§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:40:56Z

Setup 4.1 · Group case: cell pieces, IC heart, two-sided generators $M_{x,y}$

proved

Checks, for X = H under H×H, every structural hypothesis of an Iwahori summand criterion: finitely many pieces in each closure, connected stabilizers, a finite-length heart with simples IC_z, two-sided generators, and the decomposition of IC_x ∗ IC_{w_0} ∗ IC_y. Everything later rests on this list.

Setup 4.1.

Let \(H\) be split connected reductive (over \(\mathbb F_q\), \(p\) odd, for \(k=\overline{\mathbb F}_q\); or over \(\mathbb C\)), \(W\ne1\), \(X=H\) with \((h_1,h_2)\cdot x=h_1xh_2^{-1}\), so \(I=I_H\times I_H\), \(K=K_H\times K_H\). Write \(\mathrm{Fl}=LH/I_H\), \(\mathrm{Fl}_z=I_HzI_H/I_H\cong\mathbb A^{\ell(z)}\), \(\mathrm{IC}_z=j_{z!*}\underline k[\ell(z)]\) (perverse), \(\Delta_z=j_{z!}\underline k\) (unshifted).

(a) Pieces. \(I\backslash LX=I_H\backslash LH/I_H\); the \(I\)-orbits are the double cosets \(I_HzI_H\), \(z\in\widetilde W\); for \(M\gg0\) the pieces of a level are these double cosets, and \(\overline{\mathrm{Fl}_z}=\bigsqcup_{v\le z}\mathrm{Fl}_v\) is a finite union (Bruhat order). Via the right \(I_H\)-torsor \(LH\to\mathrm{Fl}\), \(\mathcal C_I\simeq D^b_{I_H,c}(\mathrm{Fl})_{\rm fs}\), equivariant complexes supported on finitely many cells.

(b) Stabilizers. \(\mathrm{Stab}_{I}(\dot z)\cong I_H\cap\dot zI_H\dot z^{-1}\) is connected; every \(I\)-equivariant local system on a piece is constant.

(c) Heart. On each \(\mathrm{Fl}_{\le u}\) the equivariant perverse heart has finite length, simple objects \(\mathrm{IC}_z\) (\(z\le u\)), pairwise non-isomorphic, \(\operatorname{End}\mathrm{IC}_z=k\); closed pushforward \(\mathrm{Fl}_{\le u}\hookrightarrow\mathrm{Fl}_{\le u'}\) is t-exact and preserves \(\mathrm{IC}_z\); \(\mathcal C_I=\operatorname{Thick}\{\mathrm{IC}_z:z\in\widetilde W\}\).

(d) Generators. \(\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}\), \((\mathcal A\boxtimes\mathcal B)\ast_I\mathcal M\simeq\mathcal A\ast\mathcal M\ast\iota^*\mathcal B\) with \(\iota(g)=g^{-1}\), \(\iota^*\mathrm{IC}_y=\mathrm{IC}_{y^{-1}}\), and \(\mathrm{IC}_0=\underline k_{K_H/I_H}=\mathrm{IC}_{w_0}[-\ell(w_0)]\). Hence $$\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{M_{x,y}:x,y\in\widetilde W\},\qquad M_{x,y}:=\mathrm{IC}_x\ast\mathrm{IC}_{w_0}\ast\mathrm{IC}_y .$$

(e) Decomposition. \(M_{x,y}=m_!(\mathrm{IC}_x\,\widetilde\boxtimes\,\mathrm{IC}_{w_0}\,\widetilde\boxtimes\,\mathrm{IC}_y)\) along the proper map \(m:\overline{\mathrm{Fl}_x}\,\widetilde\times\,\overline{\mathrm{Fl}_{w_0}}\,\widetilde\times\,\overline{\mathrm{Fl}_y}\to\mathrm{Fl}\) of projective varieties. Over \(\overline{\mathbb F}_q\), with the canonical Weil structures (\(\mathrm{IC}_z|_{\mathrm{Fl}_z}=\overline{\mathbb Q}_\ell[\ell(z)]\), Frobenius \(1\)), \(M_{x,y}\) is pure of weight \(w=\ell(x)+\ell(w_0)+\ell(y)\) and geometrically $$M_{x,y}\simeq\bigoplus_n P_n[-n],\qquad P_n={}^pH^n(M_{x,y})\simeq\bigoplus_z\mathrm{IC}_z\otimes E_{z,n},\qquad E_{z,n}=\operatorname{Hom}(\mathrm{IC}_z,P_n),$$ finitely many nonzero. The same splitting holds in the equivariant category (on the finite-type quotient stack).

  1. Orbits. \(I\) acts on \(LX=LH\) by \((i_1,i_2)x=i_1xi_2^{-1}\), so \(I\)-orbits are \(I_H\)-double cosets; \(LH=\bigsqcup_{z\in\widetilde W}I_HzI_H\) (Iwahori–Bruhat; hub page group.html, 2). Quotienting by the right \(I_H\) (a torsor) identifies \(I\)-equivariant sheaves on \(LH\) with \(I_H\)-equivariant sheaves on \(\mathrm{Fl}\); the level presentation of hypothesis.html (“Levels, pieces”) has, for \(M\gg0\), pieces \(=\) double cosets, because each double coset is one orbit at every finite level and fibres of \(\pi_{N,M}\) are right \(I_{M}\)-cosets inside one double coset. Closures: \(\overline{IzI}=\bigsqcup_{v\le z}IvI\), a finite Bruhat interval (case-group.html, 2). So H1 (finitely many pieces in each closure) holds — unlike \(\mathbb A^n\) (checklist 1).
  2. Stabilizers. \((i_1,i_2)\dot z=\dot z\iff i_1=\dot zi_2\dot z^{-1}\); so the stabilizer is \(I_H\cap\dot zI_H\dot z^{-1}=T(\mathcal O)\ltimes(\text{pro-unipotent})\), connected, with connected image at every finite level. Constancy of equivariant local systems on \(I_0\times I_0\)-orbits of \(LG\): Chen–Yi, chen-yi-v4.pdf, Lemma 71(i).
  3. Heart. \(\mathrm{Fl}_{\le u}\) is a projective variety stratified by finitely many affine spaces; perverse sheaves have finite length and the simple ones are \(j_{!*}(\mathcal L[d])\) (bbd-perverse.pdf, Thm 4.3.1); equivariance and step 2 force \(\mathcal L\) constant, giving \(\mathrm{IC}_z\). Closed pushforward is t-exact and commutes with \(j_{!*}\). Every bounded object is an iterated extension of shifts of its perverse cohomologies, each of finite length, so \(\mathcal C_I=\operatorname{Thick}\{\mathrm{IC}_z\}\) (as in summand.tex, Step 6).
  4. Generators. \(\mathcal H_{c,I}\) is the equivariant category of \(\mathrm{Fl}\times\mathrm{Fl}\) with strata \(\mathrm{Fl}_x\times\mathrm{Fl}_y\) (connected stabilizers); recollement on finitely many strata gives \(\operatorname{Thick}\{\Delta_x\boxtimes\Delta_y\}\), and the unitriangular induction of summand.tex Step 5 gives \(\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}\). The formula for \(\ast_I\): push \((\mathcal A\boxtimes\mathcal B)\widetilde\boxtimes\mathcal M\) along \((g_1,g_2,x)\mapsto g_1xg_2^{-1}\) and substitute \(h=g_2^{-1}\). \(G\mapsto G\ast_I\mathrm{IC}_0\) is exact, so \(\operatorname{Thick}\) of the images of a generating set is \(\mathcal C_I^{\mathrm{Iwa}}\) (checklist 2). \(K_H/I_H=\overline{\mathrm{Fl}_{w_0}}\) is the smooth finite flag variety, so \(\mathrm{IC}_0=\mathrm{IC}_{w_0}[-\ell(w_0)]\); shifts do not change thick spans.
  5. Decomposition. The twisted product is an iterated Zariski-locally trivial fibration with projective fibres, and \(m\) is proper. \(\mathrm{IC}_z\) with its canonical Weil structure is pure of weight \(\ell(z)\); twisted external products and proper pushforward preserve purity (Deligne, Weil II; bbd-perverse.pdf §5.1), so \(M_{x,y}\) is pure of weight \(w\). A pure complex splits geometrically as \(\bigoplus{}^pH^n[-n]\) with semisimple \({}^pH^n\) (bbd-perverse.pdf, Thm 5.4.5, Cor 5.4.6). Constituents are \(\mathrm{IC}(\mathrm{Fl}_z,\mathcal L)\); \(\mathcal L\) is a summand of \(\mathcal H^{-\ell(z)}(P_n)|_{\mathrm{Fl}_z}\) (other constituents vanish there in degree \(-\ell(z)\) by the IC support condition), which is an \(I\)-equivariant local system, constant by step 2. Equivariantly: Sun, sun-decomposition.pdf, Thm 1.2, for the proper map of finite-type quotient stacks by \(I/I_n\) (affine stabilizers), exactly as used in summand.tex. Over \(\mathbb C\): the decomposition theorem for the same proper map (decataldo-migliorini-v2.pdf, Thm 1.6.1) — checklist 3.

Builds on

  • —

Built on by

  • Proposition 4.2 · Summand criterion for the group: generator constituents decide $\mathcal C_I^{\rm Iwa}$
  • Lemma 4.3 · Sign character on IC traces: $\varepsilon(\operatorname{tr}\mathrm{IC}_z)=[z\in\Omega]$
  • Lemma 4.4 · Trace expansion of $M_{x,y}$ with pure multiplicity Frobenii
  • Audit 8.1 · Audit of the group set-up (A4-1, A2-8): pieces, stabilizers, generators, levels
Original record, as the agent wrote it at 18:40 UTC

Group case: cell pieces, IC heart, two-sided generators $M_{x,y}$

  1. Orbits. \(I\) acts on \(LX=LH\) by \((i_1,i_2)x=i_1xi_2^{-1}\), so \(I\)-orbits are \(I_H\)-double cosets; \(LH=\bigsqcup_{z\in\widetilde W}I_HzI_H\) (Iwahori–Bruhat; hub page group.html, 2). Quotienting by the right \(I_H\) (a torsor) identifies \(I\)-equivariant sheaves on \(LH\) with \(I_H\)-equivariant sheaves on \(\mathrm{Fl}\); the level presentation of hypothesis.html (“Levels, pieces”) has, for \(M\gg0\), pieces \(=\) double cosets, because each double coset is one orbit at every finite level and fibres of \(\pi_{N,M}\) are right \(I_{M}\)-cosets inside one double coset. Closures: \(\overline{IzI}=\bigsqcup_{v\le z}IvI\), a finite Bruhat interval (case-group.html, 2). So H1 (finitely many pieces in each closure) holds — unlike \(\mathbb A^n\) (checklist 1).
  2. Stabilizers. \((i_1,i_2)\dot z=\dot z\iff i_1=\dot zi_2\dot z^{-1}\); so the stabilizer is \(I_H\cap\dot zI_H\dot z^{-1}=T(\mathcal O)\ltimes(\text{pro-unipotent})\), connected, with connected image at every finite level. Constancy of equivariant local systems on \(I_0\times I_0\)-orbits of \(LG\): Chen–Yi, chen-yi-v4.pdf, Lemma 71(i).
  3. Heart. \(\mathrm{Fl}_{\le u}\) is a projective variety stratified by finitely many affine spaces; perverse sheaves have finite length and the simple ones are \(j_{!*}(\mathcal L[d])\) (bbd-perverse.pdf, Thm 4.3.1); equivariance and step 2 force \(\mathcal L\) constant, giving \(\mathrm{IC}_z\). Closed pushforward is t-exact and commutes with \(j_{!*}\). Every bounded object is an iterated extension of shifts of its perverse cohomologies, each of finite length, so \(\mathcal C_I=\operatorname{Thick}\{\mathrm{IC}_z\}\) (as in summand.tex, Step 6).
  4. Generators. \(\mathcal H_{c,I}\) is the equivariant category of \(\mathrm{Fl}\times\mathrm{Fl}\) with strata \(\mathrm{Fl}_x\times\mathrm{Fl}_y\) (connected stabilizers); recollement on finitely many strata gives \(\operatorname{Thick}\{\Delta_x\boxtimes\Delta_y\}\), and the unitriangular induction of summand.tex Step 5 gives \(\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}\). The formula for \(\ast_I\): push \((\mathcal A\boxtimes\mathcal B)\widetilde\boxtimes\mathcal M\) along \((g_1,g_2,x)\mapsto g_1xg_2^{-1}\) and substitute \(h=g_2^{-1}\). \(G\mapsto G\ast_I\mathrm{IC}_0\) is exact, so \(\operatorname{Thick}\) of the images of a generating set is \(\mathcal C_I^{\mathrm{Iwa}}\) (checklist 2). \(K_H/I_H=\overline{\mathrm{Fl}_{w_0}}\) is the smooth finite flag variety, so \(\mathrm{IC}_0=\mathrm{IC}_{w_0}[-\ell(w_0)]\); shifts do not change thick spans.
  5. Decomposition. The twisted product is an iterated Zariski-locally trivial fibration with projective fibres, and \(m\) is proper. \(\mathrm{IC}_z\) with its canonical Weil structure is pure of weight \(\ell(z)\); twisted external products and proper pushforward preserve purity (Deligne, Weil II; bbd-perverse.pdf §5.1), so \(M_{x,y}\) is pure of weight \(w\). A pure complex splits geometrically as \(\bigoplus{}^pH^n[-n]\) with semisimple \({}^pH^n\) (bbd-perverse.pdf, Thm 5.4.5, Cor 5.4.6). Constituents are \(\mathrm{IC}(\mathrm{Fl}_z,\mathcal L)\); \(\mathcal L\) is a summand of \(\mathcal H^{-\ell(z)}(P_n)|_{\mathrm{Fl}_z}\) (other constituents vanish there in degree \(-\ell(z)\) by the IC support condition), which is an \(I\)-equivariant local system, constant by step 2. Equivariantly: Sun, sun-decomposition.pdf, Thm 1.2, for the proper map of finite-type quotient stacks by \(I/I_n\) (affine stabilizers), exactly as used in summand.tex. Over \(\mathbb C\): the decomposition theorem for the same proper map (decataldo-migliorini-v2.pdf, Thm 1.6.1) — checklist 3.

Sources