mathcal O_\mu\) is open; H2(b), pushed to \(LX\) by H2(c), gives $$\Delta_\mu[\delta(\mu)]\longrightarrow\mathrm{IC}_\mu\longrightarrow i_{\mu*}i_\mu^*\mathrm{IC}_\mu\xrightarrow{+1}.$$
   • Boundary. Standard generation on the boundary (Proposition I of the workspace, step 1(a), from H1–H3) puts the right term in \(\operatorname{Thick}\{\Delta_\nu:\nu\lt\mu\}\), which equals \(\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\lt\mu\}\) by induction: the claim at every \(\nu\lt\mu\), since \(\{\nu\lt\mu\}=\bigcup_{\nu\lt\mu}\{\nu'\le\nu\}\). This is used only for \(\Delta_\mu\in\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\le\mu\}\).^[T]
   • Output. Two out of three in the triangle gives \(\Delta_\mu\in\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\le\mu\}\) and \(\mathrm{IC}_\mu\in\operatorname{Thick}\{\Delta_\nu:\nu\le\mu\}\).
6. Result All summands give everything

   If every label is a summand label, the Satake objects generate:

   $$\mathcal M=\Lambda_S^+\ \Longrightarrow\ \mathcal C_K^{\mathrm{Sat}}=\mathcal C_K.$$
   Proof

   • Inference. Step 2 puts every \(\mathrm{IC}_\nu\) in \(\mathcal C_K^{\mathrm{Sat}}\), step 5 every \(\Delta_\nu\), and standard generation \(\mathcal C_K=\operatorname{Thick}\{\Delta_\mu:\mu\in\Lambda_S^+\}\) (from H1–H3) gives \(\mathcal C_K\subseteq\mathcal C_K^{\mathrm{Sat}}\).
   • Output. With step 4 this proves (b).
   • Without H1–H3. Finite length and boundedness in PH also give \(\mathcal C_K=\operatorname{Thick}\{\mathrm{IC}_\nu\}\) directly: each perverse object is an iterated extension of simples, and each bounded object an iterated extension of its shifted perverse cohomology (run node A1-5; Stacks Lemma 13.35.7 read in the perverse t-structure).
7. Result Covered labels are summands

   The leading term of every Hecke label is a summand label:

   $$\varphi(\Lambda_T^+)\subseteq\mathcal M.$$
   Proof

   • Premises. DT, and from Proposition III: the support bound \(\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}}\) (step 2) and a nonzero stalk at \(x_{\varphi(\lambda)}=t^\lambda x_0\) (step 6, a weight count).
   • Upper bound. Write \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_b\mathrm{IC}_{\nu_b}[s_b]\). Each \(\mathcal O_{\nu_b}\) lies in the support, so \(\nu_b\le\varphi(\lambda)\).
   • Lower bound. A summand with nonzero stalk at \(x_{\varphi(\lambda)}\) has \(x_{\varphi(\lambda)}\in\overline{\mathcal O_{\nu_b}}\), i.e. \(\varphi(\lambda)\le\nu_b\) (Chen–Yi, Theorem 18).
   • Output. That summand has \(\nu_b=\varphi(\lambda)\), so \(\varphi(\lambda)\in\mathcal M\). This is (c).
8. Result H5 is sufficient

   A label at which H5 holds is a summand label, so H5 at every label gives generation by (b):

   $$\mathrm{H5}\ \Longrightarrow\ \mathcal M=\Lambda_S^+\ \Longrightarrow\ \mathcal C_K^{\mathrm{Sat}}=\mathcal C_K.$$
   Proof

   • Premises. DT; \(\lambda\) satisfying H5 at \(\mu\).
   • Inference. H5(a) gives \(\nu_b\le\mu\) for every \(b\). H5(b) gives \(a_\mu^*\mathrm{IC}_{\nu_b}\ne0\) for some \(b\), hence \(\mathcal O_\mu\subseteq\overline{\mathcal O_{\nu_b}}\) and \(\mu\le\nu_b\).
   • Output. \(\nu_b=\mu\), so \(\mu\in\mathcal M\). Only the nonvanishing in H5(b) is used; under DT the splitting is automatic.
9. Ruled out A weaker H5

   Asking only for support in the closure and a nonzero restriction does not extend generation past the covered labels:

   $$\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_\mu},\ \ a_\mu^*(V_\lambda\ast\mathrm{IC}_0)\ne0\ \Longrightarrow\ \mu=\varphi(\lambda);\qquad\Delta_\mu\notin\operatorname{Thick}\big(\operatorname{Cone}(u)\big)\ \text{on}\ B\mathbb G_m.$$
   Proof

   • (i) No Satake object qualifies at an uncovered label. The nonzero stalk at \(x_{\varphi(\lambda)}\) and support in \(\overline{\mathcal O_\mu}\) give \(\varphi(\lambda)\le\mu\); a nonzero restriction to \(\mathcal O_\mu\) and the support bound give \(\mu\le\varphi(\lambda)\). So \(\mu=\varphi(\lambda)\) is covered (the workspace's fibre proposition, parts (a)–(b)).
   • (ii) Nonvanishing does not generate. On an orbit with stabilizer \(\mathbb G_m\) (connected, so H3 holds) the orbit category is modules over the equivariant cohomology of a point, with the constant sheaf free (Bernstein–Lunts 12.7.2; ℓ-adic form in Bezrukavnikov–Yun §0.3): $$\mathcal C_\mu\simeq D^{\mathrm{fg}}(k[u]),\quad\deg u=2,\qquad\underline k_{\mathcal O_\mu}\mapsto k[u],\qquad E=\operatorname{Cone}(u:\underline k\to\underline k[2])\mapsto k[u]/(u).$$ Forgetting equivariance kills \(u\), so \(E\) has cohomology sheaves \(\underline k\) in degrees \(-2,-1\): nonzero and constant, but \(E\) is not a sum of shifted constants in \(\mathcal C_\mu\). The modules with finite-dimensional total cohomology form a thick subcategory that contains \(k[u]/(u)\) and not \(k[u]\), so \(\Delta_\mu\notin\operatorname{Thick}(E)\).
   • Consequence. Purity, through DT, is what splits the restriction; at uncovered labels summands must be detected directly, by the maximal-defect lemma below. Run node A3-4; verdict §7.
10. Method Maximal defect forces a summand

    At a stratum of maximal defect, the top cohomology of the fibre can only come from that stratum's intersection complex with the extreme shift:

    $$\operatorname{def}(\mu)=r\ \Longrightarrow\ \mathrm{IC}_\mu[-r]^{\oplus n_{\mu,r}}\ \big|\ K=m_!\underline k_Y[D],\qquad k^{n_{\mu,r}}\simeq H^{2f_\mu}\big(m^{-1}(x_\mu)\big)\ne0.$$
    Proof

    • Amplitude. Proper base change gives \(\mathcal H^i(K)_x=H^{i+D}(m^{-1}x)\), zero on \(\mathcal O_\nu\) unless \(i+D\le2f_\nu\); for such \(\nu\), \(\dim\mathcal O_\nu=D-c_\nu\le r-i\) since \(r\ge2f_\nu-c_\nu\ge i+D-c_\nu\). So \(K\in{}^pD^{\le r}\), and \(\mathbb DK\simeq K\) (\(Y\) smooth, \(m\) proper) gives \(K\in{}^pD^{\ge-r}\).
    • Decomposition. The decomposition theorem for \(Y/H\to Z/H\) (Sun, Theorem 1.2; de Cataldo–Migliorini, Theorem 1.6.1) and connected stabilizers give $$K\simeq\bigoplus_{\nu,j}\mathrm{IC}_\nu[-j]^{\oplus n_{\nu,j}},\qquad n_{\nu,j}=0\ \text{for}\ |j|\gt r.$$
    • Stalks at \(x_\mu\). For \(\nu\ne\mu\) with \(x_\mu\in\overline{\mathcal O_\nu}\), the support condition puts \(\mathrm{IC}_\nu[-j]\) in degrees \(\le-\dim\mathcal O_\mu-1+j\le-D+2f_\mu-1\). \(\mathrm{IC}_\mu[-j]\) sits in degree \(-D+c_\mu+j\le-D+2f_\mu\), with equality exactly for \(j=r\). The degree \(-D+2f_\mu\) stalk of \(K\) is \(H^{2f_\mu}(m^{-1}x_\mu)\), the top cohomology of a proper variety of dimension \(f_\mu\), of dimension the number of its \(f_\mu\)-dimensional components. For \(r=0\) this is the relevant-stratum theorem of de Cataldo–Migliorini (4.2.7).
    • Loop setting. Apply the lemma on finite-type models. For minuscule \(\omega_1,\dots,\omega_n\) the space \(Y=\operatorname{Gr}^{\omega_1}\mathbin{\widetilde\times}\cdots\mathbin{\widetilde\times}\operatorname{Gr}^{\omega_n}\mathbin{\widetilde\times}L^+X\) is smooth and \(m_!\underline k_Y[D]\) is a shift of \((V_{\omega_1}\ast\cdots\ast V_{\omega_n})\ast\mathrm{IC}_0\), a sum of shifted Satake objects; by Krull–Schmidt (\(\operatorname{End}\mathrm{IC}_\nu=k\)) each IC summand is a summand of one Satake object, so maximal-defect labels lie in \(\mathcal M\). Codimensions come from the dimension count of correction C-2 where \(LX\) is disconnected.
11. Result H5 is not necessary: GL₄/Sp₄

    For \(X=\mathrm{GL}_4/\mathrm{Sp}_4\) and \(\lambda=(1,1,0,0)\), the closed orbit has maximal defect, and its IC is a summand although H5 fails there:

    $$V_{(1,1,0,0)}\ast\mathrm{IC}_0\simeq\mathrm{IC}_{(2,2,0,0)}\oplus\mathrm{IC}_{(1,1,1,1)}\otimes\big(k[2]\oplus k\oplus k[-2]\big).$$
    Proof

    • Fibres. The convolution map is an isomorphism over \(\mathcal O_{(2,2,0,0)}\) and has fibre the Lagrangian Grassmannian \(\mathrm{LG}(2,4)\cong Q^3\) over the closed orbit \(\mathcal O_{(1,1,1,1)}\), of codimension \(4\).
    • Defects. \(\operatorname{def}=0\) on the open orbit and \(\operatorname{def}(1,1,1,1)=2\cdot3-4=2=r\). Step 9 gives \(\mathrm{IC}_{(1,1,1,1)}[-2]\) once (\(Q^3\) is irreducible); Verdier duality gives \([2]\), and the middle term completes the stalk count.
    • H5 fails. In the coordinates of Theorem 2, \(\varphi(\lambda)=(\lambda_1+\lambda_2,\lambda_1+\lambda_2,\lambda_3+\lambda_4,\lambda_3+\lambda_4)\), which is never \((1,1,1,1)\) for dominant \(\lambda\); H5 forces a dominant lift, so it fails at \((1,1,1,1)\).
    • Output. \((1,1,1,1)\in\mathcal M\); central translates give every \((a,a,a,a)\), and \(\mathcal M=\Lambda_S^+\) (Proposition IV, over \(\overline{\mathbb F}_q\) under D0–D2). This is (d).
12. Conclusion Summand criterion

    The steps above prove (a)–(e). Under H1–H3, PH and DT, generation is the statement that every orbit's intersection complex is a summand of a Satake object:

    $$\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\iff\mathcal M=\Lambda_S^+,\qquad\varphi(\Lambda_T^+)\subseteq\mathcal M.$$

    Proposition II (trace.html) reaches \(\mathcal M=\Lambda_S^+\) from function-level generation; Proposition IV (symplectic.html) and Proposition VIII (e6-f4.html) reach it at uncovered labels through fibre cohomology.