§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:45:23Z

Lemma 2.5 · Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$

proved

Reduces the decomposition hypothesis DT_I to the existence of a finite-type proper model of the convolution; over F_q-bar it then follows from Gabber purity, Deligne's weight bounds and Sun's equivariant decomposition theorem for stacks; over C it is the cited equivariant decomposition theorem.

Lemma 2.5.

(FM$w$) finite-type model. There are a level $(N,M)$ with $\overline{IwI}\cdot L^+X\subseteq Y_N$, a variety $Z$ of finite type with an action of $H''=I/I''$ ($I''\subseteq I{N,M}$ normal, $I_{N,M}/I''$ unipotent), an $H''$-equivariant proper morphism $p\colon Z\to Y_{N,M}$ and a morphism $r\colon Z\to\mathrm{Fl}_w:=\overline{IwI}/I$, all defined over $\mathbb F_q$ when $k=\overline{\mathbb F}_q$, such that the convolution space is the base change $$\overline{IwI}\times^IL^+X\ \simeq\ Z\times_{Y_{N,M}}Y_N\quad\text{over }Y_N,\qquad \mathrm{IC}_w\tilde\boxtimes k_{L^+X}\ \leftrightarrow\ r^*\mathrm{IC}_w .$$

Lemma. Assume (L1)–(L3) and (FM$w$). Then $G_w\simeq\pi{N,M}^\,p_!\,r^\mathrm{IC}_w$ (extended by zero), so $G_w\in\mathcal C_I^{(N,M)}$. Over $\overline{\mathbb F}_q$, with the Weil structure induced from $\mathrm{IC}_w$, $G_w$ is pure and $$G_w\simeq\bigoplus_i{}^pH^i(G_w)[-i]\ \text{in }\mathcal C_I,\qquad {}^pH^i(G_w)\ \text{semisimple},$$ i.e. (DT$_I$) holds for $w$. Over $\mathbb C$, (DT$_I$) for $w$ holds by the cited equivariant decomposition theorem (Step 5).

  1. Base change. The action map $m$ factors through the closed embedding $[g,y]\mapsto(gI,gy)$ of $\overline{IwI}\times^IL^+X$ into $\mathrm{Fl}w\times Y_N$, followed by the projection, which is proper since $\mathrm{Fl}_w$ is projective (checklist item 3: properness on bounded supports). By (FM$_w$) this is the base change of $p$ along $\pi{N,M}$, so proper base change gives $m_!(\mathrm{IC}w\tilde\boxtimes k)\simeq\pi{N,M}^p_!r^\mathrm{IC}_w$.
  2. Purity of the input. $\mathrm{IC}w=j{!}(\overline{\mathbb Q}\ell[\ell(w)])$ is pure of weight $\ell(w)$ (Gabber; BBD 5.3.2, quoted as such in sun-decomposition.pdf, Lemma 3.3). In the models used here (Proposition 2.6, Proposition 2.8) $r$ is smooth (a vector bundle, resp. an iterated Schubert fibration), so $r^\mathrm{IC}_w$ is pure: $r^$ respects $D{\le w}$ and $r^!=r^[2d](d)$ respects $D_{\ge w}$ (BBD 5.1.14; Sun, Prop. 3.9(ii)).
  3. Purity of the output. $p$ is proper, so $p_!=p_*$ preserves both weight bounds (BBD 5.1.14; for the quotient stacks, Sun, Prop. 3.9(iii), which needs affine stabilizers, true for a quotient by a linear algebraic group).
  4. Equivariant splitting. Apply Sun, Thm. 1.2 (sun-decomposition.pdf) to the proper representable (hence finite-diagonal) morphism $[H''\backslash Z]\to[H''\backslash Y_{N,M}]$ of $\mathbb F_q$-stacks of finite type with affine stabilizers and the pure complex $r^*\mathrm{IC}w$: $pr^\mathrm{IC}w\simeq\bigoplus_i({}^pR^ip)[-i]$ geometrically, each ${}^pR^ip_$ semisimple. Inflation along the unipotent kernel $I_{N,M}/I''$ and $\pi_{N,M}^*$ are fully faithful and t-exact up to shift (Setup 2.1 (L3), Lemma 2.2(b)), so the splitting and semisimplicity hold in $\mathcal C_I$; a semisimple object of a finite-length heart is a finite sum of simple objects. This is (DT$_I$).
  5. Over $\mathbb C$. Replace Step 4 by the equivariant decomposition theorem for proper maps of quotient stacks (Bernstein–Lunts, LNM 1578, §5.3, via finite-dimensional approximations; stated as Sun, Thm. 3.15, without proof in sun-decomposition.pdf); its non-equivariant form is de Cataldo–Migliorini, Thm. 1.6.1. Status over $\mathbb C$: cited, not re-proved here.

Builds on

  • Setup 2.1 · Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.2 · Lemma (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$

Built on by

  • Proposition 2.6 · $\mathbb A^n$: all hypotheses hold; simple objects are lattice sheaves $k_\Lambda$
  • Proposition 2.8 · Group: all hypotheses hold ($\Omega$-truncated levels); $\mathcal S_I=\{\mathrm{IC}_v\}$
  • Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction
  • Audit 7.4 · Audit of A2-5 and A5-4: the finite-type proper model gives (DT$_I$); version 2 of (I$_I$) needs $D^b$-full faithfulness in (L)
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 18:45 UTC

Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$

  1. Base change. The action map $m$ factors through the closed embedding $[g,y]\mapsto(gI,gy)$ of $\overline{IwI}\times^IL^+X$ into $\mathrm{Fl}w\times Y_N$, followed by the projection, which is proper since $\mathrm{Fl}_w$ is projective (checklist item 3: properness on bounded supports). By (FM$_w$) this is the base change of $p$ along $\pi{N,M}$, so proper base change gives $m_!(\mathrm{IC}w\tilde\boxtimes k)\simeq\pi{N,M}^p_!r^\mathrm{IC}_w$.
  2. Purity of the input. $\mathrm{IC}w=j{!}(\overline{\mathbb Q}\ell[\ell(w)])$ is pure of weight $\ell(w)$ (Gabber; BBD 5.3.2, quoted as such in sun-decomposition.pdf, Lemma 3.3). In the models used here (Proposition 2.6, Proposition 2.8) $r$ is smooth (a vector bundle, resp. an iterated Schubert fibration), so $r^\mathrm{IC}_w$ is pure: $r^$ respects $D{\le w}$ and $r^!=r^[2d](d)$ respects $D_{\ge w}$ (BBD 5.1.14; Sun, Prop. 3.9(ii)).
  3. Purity of the output. $p$ is proper, so $p_!=p_*$ preserves both weight bounds (BBD 5.1.14; for the quotient stacks, Sun, Prop. 3.9(iii), which needs affine stabilizers, true for a quotient by a linear algebraic group).
  4. Equivariant splitting. Apply Sun, Thm. 1.2 (sun-decomposition.pdf) to the proper representable (hence finite-diagonal) morphism $[H''\backslash Z]\to[H''\backslash Y_{N,M}]$ of $\mathbb F_q$-stacks of finite type with affine stabilizers and the pure complex $r^*\mathrm{IC}w$: $pr^\mathrm{IC}w\simeq\bigoplus_i({}^pR^ip)[-i]$ geometrically, each ${}^pR^ip_$ semisimple. Inflation along the unipotent kernel $I_{N,M}/I''$ and $\pi_{N,M}^*$ are fully faithful and t-exact up to shift (Setup 2.1 (L3), Lemma 2.2(b)), so the splitting and semisimplicity hold in $\mathcal C_I$; a semisimple object of a finite-length heart is a finite sum of simple objects. This is (DT$_I$).
  5. Over $\mathbb C$. Replace Step 4 by the equivariant decomposition theorem for proper maps of quotient stacks (Bernstein–Lunts, LNM 1578, §5.3, via finite-dimensional approximations; stated as Sun, Thm. 3.15, without proof in sun-decomposition.pdf); its non-equivariant form is de Cataldo–Migliorini, Thm. 1.6.1. Status over $\mathbb C$: cited, not re-proved here.

Sources