§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:46:13Z

Proposition 1.5 · Symmetric rows: levels, heart and decomposition from Chen–Yi; I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant. decided by a torus

proved ★ key step

For the symmetric rows the finiteness, perverse heart, decomposition and local-system inputs of the Iwahori criterion are theorems of Chen–Yi; connected stabilizers reduce to connectedness of \(T_0^{\psi_v}\), which holds for the GL, symplectic and sphere families and stays open for \(E_6/F_4\).

Proposition 1.5.

Let \(X=G/G^\theta\) be a symmetric row of Table 1 other than the group and \(\mathbb G_m\), with \(G^\theta\) connected, over \(\overline{\mathbb F}_q\), \(p\) odd, \(\overline{\mathbb Q}_\ell\)-coefficients; \((B_0,T_0)\) a \(\theta\)-stable Borel pair, \(I_0\) its Iwahori, \(O_v=I_0x_v\), \(v\in\mathcal V\), the \(I_0\)-orbits, \(\psi_v=\operatorname{Ad}(n_v)\circ\theta\) with \(n_v=\tau(\dot v)\), \(\tau(g)=g\theta(g)^{-1}\), so that \(\operatorname{Stab}_{LG}(x_v)=(LG)^{\psi_v}\); \(T_v:=T_0^{\psi_v}\).

(a) (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).) Every \(I_0\)-orbit closure is placid and contains finitely many orbits; hence \(LX\) has a level presentation whose pieces are orbits at large level, finitely many per level. (b) (PH\(_I\)) \(\mathcal C_I\) carries the \(\ast\)-adapted perverse \(t\)-structure normalized by a dimension theory \(\delta\), compatible with the transition functors; its simple objects are the \(\mathrm{IC}_{\xi,v}\), \((\xi,v)\in\mathcal D\), \(\xi\in X(\pi_0T_v)\). (c) (DT\(_I\)) For semisimple \(\mathcal G\in\mathcal H_{c,I}\) and semisimple \(F\in\mathcal C_I\), \(\mathcal G\ast_IF\) is semisimple; \(\mathrm{IC}_{\xi,v}\) is pointwise pure. (d) (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) \(\pi_0\operatorname{Stab}_{I_0}(x_v)\simeq\pi_0(T_v)\), and \(T_v\) connected for all \(v\) implies connected stabilizers at every finite level; if some \(T_v\) is disconnected, \(\mathcal C_I\) contains non-constant \(\mathrm{IC}_{\xi,v}\) and the criterion must count pairs \((\xi,v)\). (e) \(T_v\) is connected for every \(v\) when \(X\) is \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) (including \(\mathrm{GL}_2/T\)), \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), or \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\). For \(E_6/F_4\) it is open. (f) \(L^+X=K\cdot x_0\) is one \(K\)-orbit, closed and pro-smooth, and \(\mathrm{IC}_0\) is its constant sheaf (shifted).

Proof.

  1. (a). chen-yi-v4.pdf, Corollary 27(i) (closures contain finitely many \(I_0\)-orbits) and Theorem 36 (closures are placid, the embeddings are finitely presented); Lemma 23(iii) identifies \(I_0\backslash LX\) with \(L^+T_0\backslash V/LK\). Take \(Y_N\) a finite union of orbit closures; finitely presented embeddings make each orbit the preimage of its image at large \(M\). Chen–Yi assume \(G^\theta\) connected from §4 on (stated at the start of §4): true for every row.
  2. (b). chen-yi-v4.pdf, App. A.7, (45)–(47): smooth \(\ast\)-pullback is \(t\)-exact for the \(\ast\)-adapted structure and \(!\)-pushforward along closed embeddings shifts by \(\delta\); Proposition 41 gives \(\delta\); Lemma 107 gives Verdier self-duality. Finite length holds level by level (finite-type quotients with finitely many orbits). The simple objects are classified by Lemma 71(ii) and Definition 72.
  3. (c). chen-yi-v4.pdf, Proposition 70(ii) (proved with the decomposition theorem on finite-type models of the convolution), Theorem 81 (pointwise purity of \(\mathrm{IC}_{\xi,v}\)).
  4. (d). Lemma 71 and §8.1.3 of chen-yi-v4.pdf: \(\pi_{T_0}(I_0\cap(LG)^{\psi_v})=T_v\) with connected fibres \(I_0^+\cap(LG)^{\psi_v}\), so \(\pi_0\operatorname{Stab}_{I_0}(x_v)\simeq\pi_0T_v\). At a level where the piece through \(x_v\) is the orbit, \(\operatorname{Stab}(\pi x_v)/\operatorname{Stab}(x_v)\) is the fibre of \(\pi\) in \(O_v\), an affine space, hence connected; so \(\pi_0\operatorname{Stab}(x_v)\to\pi_0\operatorname{Stab}(\pi x_v)\) is onto.
  5. A uniform description of \(\psi_v\). \(\operatorname{Ad}(n_v)\circ\theta=\operatorname{Ad}(\dot v)\theta\operatorname{Ad}(\dot v)^{-1}\) is the involution \(\theta_{x_v}\) whose fixed group is the stabilizer of \(x_v\); on \(T_0\) it acts by the image \(\bar n_v\in N_G(T_0)\) composed with \(\theta\) (as \(\operatorname{Ad}(t^\lambda)\) is trivial on \(T_0\)).
  6. \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\). \(\theta=\operatorname{Ad}(s)\), \(s=\mathrm{diag}(1_p,-1_q)\in T_0\); \(\psi_v=\operatorname{Ad}(\dot vs\dot v^{-1})\) normalizes the diagonal torus, so acts on \(T_0=\mathbb G_m^n\) by an involutive permutation \(\pi\). Then \(T_v\simeq\mathbb G_m^{\#\{\text{fixed points}\}}\times\mathbb G_m^{\#\{2\text{-cycles}\}}\), connected.
  7. \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\). \(\theta(g)=J(g^t)^{-1}J^{-1}\) with \(J\) antidiagonal alternating (so the upper-triangular pair is \(\theta\)-stable). \(n_vJ=\dot vJ\dot v^t\) is alternating and monomial, so its permutation \(\pi\) is a fixed-point-free involution (zero diagonal, \((i,\pi i)\ne0\Rightarrow(\pi i,i)\ne0\)). \(\psi_v(t)=(n_vJ)t^{-1}(n_vJ)^{-1}\), i.e. \(t_{\pi(k)}\mapsto t_k^{-1}\); so \(T_v=\{t_{\pi(k)}=t_k^{-1}\}\simeq\mathbb G_m^n\), connected, and \(\det=1\) holds automatically for \(\mathrm{SL}_{2n}\).
  8. \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\). \(X\) is the quadric \(\{Q(x)=1\}\), \(\theta=\operatorname{Ad}(r_{e})\) for a reflection, and \(\theta_{x}=\operatorname{Ad}(r_x)\). If \(r_x\) normalizes the split torus \(T_0\) it permutes the weight lines \(\langle e_i\rangle,\langle f_i\rangle\) of \(V=\bigoplus_i(\langle e_i\rangle\oplus\langle f_i\rangle)\) while fixing a hyperplane, so \(x\in\langle e_i,f_i\rangle\) for one \(i\) and \(\psi_v\) acts on \(X_*(T_0)\) by the single sign change \(\sigma_i:x_i\mapsto-x_i\). With \(L=X_*(T_{\mathrm{Spin}})=\{x\in\mathbb Z^n:\sum x_k\text{ even}\}\): \(\pi_0(T^{\sigma})\simeq H^1(\langle\sigma\rangle,L)=\ker(1+\sigma)/\operatorname{im}(1-\sigma)=2\mathbb Ze_i/2\mathbb Ze_i=0\) (any \(x_i\) occurs in \(L\) by adjusting another coordinate, \(n\ge2\)). So \(T_v\) is connected.
  9. \(E_6/F_4\). Here \(\theta\) acts on \(X_*(T_0)\) by \(-w_0\) and \(\psi_v\) by \(-\bar w_vw_0\); e.g. \(-1\) has \(H^1(\langle-1\rangle,L)=L/2L\ne0\). Whether an involution \(\sigma=-u\) with \(H^1(\langle\sigma\rangle,L)\ne0\) occurs as \(\psi_v\) for an \(I_0\)-orbit requires the image of \(p:\mathcal V\to\mathcal I\) (chen-yi-v4.pdf, Lemmas 25–26). Open.
  10. (f). \(L^+X\to X(\mathcal O/t^m)\) smooth surjective (\(X\) smooth); \(G(\mathcal O)\to X(\mathcal O)\) surjective because \(G\to X\) is a \(G^\theta\)-torsor with \(G^\theta\) smooth connected and the residue field algebraically closed (Hensel).

The formula \(\pi_0(T^\sigma)\simeq H^1(\langle\sigma\rangle,X_*(T))\) for an involution \(\sigma\) of a torus: over \(\mathbb C\), apply \(\sigma\)-invariants to \(0\to X_*(T)\to X_*(T)\otimes\mathbb C\xrightarrow{\exp}T\to0\); since \(H^1(\langle\sigma\rangle,X_*\otimes\mathbb C)=0\), \(T^\sigma\) modulo the connected image of \((X_*\otimes\mathbb C)^\sigma\) is \(H^1(\langle\sigma\rangle,X_*)\), a \(2\)-group. Over \(\overline{\mathbb F}_q\), \(p\) odd, the same holds with \(\exp\) replaced by the prime-to-\(p\) torsion sequence. Check: \(\sigma=-1\) on \(\mathbb Z\) gives \(\mu_2\) and \(\mathbb Z/2\); the swap on \(\mathbb Z^2\) gives the diagonal \(\mathbb G_m\) and \(0\).

Builds on

  • Setup 1.1 · Setting audit: D-modules, Betti and ℓ-adic give the same \(\mathcal C_I\) and \(\ast_I\)

Built on by

  • Lemma 1.6 · Dictionary inputs at Iwahori level: rational orbits, Lang, trace bases, convolution
  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument
  • Summary 1.10 · Verdict of the audit: rows 1–3 agree, the rest reduce to one module computation
Original record, as the agent wrote it at 18:46 UTC

Symmetric rows: levels, heart and decomposition from Chen–Yi; I2 decided by a torus

Proof.

  1. (a). chen-yi-v4.pdf, Corollary 27(i) (closures contain finitely many \(I_0\)-orbits) and Theorem 36 (closures are placid, the embeddings are finitely presented); Lemma 23(iii) identifies \(I_0\backslash LX\) with \(L^+T_0\backslash V/LK\). Take \(Y_N\) a finite union of orbit closures; finitely presented embeddings make each orbit the preimage of its image at large \(M\). Chen–Yi assume \(G^\theta\) connected from §4 on (stated at the start of §4): true for every row.
  2. (b). chen-yi-v4.pdf, App. A.7, (45)–(47): smooth \(\ast\)-pullback is \(t\)-exact for the \(\ast\)-adapted structure and \(!\)-pushforward along closed embeddings shifts by \(\delta\); Proposition 41 gives \(\delta\); Lemma 107 gives Verdier self-duality. Finite length holds level by level (finite-type quotients with finitely many orbits). The simple objects are classified by Lemma 71(ii) and Definition 72.
  3. (c). chen-yi-v4.pdf, Proposition 70(ii) (proved with the decomposition theorem on finite-type models of the convolution), Theorem 81 (pointwise purity of \(\mathrm{IC}_{\xi,v}\)).
  4. (d). Lemma 71 and §8.1.3 of chen-yi-v4.pdf: \(\pi_{T_0}(I_0\cap(LG)^{\psi_v})=T_v\) with connected fibres \(I_0^+\cap(LG)^{\psi_v}\), so \(\pi_0\operatorname{Stab}_{I_0}(x_v)\simeq\pi_0T_v\). At a level where the piece through \(x_v\) is the orbit, \(\operatorname{Stab}(\pi x_v)/\operatorname{Stab}(x_v)\) is the fibre of \(\pi\) in \(O_v\), an affine space, hence connected; so \(\pi_0\operatorname{Stab}(x_v)\to\pi_0\operatorname{Stab}(\pi x_v)\) is onto.
  5. A uniform description of \(\psi_v\). \(\operatorname{Ad}(n_v)\circ\theta=\operatorname{Ad}(\dot v)\theta\operatorname{Ad}(\dot v)^{-1}\) is the involution \(\theta_{x_v}\) whose fixed group is the stabilizer of \(x_v\); on \(T_0\) it acts by the image \(\bar n_v\in N_G(T_0)\) composed with \(\theta\) (as \(\operatorname{Ad}(t^\lambda)\) is trivial on \(T_0\)).
  6. \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\). \(\theta=\operatorname{Ad}(s)\), \(s=\mathrm{diag}(1_p,-1_q)\in T_0\); \(\psi_v=\operatorname{Ad}(\dot vs\dot v^{-1})\) normalizes the diagonal torus, so acts on \(T_0=\mathbb G_m^n\) by an involutive permutation \(\pi\). Then \(T_v\simeq\mathbb G_m^{\#\{\text{fixed points}\}}\times\mathbb G_m^{\#\{2\text{-cycles}\}}\), connected.
  7. \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\). \(\theta(g)=J(g^t)^{-1}J^{-1}\) with \(J\) antidiagonal alternating (so the upper-triangular pair is \(\theta\)-stable). \(n_vJ=\dot vJ\dot v^t\) is alternating and monomial, so its permutation \(\pi\) is a fixed-point-free involution (zero diagonal, \((i,\pi i)\ne0\Rightarrow(\pi i,i)\ne0\)). \(\psi_v(t)=(n_vJ)t^{-1}(n_vJ)^{-1}\), i.e. \(t_{\pi(k)}\mapsto t_k^{-1}\); so \(T_v=\{t_{\pi(k)}=t_k^{-1}\}\simeq\mathbb G_m^n\), connected, and \(\det=1\) holds automatically for \(\mathrm{SL}_{2n}\).
  8. \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\). \(X\) is the quadric \(\{Q(x)=1\}\), \(\theta=\operatorname{Ad}(r_{e})\) for a reflection, and \(\theta_{x}=\operatorname{Ad}(r_x)\). If \(r_x\) normalizes the split torus \(T_0\) it permutes the weight lines \(\langle e_i\rangle,\langle f_i\rangle\) of \(V=\bigoplus_i(\langle e_i\rangle\oplus\langle f_i\rangle)\) while fixing a hyperplane, so \(x\in\langle e_i,f_i\rangle\) for one \(i\) and \(\psi_v\) acts on \(X_*(T_0)\) by the single sign change \(\sigma_i:x_i\mapsto-x_i\). With \(L=X_*(T_{\mathrm{Spin}})=\{x\in\mathbb Z^n:\sum x_k\text{ even}\}\): \(\pi_0(T^{\sigma})\simeq H^1(\langle\sigma\rangle,L)=\ker(1+\sigma)/\operatorname{im}(1-\sigma)=2\mathbb Ze_i/2\mathbb Ze_i=0\) (any \(x_i\) occurs in \(L\) by adjusting another coordinate, \(n\ge2\)). So \(T_v\) is connected.
  9. \(E_6/F_4\). Here \(\theta\) acts on \(X_*(T_0)\) by \(-w_0\) and \(\psi_v\) by \(-\bar w_vw_0\); e.g. \(-1\) has \(H^1(\langle-1\rangle,L)=L/2L\ne0\). Whether an involution \(\sigma=-u\) with \(H^1(\langle\sigma\rangle,L)\ne0\) occurs as \(\psi_v\) for an \(I_0\)-orbit requires the image of \(p:\mathcal V\to\mathcal I\) (chen-yi-v4.pdf, Lemmas 25–26). Open.
  10. (f). \(L^+X\to X(\mathcal O/t^m)\) smooth surjective (\(X\) smooth); \(G(\mathcal O)\to X(\mathcal O)\) surjective because \(G\to X\) is a \(G^\theta\)-torsor with \(G^\theta\) smooth connected and the residue field algebraically closed (Hensel).

The formula \(\pi_0(T^\sigma)\simeq H^1(\langle\sigma\rangle,X_*(T))\) for an involution \(\sigma\) of a torus: over \(\mathbb C\), apply \(\sigma\)-invariants to \(0\to X_*(T)\to X_*(T)\otimes\mathbb C\xrightarrow{\exp}T\to0\); since \(H^1(\langle\sigma\rangle,X_*\otimes\mathbb C)=0\), \(T^\sigma\) modulo the connected image of \((X_*\otimes\mathbb C)^\sigma\) is \(H^1(\langle\sigma\rangle,X_*)\), a \(2\)-group. Over \(\overline{\mathbb F}_q\), \(p\) odd, the same holds with \(\exp\) replaced by the prime-to-\(p\) torsion sequence. Check: \(\sigma=-1\) on \(\mathbb Z\) gives \(\mu_2\) and \(\mathbb Z/2\); the swap on \(\mathbb Z^2\) gives the diagonal \(\mathbb G_m\) and \(0\).

Sources