I · Conditional generation
Proved under H1–H5Under five orbit and convolution hypotheses H1–H5, the goal holds:
$$\mathcal C^{\mathrm{Sat}}=\mathcal C.$$The Satake objects also detect zero in the ind-completion.
Hypotheses, statement and proof →1 · The goal
Let \(G\) be a connected complex reductive group with an involution \(\theta\), and let the arc group \(L^+G\) act on the loop space \(LX\) of \(X=G/G^\theta\). The project draft studies the equivariant constructible category of \(LX\):
$$\mathcal C:=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX).$$\(\mathcal C\) is the small, idempotent-complete, stable \(\infty\)-category of bounded \(L^+G\)-equivariant constructible complexes on \(LX\) with coefficients in \(k=\mathbb C\). The goal is that the Satake objects thickly generate it:
$$\mathcal C^{\mathrm{Sat}}:=\operatorname{Thick}_{\mathcal C}\{V_\lambda\ast\mathrm{IC}_0:\lambda\in\Lambda_T^+\},\qquad\boxed{\mathcal C^{\mathrm{Sat}}=\mathcal C.}$$Proposition I deduces this from five hypotheses, H1–H5. Proposition II verifies them for the group symmetric space, and Proposition III reduces them, for every symmetric space, to two conditions. Proposition IV leaves symmetric spaces: for \(\mathbb A^n\) under \(\mathrm{GL}_n\), H1 fails, and a level-wise form of H1–H5 recovers the argument with the central Satake objects alone.
2 · Definitions
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 \(S\) \(X\) \(T\) \(S\) \(\operatorname{rank}X\) \((H\times H)/H\) \(T_H\times T_H\) \(\{(s,s^{-1}):s\in T_H\}\) \(\operatorname{rank}H\) \(\mathrm{GL}_2/\mathrm{Sp}_2\) \(\{\operatorname{diag}(t_1,t_2)\}\) \(\{\operatorname{diag}(a,a)\}\) \(1\) 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
- 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 \(\lambda\mapsto\lambda-\theta(\lambda)\): $$T\longrightarrow S,\quad t\longmapsto t\theta(t)^{-1},\qquad\lambda-\theta(\lambda)\in\Lambda_S.$$
- 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: $$\theta\alpha=\alpha\quad(\alpha\in\Phi_L),\qquad-\theta\alpha\in\Phi^+\quad(\alpha\in\Phi^+\setminus\Phi_L).$$
- 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: $$\boxed{\varphi:\Lambda_T^+\longrightarrow\Lambda_S^+,\quad\lambda\longmapsto\lambda-\theta(\lambda).}$$
- 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.
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.1 Two of its properties are used:
$$\text{(T1)}\quad A\to B\to C\xrightarrow{+1}\ \text{exact},\ \ A,C\in\operatorname{Thick}_{\mathcal D}(\mathcal S)\ \Longrightarrow\ B\in\operatorname{Thick}_{\mathcal D}(\mathcal S),$$ $$\text{(T2)}\quad F:\mathcal D\to\mathcal D'\ \text{exact}\ \Longrightarrow\ F\big(\operatorname{Thick}_{\mathcal D}(\mathcal S)\big)\subseteq\operatorname{Thick}_{\mathcal D'}\big(F(\mathcal S)\big).$$Thick versus the draft's two spans → The draft's span after Gaitsgory–Rozenblyum is undefined on a small category, and its double orthogonal can be strictly larger: skyscrapers on \(\mathbb A^1\) detect zero without thickly generating.
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.
$$\Map_{\mathcal D}:\mathcal D^{\mathrm{op}}\times\mathcal D\longrightarrow D(k),$$ $$\Map_{\mathcal D}(A,B)=0\quad\Longleftrightarrow\quad\operatorname{Hom}_{h\mathcal D}(A,B[n])=0\ \text{ for every }n\in\mathbb Z.$$
3 · The four propositions
II · Group-case verification
ProvedFor a connected complex reductive group \(H\), the group symmetric space satisfies H1–H5:
$$X=(H\times H)/\operatorname{diag}(H)\simeq H.$$Every hypothesis reduces to the affine Grassmannian of \(H\).
Verification and its proof graph →III · Symmetric-space criterion
Proved up to the transfer to ℂLet \(X=G/G^\theta\) with \(G^\theta\) connected. H1, H2 and H4 always hold, and the other two hypotheses reduce to two conditions:
$$\mathrm{H1}\wedge\cdots\wedge\mathrm{H5}\iff Z_{G^\theta}(S)\ \text{is connected and}\ \varphi(\Lambda_T^+)=\Lambda_S^+.$$Both hold for \((H\times H)/H\), \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\) and \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\). The second fails for \(E_6/F_4\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\), although Chen–Yi's semisimplicity criterion applies to them.
Statement, proof and proof graph →IV · Affine space
Proved, given the level presentationFor \(X=\mathbb A^n\) under \(\mathrm{GL}_n\), H1 fails: every orbit closure \(t^m\mathcal O^n\) contains infinitely many orbits and the point \(0\). Read on the pieces of a finite level \(t^{-N}\mathcal O^n/t^M\mathcal O^n\), H1–H5 hold again, and the proof of Proposition I gives
$$\mathcal C=\operatorname{Thick}\{V_{m\cdot\mathbf 1}\ast\mathrm{IC}_0=\underline k_{t^m\mathcal O^n}:m\in\mathbb Z\}=\mathcal C^{\mathrm{Sat}}.$$The central Satake objects alone generate; the orbit \(\{0\}\) has no standard and needs none.
What breaks, the level form, and the proof →