Standard generation · statement and proof

This page proves the input to step 1 of Proposition I. The setting defines every symbol the proposition uses.

Setting · the space, the category \(\mathcal C\), orbits, and standards
  • Space and group. Work over \(k=\mathbb C\) with coefficients in \(k\). Let \(G\) be a connected reductive group with an involution \(\theta\), and let the arc group act on the loop space of the symmetric space \(X=G/G^\theta\): $$L^+G\ \curvearrowright\ LX.$$
  • Orbits and their order. By Chen–Yi Proposition 17, the \(L^+G\)-orbits on \(LX\) are indexed by the dominant coweights of a maximal \(\theta\)-split torus \(S\subseteq T\). Order them by the dominance order restricted from \(\Lambda_T\): $$\mathcal O_\mu=L^+G\cdot x_\mu\quad(\mu\in\Lambda_S^+),\qquad\nu\le\mu\iff\mu-\nu\in\textstyle\sum_{\check\alpha\in\check\Phi^+}\mathbb Z_{\ge0}\,\check\alpha.$$
  • Closed finite unions. A closed finite union of orbits is a closed \(L^+G\)-stable subset \(Z\subseteq LX\) that is the union of finitely many orbits. We record its orbits and their number: $$I(Z)=\{\mu\in\Lambda_S^+:\mathcal O_\mu\subseteq Z\},\qquad n(Z)=\#I(Z).$$
  • 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 \(k\). The category \(\mathcal C\) of Proposition I is the one on all of \(LX\): $$\mathcal C=\mathcal C_{LX}=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX),\qquad\mathcal C_Z=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash Z),\qquad\mathcal C_\mu=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash\mathcal O_\mu).$$
  • Embeddings and standards. An orbit in a closed finite union \(Z\) embeds locally closed in \(Z\), and \(Z\) embeds closed in \(LX\). The standard object of an orbit is its constant sheaf extended by zero, to \(Z\) or to all of \(LX\): $$a_{\mu,Z}:\mathcal O_\mu\hookrightarrow Z,\qquad r_Z:Z\hookrightarrow LX,\qquad a_\mu=r_Z\circ a_{\mu,Z}:\mathcal O_\mu\hookrightarrow LX,$$ $$\Delta_{\mu,Z}=a_{\mu,Z!}\,\underline k_{\mathcal O_\mu}\in\mathcal C_Z,\qquad\Delta_\mu=a_{\mu!}\,\underline k_{\mathcal O_\mu}\in\mathcal C.$$
  • Support. An object \(M\in\mathcal C\) is supported on a closed finite union of orbits \(Z\) when the unit of the adjunction \(r_Z^*\dashv r_{Z*}\) is an isomorphism on \(M\): $$\operatorname{supp}(M)\subseteq Z\quad:\Longleftrightarrow\quad M\xrightarrow{\ \sim\ }r_{Z*}r_Z^*M.$$
  • 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 of \(\mathcal D\) that contains \(0\) and \(\mathcal S\) and is closed under isomorphisms, shifts, cofibers, and retracts. We use two of its closure properties: $$\text{(T1)}\quad A\to B\to C\xrightarrow{+1}\ \text{exact},\quad A,C\in\operatorname{Thick}_{\mathcal D}(\mathcal S)\ \Longrightarrow\ B\in\operatorname{Thick}_{\mathcal D}(\mathcal S),$$ $$\text{(T2)}\quad F:\mathcal D\to\mathcal D'\ \text{exact}\ \Longrightarrow\ F\big(\operatorname{Thick}_{\mathcal D}(\mathcal S)\big)\subseteq\operatorname{Thick}_{\mathcal D'}\big(F(\mathcal S)\big).$$
    Why (T1) and (T2) hold
    • (T1) Rotating the triangle presents \(B\) as the cofiber of a map between objects of the span, and the span is closed under shifts and cofibers: $$C[-1]\longrightarrow A\longrightarrow B\xrightarrow{+1},\qquad B\simeq\operatorname{cofib}(C[-1]\to A).$$ The zero map gives finite direct sums, \(A\oplus C\simeq\operatorname{cofib}(0:C[-1]\to A)\).
    • (T2) The objects that \(F\) sends into \(\operatorname{Thick}_{\mathcal D'}(F(\mathcal S))\) form a full subcategory containing \(0\) and \(\mathcal S\). It is closed under shifts and cofibers because \(F\) is exact, and under retracts because every functor preserves retract diagrams, so it contains \(\operatorname{Thick}_{\mathcal D}(\mathcal S)\).

Hypotheses

  • 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.

Proposition · Constant standards generate \(\mathcal C\)

Let \(L^+G\) act on \(LX\) and let \(\mathcal C=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX)\), with orbits \(\mathcal O_\mu\), closed finite unions of orbits \(Z\), and standards \(\Delta_{\mu,Z}\), \(\Delta_\mu\) as in the setting. Assume H1–H3.

(a) For every closed finite union of orbits \(Z\), the standards of its orbits generate \(\mathcal C_Z\):

$$\boxed{\mathcal C_Z=\operatorname{Thick}_{\mathcal C_Z}\{\Delta_{\mu,Z}:\mu\in I(Z)\}.}$$

(b) The standards of all orbits generate \(\mathcal C\):

$$\boxed{\mathcal C=\operatorname{Thick}_{\mathcal C}\{\Delta_\mu:\mu\in\Lambda_S^+\}.}$$

Proposition I uses (b) in step 1, and (a) for the boundary of an orbit closure in step 3. Thick generation implies generation in the sense of draft Definition 3.32, so (b) proves draft Proposition 3.37 for symmetric \(X\) and \(L^+G\) under H1–H3.

Without H1(a), (b) fails: for \(\mathbb A^n\) the orbit standards span only the objects with zero stalk at \(0\). The standards of the pieces of each level, e.g. \(\{\Delta_m\}\cup\{\mathrm{IC}_0\}\), still generate (Proposition IV, section 4; reader question).[T]

Proof · three steps

  1. Under H3, the constant sheaf generates each orbit category:

    $$\mathcal C_\mu=\operatorname{Thick}_{\mathcal C_\mu}(\underline k_{\mathcal O_\mu}).$$
    Proof
    • Fix \(M\in\mathcal C_\mu\) and write \(\mathcal T\) for the thick span of the constant sheaf. Because the t-structure is bounded, there are integers \(r\le s\) with $$\mathcal H^q(M)=0\ \ (q\notin[r,s]),\qquad\tau_{\le r-1}M=0,\qquad\tau_{\le s}M=M.$$
    • Each degree contributes one truncation triangle, whose third term is a shifted object of the heart: $$\tau_{\le q-1}M\longrightarrow\tau_{\le q}M\longrightarrow\mathcal H^q(M)[-q]\xrightarrow{+1}.$$
    • H3 gives \(\mathcal H^q(M)\simeq\underline k_{\mathcal O_\mu}^{\oplus n_q}\) with \(n_q\ge0\), an object of \(\mathcal T\). Starting from \(\tau_{\le r-1}M=0\), (T1) puts each \(\tau_{\le q}M\) in \(\mathcal T\), and \(q=s\) gives \(M\in\mathcal T\).
    • The triangles need not split, since \(\operatorname{Ext}^n_{\mathcal C_\mu}(\underline k_{\mathcal O_\mu},\underline k_{\mathcal O_\mu})=H^n(B\operatorname{Stab}_{L^+G}(x_\mu);k)\) can be nonzero for \(n\ge2\). The argument uses only the triangles, never a splitting of \(M\).

  2. Induction on the number of orbits proves part (a):

    $$\mathcal C_Z=\operatorname{Thick}_{\mathcal C_Z}\{\Delta_{\nu,Z}:\nu\in I(Z)\}.$$
    Proof
    • Base. If \(n(Z)=0\), then \(Z=\varnothing\) and \(\mathcal C_Z=0\), which is the thick span of the empty set.
    • Choose a maximal orbit. Let \(n(Z)\ge1\) and choose \(\mu\) maximal in the finite set \(I(Z)\). Since \(Z\) is closed, it contains the closure of each of its orbits: $$Z=\bigcup_{\nu\in I(Z)}\overline{\mathcal O_\nu}.$$
    • Its complement is closed. By H1(a), \(\overline{\mathcal O_\nu}\) meets \(\mathcal O_\mu\) only if \(\mu\le\nu\), and maximality allows this only for \(\nu=\mu\). The rest of \(\overline{\mathcal O_\mu}\) consists of orbits \(\mathcal O_{\nu'}\) with \(\nu'\lt\mu\), each contained in its own closure with \(\nu'\in I(Z)\setminus\{\mu\}\): $$D:=Z\setminus\mathcal O_\mu=\bigcup_{\nu\in I(Z)\setminus\{\mu\}}\overline{\mathcal O_\nu},\qquad I(D)=I(Z)\setminus\{\mu\},\qquad n(D)=n(Z)-1.$$ Thus \(D\) is a closed finite union of orbits and \(j=a_{\mu,Z}\) is an open embedding. For \(Z=\overline{\mathcal O_\mu}\), the complement is the boundary \(\overline{\mathcal O_\mu}\setminus\mathcal O_\mu\), whose orbits are those with \(\nu\lt\mu\).
    • Localization. For \(M\in\mathcal C_Z\), H2(b) gives $$j_!j^*M\longrightarrow M\longrightarrow i_*i^*M\xrightarrow{+1}.$$
    • Open term. Step 1 places \(j^*M\) in the thick span of \(\underline k_{\mathcal O_\mu}\). The functor \(j_!\) is exact and \(j_!\underline k_{\mathcal O_\mu}=\Delta_{\mu,Z}\), so (T2) gives $$j_!j^*M\in\operatorname{Thick}_{\mathcal C_Z}(\Delta_{\mu,Z}).$$
    • Closed term. The induction hypothesis for \(D\) places \(i^*M\) in the thick span of \(\{\Delta_{\nu,D}:\nu\in I(D)\}\). The functor \(i_*\) is exact and \(i_*\Delta_{\nu,D}\simeq\Delta_{\nu,Z}\) by H2(c), so (T2) gives $$i_*i^*M\in\operatorname{Thick}_{\mathcal C_Z}\{\Delta_{\nu,Z}:\nu\in I(D)\}.$$
    • Conclusion. Both outer terms lie in the thick span of \(\{\Delta_{\nu,Z}:\nu\in I(Z)\}\), so (T1) puts \(M\) there too. The reverse inclusion holds because each \(\Delta_{\nu,Z}\) is an object of \(\mathcal C_Z\).

  3. Finite support transfers part (a) to all of \(\mathcal C\), which is part (b).

    Proof
    • Let \(M\in\mathcal C\). H1(b) gives a finite \(J\) such that \(M\) is supported on the union \(Z\) of the closures \(\overline{\mathcal O_\mu}\), \(\mu\in J\). By H1(a), \(Z\) is a closed finite union of orbits: $$Z=\bigcup_{\mu\in J}\overline{\mathcal O_\mu}=\bigcup_{\mu\in J}\ \bigsqcup_{\nu\le\mu}\mathcal O_\nu.$$
    • The support condition gives \(M\simeq r_{Z*}r_Z^*M\). Part (a) for \(Z\) places \(r_Z^*M\) in the thick span of \(\{\Delta_{\nu,Z}:\nu\in I(Z)\}\).
    • The functor \(r_{Z*}\) is exact and \(r_{Z*}\Delta_{\nu,Z}\simeq\Delta_\nu\) by H2(c). Hence (T2) gives $$M\simeq r_{Z*}r_Z^*M\in\operatorname{Thick}_{\mathcal C}\{\Delta_\nu:\nu\in I(Z)\}\subseteq\operatorname{Thick}_{\mathcal C}\{\Delta_\nu:\nu\in\Lambda_S^+\}.$$ The reverse inclusion is immediate, which proves (b).

Proposition II checks H1–H3 in the group case, and Proposition III decides them for every symmetric space.