Proof

Step 1 splits the box into three claims. Steps 2–3 prove the support bound and the three hypotheses that hold for every \(X\), step 4 decides H3, steps 5–6 decide H5, and step 7 sorts the families.

1. Purpose Reduce to three claims

   The box follows from three claims about \(X\):

   1. H1, H2 and H4 hold;
   2. H3 holds exactly when \(Z_{G^\theta}(S)\) is connected;
   3. under H3, H5 holds exactly when \(\varphi(\Lambda_T^+)=\Lambda_S^+\).
   Proof

   • (⟹) H3 gives connectedness by (b), and with H3, (c) turns H5 into \(\varphi(\Lambda_T^+)=\Lambda_S^+\).
   • (⟸) (a) and (b) give H1–H4, and (c) gives H5. Steps 2–3 prove (a), step 4 proves (b), and steps 5–6 prove (c).
2. Result Stratify and bound supports

   Claim (a), first half: H1 holds for every \(X\), and so does the support bound that step 6 uses.

   $$\overline{\mathcal O_\mu}=\bigsqcup_{\nu\le\mu}\mathcal O_\nu,\qquad\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}}.$$
   Proof

   • H1. Chen–Yi Theorem 18 gives the closure relation; over \(\mathbb C\) it is also Nadler's (Remark 19). Dominant \(\nu\le\mu\) lie in the convex hull of \(W\mu\), so lower intervals are finite, and objects of \(\mathcal C\) are supported on closed finite unions of orbits by definition.
   • Translates. The convolution lives on translates of the arc space, and \(\tau\) maps each translate into three double cosets: $$\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\bigcup_{\lambda'\le\lambda}L^+G\,t^{\lambda'}L^+X,$$ $$\tau\big(L^+G\,t^{\lambda'}L^+X\big)\subseteq L^+G\,t^{\lambda'}L^+G\,t^{-\theta\lambda'}L^+G.$$
   • Bound. Since \(\tau(\mathcal O_\nu)\subseteq L^+G\,t^\nu L^+G\), convolution on \(\operatorname{Gr}_G\) bounds every \(\nu\) whose orbit meets the support. The dominant representative of \(-\theta\lambda'\) differs from it by coroots of \(L\): $$\nu\le\lambda'+(-\theta\lambda')^+,\qquad(-\theta\lambda')^+-(-\theta\lambda')\in\textstyle\sum_{\check\alpha\in\check\Phi_L^+}\mathbb Z_{\ge0}\,\check\alpha.$$
   • Projection. \(p=\tfrac12(1-\theta)\) fixes \(\nu\) and \(\varphi(\lambda')\), annihilates the coroots of \(L\), and sends every other positive coroot into the positive cone: $$\varphi(\lambda')-\nu=p\big(\lambda'+(-\theta\lambda')^+-\nu\big)\in\mathbb Z\check\Phi\cap\textstyle\sum_{\check\alpha\in\check\Phi^+}\mathbb Q_{\ge0}\,\check\alpha=\sum_{\check\alpha\in\check\Phi^+}\mathbb Z_{\ge0}\,\check\alpha.$$ The same projection gives \(\varphi(\lambda')\le\varphi(\lambda)\), and Theorem 18 turns \(\nu\le\varphi(\lambda)\) into the support bound.
   • The draft. Draft Proposition 3.43 reads its bound off the single point \(t^\lambda\cdot_\theta t^\mu=t^{\bar\lambda+\mu}\); the last two bullets bound the whole double coset. The gap is invisible in the group case, where \(L=T\) and \(-\theta\lambda\) is already dominant.
3. Result Build recollement and convolution

   Claim (a), second half: H2 and H4 hold in Chen–Yi's sheaf theory. An open orbit \(\mathcal O_\mu\) of a closed finite union \(Z\), with closed complement \(i:D\hookrightarrow Z\), gives the triangle, and convolution stays constructible:

   $$a_{\mu,Z!}a_{\mu,Z}^*M\longrightarrow M\longrightarrow i_*i^*M\xrightarrow{+1},\qquad V_\lambda\ast\mathrm{IC}_0\in\mathcal C.$$
   Proof

   • Recollement. Orbit closures in \(LX\) are placid, and the inclusions of nested closed finite unions of orbits and of their open orbits are finitely presented (Chen–Yi Theorem 36). Appendix A gives restriction and extension along such maps on the placid quotients; open–closed localization gives the triangle, and composition gives H2(c).
   • Convolution. For a closed finite union \(Z\), the map \([g,x]\mapsto(gL^+G,gx)\) embeds \(\operatorname{Gr}_{\le\lambda}\mathbin{\widetilde\times}Z\) as a closed subspace of \(\operatorname{Gr}_{\le\lambda}\times LX\). Since \(\operatorname{Gr}_{\le\lambda}\) is projective, \(m\) is proper there, and proper pushforward keeps the convolution constructible: this is H4.
   • Over \(\mathbb C\). Both bullets use the Betti form of the theory. Chen–Yi §1.2.6 states that the arguments carry over through mixed Hodge modules; writing this out is open item 1.
4. Result Decide H3 by stabilizers

   Claim (b). Each orbit stabilizer has the component group of a centralizer in \(G^\theta\), and one centralizer decides them all:

   $$\pi_0\big(\operatorname{Stab}_{L^+G}(x_\mu)\big)\cong\pi_0\big(Z_{G^\theta}(\mu)\big),\qquad\mathrm{H3}\iff Z_{G^\theta}(S)\ \text{connected}.$$
   Proof

   • Stabilizer. Through \(\tau\), \(L^+G\) acts by \(g\cdot y=gy\theta(g)^{-1}\). The stabilizer of \(t^\mu\) is the fixed group of \(\theta_\mu=\operatorname{Ad}(t^\mu)\circ\theta\) on a group with pro-unipotent kernel \(J_\mu^+\): $$\operatorname{Stab}_{L^+G}(x_\mu)=J_\mu^{\theta_\mu},\qquad J_\mu=L^+G\cap t^\mu L^+G\,t^{-\mu}=Z_G(\mu)\ltimes J_\mu^+.$$
   • Component group. Both factors are \(\theta_\mu\)-stable, and \(\theta_\mu=\theta\) on \(Z_G(\mu)\), which commutes with \(t^\mu\). Fixed points of an involution of a pro-unipotent group are connected in characteristic \(0\), which gives the first display.
   • One centralizer. \(Z_{G^\theta}(\mu)^\circ\) conjugates the maximal θ-split tori of \(Z_G(\mu)\) and realizes their little Weyl group (Kostant–Rallis; Richardson), so \(Z_{G^\theta}(\mu)=Z_{G^\theta}(\mu)^\circ\cdot Z_{G^\theta}(S)\). A label with \(Z_G(\mu)=L\), such as \(\varphi(\lambda)\) for regular \(\lambda\), has \(Z_{G^\theta}(\mu)=Z_{G^\theta}(S)\): $$\pi_0\big(\operatorname{Stab}_{L^+G}(x_\mu)\big)=1\ \text{ for all }\mu\in\Lambda_S^+\iff Z_{G^\theta}(S)\ \text{connected}.$$
   • H3. With connected stabilizers the ordinary heart of \(\mathcal C_\mu\) is \(\operatorname{Rep}(\pi_0)=\operatorname{Vect}\), which is H3. Conversely, H3 puts \(\mathcal C_\mu\) inside \(\operatorname{Thick}(\underline k_{\mathcal O_\mu})\), whose cohomology sheaves carry trivial \(\pi_0\)-action, while a nontrivial representation of \(\pi_0\) gives a local system on \(\mathcal O_\mu\).
   • A finite check. \(S\) is maximal θ-split and central in \(L\), so \(\theta\) acts trivially on \(L/S\). Hence \(L=S\cdot Z_{G^\theta}(S)\) with \(S\cap Z_{G^\theta}(S)=S[2]\), and \(Z_{G^\theta}(S)\) is connected exactly when \(S[2]\subseteq Z_{G^\theta}(S)^\circ\).
5. Assumption Lift each label dominantly

   Claim (c) turns on coverage: every orbit label has a dominant lift. Step 6 shows that H5 forces it, and for the converse we assume it:

   $$\forall\mu\in\Lambda_S^+\ \exists\lambda\in\Lambda_T^+:\quad\varphi(\lambda)=\mu.$$
   Proof

   • Coverage depends on \((G,\theta)\) and on the isogeny type; step 7 checks it row by row.
   • Chen–Yi's Theorem 87 does not supply it. Its Lemma 90 lifts labels to \(\Lambda_T\) but not to \(\Lambda_T^+\), and \(E_6/F_4\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) meet its assumptions without coverage. Result · Find labels without dominant lifts
6. Method Decide H5 by lifts

   Claim (c), second half: assume H3. A dominant \(\lambda\) satisfies H5 at \(\mu\) exactly when it lifts \(\mu\), and a lift's top restriction splits:

   $$\varphi(\lambda)=\mu\quad\Longrightarrow\quad a_\mu^*(V_\lambda\ast\mathrm{IC}_0)\simeq\bigoplus_{s=1}^{N_\mu}\underline k_{\mathcal O_\mu}[d_s]\ \text{ in }\mathcal C_\mu,$$ $$0\lt N_\mu\lt\infty.$$
   Proof

   • Decompose. \(V_\lambda\mathbin{\widetilde\boxtimes}\mathrm{IC}_0\) is pure and \(m\) is proper on its support (step 3), so the decomposition theorem splits the convolution (Chen–Yi Proposition 70, as used for Theorem 93): $$V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_i\mathrm{IC}(\mathcal O_{\nu_i},\mathcal L_i)[n_i].$$
   • Restrict. The support bound of step 2 gives \(\nu_i\le\varphi(\lambda)=\mu\), which is H5(a). Only the summands with \(\nu_i=\mu\) meet \(\mathcal O_\mu\), and H3 makes their local systems constant; this gives the splitting with \(N_\mu\lt\infty\).
   • Locate a stalk. Proper base change computes the stalk at \(x_\mu=t^\lambda x_0\) on a projective fibre that meets the open Schubert cell: $$F=\{g\in\operatorname{Gr}_{\le\lambda}:g^{-1}x_\mu\in L^+X\},\qquad U=F\cap\operatorname{Gr}_\lambda\ni t^\lambda,\qquad e=\dim U.$$
   • Read the lowest degree. With \(d=\langle2\rho,\lambda\rangle\), \(\mathcal H^{-d}V_\lambda\supseteq\underline k\) by Deligne's formula, so the stalk is nonzero in degree \(-d\), and \(N_\mu\gt0\): $$H^{-d}\big((V_\lambda\ast\mathrm{IC}_0)_{x_\mu}\big)\supseteq H^0(F,k)\ne0.$$ Weights give the same: \(0\ne H^{2e}_c(U)\hookrightarrow H^{2e-d}\) of the stalk. Expanded argument →^[T]
   • Only lifts. The last two bullets do not use H3 and apply to every dominant \(\lambda\), at \(t^\lambda x_0\in\mathcal O_{\varphi(\lambda)}\). So if \(\lambda\) satisfies H5 at \(\mu\), H5(a) puts \(\mathcal O_{\varphi(\lambda)}\) inside \(\overline{\mathcal O_\mu}\), H5(b) and step 2 put \(\mathcal O_\mu\) inside \(\overline{\mathcal O_{\varphi(\lambda)}}\), and H1(a) gives \(\varphi(\lambda)=\mu\). Fibre proposition →
7. Conclusion Sort the Chen–Yi families

   Steps 2–6 prove the box. Applied row by row, it sorts the symmetric varieties in Chen–Yi:

   Table 2. \(X\) and In Chen–Yi and \(Z_{G^\theta}(S)\)

   \(X\)	In Chen–Yi	\(Z_{G^\theta}(S)\)	Dominant lift	H1–H5
   \((H\times H)/H\)	§1.2.2, §10.1	\(T_H\) ✓	\((\mu,-\mu)=\varphi(\mu,0)\) ✓	Hold
   \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(n\ge2\)	Example 88	\(\mathrm{Spin}_{2n-2}\) ✓	\(2m\varepsilon_1=\varphi(m\varepsilon_1+m\varepsilon_2)\) ✓	Hold
   \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(p\le q\)	Remark 91	\(\mathbb G_m^p\times\mathrm{GL}_{q-p}\) ✓	\(\mu=\varphi(a_1,\dots,a_p,0,\dots,0)\) ✓	Hold
   \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\)	Example 88	\(\mathrm{SL}_2^n\) ✓	none for \((1,1,1,1,0,\dots,0,-2,-2)\), \(n\ge3\)	Coverage fails, \(n\ge3\)
   \(E_6/F_4\)	Example 88	\(\mathrm{Spin}_8\) ✓	none for \(3k\omega_1^\vee\), \(3k\omega_6^\vee\), \(k\) odd	Coverage fails
   \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\)	§6.2	\(\mathrm{SL}_2^m\) ✓	none for \((1,1,1,1,0,\dots,0)\), \(m\ge2\)	Coverage fails, \(m\ge2\)
   \(\mathrm{SL}_2/\mathrm{SO}_2\)	Remark 89	\(\mu_2\) ✗	none for \(\alpha^\vee\); \(\varphi\) doubles	Both fail
   \(\mathrm{GL}_n/\mathrm{O}_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\)	§3.2.1, §6.2, §1.2.4	\(G^\theta\) disconnected	\(\varphi\) doubles	Outside the setting

   Proof

   • Each row has its own page of explicit matrices: the model of \(G\), \(\theta\), \(T\) and \(S\); the matrices of \(\theta\) and \(\varphi\) on \(\Lambda_T\); the centralizer \(Z_{G^\theta}(S)\); and a dominant lift of every label, or a label with none. Group case · Spin · \(\mathrm{GL}_p\times\mathrm{GL}_q\) · Symplectic · \(E_6/F_4\) · Split forms.
   • Search. A search over boxes of dominant coweights confirms the lift column for \(\mathrm{Spin}_{2n}\) (\(n\le5\)), \(\mathrm{GL}_n\) (\(n\le6\)), \(\mathrm{SL}_{2n}\) (\(n\le4\)) and \(E_6\).