• Symmetric space

  Work over \(k=\mathbb C\) with complex coefficients. Fix a connected reductive group \(G\) and an involution \(\theta\) of \(G\) whose fixed subgroup \(G^\theta\) is connected.

  $$X=G/G^\theta,\qquad x_0=eG^\theta.$$

  Two tori, two jobs. \(T\) belongs to the group: its dominant cocharacters label the Satake sheaves. \(S\subseteq T\) belongs to the space: its dominant cocharacters label the orbits on \(LX\) (see the index sets).

  The θ-split torus \(S\). A torus is θ-split when \(\theta\) inverts it, and \(S\) is a maximal one. All such are \(G^\theta\)-conjugate, and their dimension is the rank of \(X\). \(S\) is the torus that \(X\) sees: the Cartan embedding \(\tau\) lands in the elements that \(\theta\) inverts, and each orbit has a representative with \(\tau(x_\mu)=t^\mu\in S(F)\).

  $$\theta(s)=s^{-1}\quad(s\in S),\qquad\dim S=\operatorname{rank}X.$$

  The maximal torus \(T\). \(T\) is a maximal torus of \(G\) that contains \(S\) and is θ-stable. \(\theta\) splits it into a fixed part and the split part, which is \(S\):

  $$S=\{t\in T:\theta(t)=t^{-1}\}^\circ,\qquad T=(T^\theta)^\circ\cdot S.$$

  Positive roots. A Borel \(B\supseteq T\) inside a minimal θ-split parabolic \(P\) fixes the positive roots \(\Phi^+\). The Levi \(L\) centralizes \(S\).

  $$P\cap\theta(P)=L=Z_G(S).$$

  Loops and arcs. Loop spaces use Laurent series \(F=k(\!(t)\!)\) and arc spaces power series. The arc group \(L^+G\) acts on the loop space \(LX\).

  $$LX(k)=X(F),\qquad L^+X(k)=X(k[\![t]\!]).$$

  Examples. In the group case \(G=H\times H\), \(\theta(g_1,g_2)=(g_2,g_1)\), and \(T_H\) is a maximal torus of \(H\). In \(\mathrm{GL}_2/\mathrm{Sp}_2\), \(\theta(g)=g/\det g\) (worked out here).

  Table 1. \(X\) and \(T\) and \(S\)

  \(X\)	\(T\)	\(S\)	\(\operatorname{rank}X\)
  \((H\times H)/H\)	\(T_H\times T_H\)	\(\{(s,s^{-1}):s\in T_H\}\)	\(\operatorname{rank}H\)
  \(\mathrm{GL}_2/\mathrm{Sp}_2\)	\(\{\operatorname{diag}(t_1,t_2)\}\)	\(\{\operatorname{diag}(a,a)\}\)	\(1\)

• The group case and its dictionary

  The factor \(H\). \(H\) is a connected reductive group over \(k\) with maximal torus \(T_H\), Borel subgroup \(B_H\supseteq T_H\), opposite Borel subgroup \(B_H^-\), positive roots \(\Phi_H^+\), simple coroots \(\alpha_1^\vee,\dots,\alpha_r^\vee\), half-sum of positive roots \(\rho_H\) and longest Weyl element \(w_0\). Its cocharacters carry the dominance order:

  $$\Lambda_H=X_*(T_H),\qquad\Lambda_H^+=\{\eta\in\Lambda_H:\langle\alpha,\eta\rangle\ge0\ \text{for all }\alpha\in\Phi_H^+\},\qquad\eta'\le_H\eta\iff\eta-\eta'\in\textstyle\sum_i\mathbb Z_{\ge0}\,\alpha_i^\vee.$$

  The affine Grassmannian of \(H\). \(p\) is the quotient map. For \(\eta\in\Lambda_H^+\), the Schubert cell \(\operatorname{Gr}_H^\eta\) is the \(L^+H\)-orbit of \(t^\eta\), the Schubert variety \(\operatorname{Gr}_H^{\le\eta}\) is its closure, and \(\mathrm{IC}^H_\eta\) is the intersection complex of \(\operatorname{Gr}_H^{\le\eta}\), normalized to be perverse:

  $$p:LH\longrightarrow\operatorname{Gr}_H=LH/L^+H,\qquad\operatorname{Gr}_H^\eta=L^+H\,t^\eta L^+H/L^+H,\qquad\mathrm{IC}^H_\eta\big|_{\operatorname{Gr}_H^\eta}\simeq\underline k[\langle2\rho_H,\eta\rangle].$$

  \(\star\) is convolution of \(L^+H\)-equivariant complexes on \(\operatorname{Gr}_H\). Its unit is \(\mathrm{IC}^H_0\), the constant sheaf on the point \(\operatorname{Gr}_H^0\).

  Jet groups. The arc group is the limit of the jet groups, and the kernels \(K_n\) are pro-unipotent:

  $$J_nH=H\big(k[t]/(t^n)\big),\qquad L^+H=\varprojlim_nJ_nH,\qquad K_n=\ker\big(L^+H\to J_nH\big).$$

  Dictionary. Unadorned symbols are those of H1–H5 for \(G=H\times H\); an orbit label is written \(\mu=(\mu_H,-\mu_H)\). Step 1 proves each row.

  Table 2. For \(G=H\times H\), as in H1–H5 and On the affine Grassmannian of \(H\)

  For \(G=H\times H\), as in H1–H5	On the affine Grassmannian of \(H\)
  \(X=G/G^\theta\), base point \(x_0\)	\(H\) with \((g_1,g_2)\cdot x=g_1xg_2^{-1}\), base point \(e\)
  \(L^+G\) acting on \(LX\)	\(L^+H\times L^+H\) acting on \(LH\)
  \(\lambda=(\lambda_1,\lambda_2)\in\Lambda_T^+\)	\(\lambda_1\in\Lambda_H^+\), \(-\lambda_2\in\Lambda_H^+\)
  \(\mu=(\mu_H,-\mu_H)\in\Lambda_S^+\)	\(\mu_H\in\Lambda_H^+\)
  \(\nu\le\mu\)	\(\nu_H\le_H\mu_H\)
  \(\mathcal O_\mu\) and \(\overline{\mathcal O_\mu}\)	\(p^{-1}(\operatorname{Gr}_H^{\mu_H})\) and \(p^{-1}(\operatorname{Gr}_H^{\le\mu_H})\)
  \(\mathcal C\)	\(\operatorname{Shv}_{\mathrm{cons}}(L^+H\backslash\operatorname{Gr}_H)\), via \(p^*\)
  \(V_\lambda\ast\mathrm{IC}_0\)	\(\mathrm{IC}^H_{\lambda_1}\star\mathrm{IC}^H_{-\lambda_2}\)

  Example. Nothing here fixes \(H\). For \(H=\mathrm{GL}_2\), \(\Lambda_H^+=\{(a_1,a_2)\in\mathbb Z^2:a_1\ge a_2\}\); the orbit with \(\mu_H=(a_1,a_2)\) is the double coset of \(\operatorname{diag}(t^{a_1},t^{a_2})\), its Schubert cell has dimension \(a_1-a_2\), and step 6 uses the label \(\lambda=\big((a_1,a_2),(0,0)\big)\).

• Two index sets, two roles

  Hecke labels. Cocharacters of \(T\) label the Satake sheaves \(V_\lambda\). A dominant \(\lambda\) is a highest weight of the dual group.

  $$\Lambda_T=X_*(T),\qquad\Lambda_T^+=\{\lambda\in\Lambda_T:\langle\alpha,\lambda\rangle\ge0\text{ for all }\alpha\in\Phi^+\}.$$

  Orbit labels. Dominant cocharacters of \(S\) label the arc-group orbits on \(LX\). The Cartan embedding \(\tau\) picks a representative \(x_\mu\) of each orbit (Chen–Yi, Proposition 17).

  $$\Lambda_S=X_*(S)\subseteq\Lambda_T,\qquad\Lambda_S^+=\Lambda_S\cap\Lambda_T^+,$$ $$\tau:X\hookrightarrow G,\quad gG^\theta\longmapsto g\theta(g)^{-1},\qquad\tau(x_\mu)=t^\mu,$$ $$LX=\bigsqcup_{\mu\in\Lambda_S^+}\mathcal O_\mu,\qquad\mathcal O_\mu=L^+G\cdot x_\mu.$$

  The leading-term map. Translating the base point by a Hecke label lands in one orbit, and \(\varphi\) names it.

  $$\varphi:\Lambda_T^+\longrightarrow\Lambda_S^+,\qquad\lambda\longmapsto\lambda-\theta(\lambda),$$ $$\tau(t^\lambda x_0)=t^\lambda\theta(t^\lambda)^{-1}=t^{\varphi(\lambda)},\qquad t^\lambda x_0\in\mathcal O_{\varphi(\lambda)}.$$

  Why the leading-term map is well defined

  1. For \(t\in T\), \(\theta\) inverts \(t\theta(t)^{-1}\), and the image of \(T\) is connected; so the image lies in \((T^{-\theta})^\circ\), which is \(S\) by maximality of \(S\). On cocharacters the map is \(\lambda\mapsto\lambda-\theta(\lambda)\): $$T\longrightarrow S,\quad t\longmapsto t\theta(t)^{-1},\qquad\lambda-\theta(\lambda)\in\Lambda_S.$$
  2. Roots of \(L\) vanish on \(S\), and \(T=(T^\theta)^\circ\cdot S\), so \(\theta\) fixes them. Since \(B\subseteq P\) and \(P\cap\theta(P)=L\), the parabolic \(\theta(P)\) is opposite to \(P\), so \(\theta\) sends the roots \(\Phi^+\setminus\Phi_L\) of the unipotent radical of \(P\) to negative roots: $$\theta\alpha=\alpha\quad(\alpha\in\Phi_L),\qquad-\theta\alpha\in\Phi^+\quad(\alpha\in\Phi^+\setminus\Phi_L).$$
  3. Since \(\langle\alpha,\theta\lambda\rangle=\langle\theta\alpha,\lambda\rangle\), a dominant input gives an output that pairs nonnegatively with every positive root: $$\langle\alpha,\lambda-\theta\lambda\rangle=\begin{cases}0,&\alpha\in\Phi_L^+,\\\langle\alpha,\lambda\rangle+\langle-\theta\alpha,\lambda\rangle\ge0,&\alpha\in\Phi^+\setminus\Phi_L.\end{cases}$$ Therefore the map has the stated domain and codomain: $$\boxed{\varphi:\Lambda_T^+\longrightarrow\Lambda_S^+,\quad\lambda\longmapsto\lambda-\theta(\lambda).}$$
  4. The Cartan embedding is injective, and \(\tau(t^\lambda x_0)=t^{\varphi(\lambda)}=\tau(x_{\varphi(\lambda)})\). So the translate is the orbit representative: $$t^\lambda x_0=x_{\varphi(\lambda)}\in\mathcal O_{\varphi(\lambda)}.$$

  For a symmetric space, H5 forces \(\varphi\) to be onto, and under H3 Proposition III proves the converse. The example page computes \(\varphi\) for the group case of \(\mathrm{SL}_2\), where it is onto, and for \(\mathrm{SL}_2/\mathrm{SO}_2\), where it doubles.

  Leading-term map · two minimal examples →

• Categories

  For an \(L^+G\)-stable locally closed \(W\subseteq LX\), let \(\mathcal C_W\) be the small, idempotent-complete, stable \(\infty\)-category of bounded \(L^+G\)-equivariant constructible complexes on \(W\) with coefficients in \(k\). Proposition I concerns the one on all of \(LX\):

  $$\mathcal C=\mathcal C_{LX}=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX),\qquad\mathcal C_Z=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash Z),$$ $$\mathcal C_\mu=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash\mathcal O_\mu),\qquad\widehat{\mathcal C}=\operatorname{Ind}(\mathcal C).$$
• Orbit order and supports

  Order \(\Lambda_S^+\) by the dominance order of \(\Lambda_T\):

  $$\nu\le\mu\iff\mu-\nu\in\textstyle\sum_{\check\alpha\in\check\Phi^+}\mathbb Z_{\ge0}\,\check\alpha.$$

  A closed finite union of orbits is a closed \(L^+G\)-stable \(Z\subseteq LX\) that is the union of finitely many orbits. \(I(Z)\) lists its orbits, and \(r_Z\) is its inclusion:

  $$I(Z)=\{\mu:\mathcal O_\mu\subseteq Z\},\qquad r_Z:Z\hookrightarrow LX.$$

  An object \(M\in\mathcal C\) is supported on \(Z\) when the unit of \(r_Z^*\dashv r_{Z*}\) is an isomorphism on \(M\):

  $$\operatorname{supp}(M)\subseteq Z\quad:\Longleftrightarrow\quad M\xrightarrow{\ \sim\ }r_{Z*}r_Z^*M.$$
• Boundary, orbit inclusions and standards

  Boundary. The boundary of the orbit \(\mathcal O_\mu\) is its closure minus the orbit. By H1(a) it is the union of the orbits below \(\mu\), a closed finite union of orbits:

  $$\overline{\mathcal O_\mu}\setminus\mathcal O_\mu=\bigsqcup_{\nu\lt\mu}\mathcal O_\nu.$$

  Inclusions. The orbit inclusion \(a_\mu\) is locally closed and the boundary inclusion \(i_\mu\) is closed. Inside a closed finite union \(Z\), the orbit includes by \(a_{\mu,Z}\).

  $$a_\mu:\mathcal O_\mu\hookrightarrow LX,\qquad i_\mu:\overline{\mathcal O_\mu}\setminus\mathcal O_\mu\hookrightarrow LX,\qquad a_{\mu,Z}:\mathcal O_\mu\hookrightarrow Z.$$

  Top and boundary parts. The top part of \(M\in\mathcal C\) restricts it to the orbit and extends by zero. The boundary part restricts it to the boundary and pushes forward:

  $$a_{\mu!}a_\mu^*M\ \ \text{(top part)},\qquad i_{\mu*}i_\mu^*M\ \ \text{(boundary part)}.$$

  Standards. A standard object extends the constant sheaf of an orbit by zero. \(\mathrm{IC}_0\) is the standard of \(\mathcal O_0=L^+X\), the orbit of the base point, which is closed.

  $$\Delta_\mu=a_{\mu!}\underline k_{\mathcal O_\mu}\in\mathcal C,\qquad\Delta_{\mu,Z}=a_{\mu,Z!}\underline k_{\mathcal O_\mu}\in\mathcal C_Z,\qquad\mathrm{IC}_0=\Delta_0.$$

  The constant objects are unshifted; a fixed normalization shift does not affect thick generation.

• Thick span

  For a set \(\mathcal S\) of objects of a stable \(\infty\)-category \(\mathcal D\), \(\operatorname{Thick}_{\mathcal D}(\mathcal S)\) is the smallest full subcategory that contains \(0\) and \(\mathcal S\) and is closed under isomorphisms, shifts, cofibers and retracts. Two of its properties are used:

  $$\text{(T1)}\quad A\to B\to C\xrightarrow{+1}\ \text{exact},\ \ A,C\in\operatorname{Thick}_{\mathcal D}(\mathcal S)\ \Longrightarrow\ B\in\operatorname{Thick}_{\mathcal D}(\mathcal S),$$ $$\text{(T2)}\quad F:\mathcal D\to\mathcal D'\ \text{exact}\ \Longrightarrow\ F\big(\operatorname{Thick}_{\mathcal D}(\mathcal S)\big)\subseteq\operatorname{Thick}_{\mathcal D'}\big(F(\mathcal S)\big).$$

  Proofs of (T1) and (T2) → · Thick versus the draft's two spans →

• Spherical Hecke action

  The action uses the contracted product, where the arc group acts on the right on the first factor and on the left on the second.

  $$\operatorname{Gr}_G=LG/L^+G,\qquad\mathcal H=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LG/L^+G),$$ $$m:LG\mathbin{\times^{L^+G}}LX\longrightarrow LX,\qquad[g,x]\longmapsto gx,$$ $$\boxed{\ast:\mathcal H\times\mathcal C\longrightarrow\mathcal C,\qquad\mathcal A\ast\mathcal F=m_!(\mathcal A\mathbin{\widetilde\boxtimes}\mathcal F).}$$

  The twisted external product descends the equivariant external product to the contracted product. Hypothesis H4 requires the result to stay in the constructible category.

• Satake objects

  Geometric Satake identifies representations of the dual group with spherical perverse sheaves. \(V_\lambda\) is the sheaf of the irreducible representation with highest weight \(\lambda\).

  $$\operatorname{Sat}:\operatorname{Rep}_k(\check G)\xrightarrow{\sim}\operatorname{Perv}(L^+G\backslash\operatorname{Gr}_G)\subseteq\mathcal H,\qquad V_\lambda=\operatorname{Sat}(V_\lambda^{\mathrm{rep}}).$$

  The Satake objects are the convolutions \(V_\lambda\ast\mathrm{IC}_0\), \(\lambda\in\Lambda_T^+\). Chen–Yi (2025), §10.2 defines the spherical action in its sheaf theory; we assume its realization in the category specified here.

• Mapping convention

  Every Map denotes the full derived mapping complex, with its category as a subscript.

  $$\Map_{\mathcal D}:\mathcal D^{\mathrm{op}}\times\mathcal D\longrightarrow D(k),$$ $$\Map_{\mathcal D}(A,B)=0\quad\Longleftrightarrow\quad\operatorname{Hom}_{h\mathcal D}(A,B[n])=0\ \text{ for every }n\in\mathbb Z.$$

On this page every hypothesis is read for \(G=H\times H\).

• 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 \(\mathcal C\) is supported on a finite union of orbit closures: $$\forall M\in\mathcal C\ \ \exists J\subseteq\Lambda_S^+\ \text{finite}:\qquad\operatorname{supp}(M)\subseteq\bigcup_{\mu\in J}\overline{\mathcal O_\mu}.$$
• 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.
• 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.
• H4 · Constructible Hecke action.

  The action is defined and takes values in \(\mathcal C\): $$\ast:\mathcal H\times\mathcal C\longrightarrow\mathcal C,\qquad V_\lambda\ast\mathrm{IC}_0\in\mathcal C\quad(\lambda\in\Lambda_T^+).$$
• H5 · Nonzero top splitting.

  Let \(a_\mu:\mathcal O_\mu\hookrightarrow LX\) be the orbit inclusion. For every \(\mu\in\Lambda_S^+\) there is \(\lambda\in\Lambda_T^+\) whose Satake object \(V_\lambda\ast\mathrm{IC}_0\) has two properties:

  1. It lives on the closure of \(\mathcal O_\mu\): $$\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_\mu}.$$
  2. Its restriction to \(\mathcal O_\mu\) is a finite, nonzero sum of shifted constant sheaves: $$a_\mu^*(V_\lambda\ast\mathrm{IC}_0)\simeq\bigoplus_{s=1}^{N_\mu}\underline k_{\mathcal O_\mu}[d_s]\ \text{ in }\mathcal C_\mu,$$ $$0\lt N_\mu\lt\infty,\qquad d_s\in\mathbb Z.$$