§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:40:44Z

Lemma 3.4 · Unitriangular IC-trace basis of $C_I$ and the decomposition matrix $m_{S,w}$

proved

Traces of the simple objects form a basis of $C_I\otimes\overline{\mathbb Q}\ell$ (unitriangular at a common level), the trace of each generator expands in it with coefficients $m{S,w}$ read off the Frobenius on multiplicity spaces, and (IG) is equivalent to the columns of $m$ spanning.

Lemma 3.4.

Assume Setup 3.1 ((L1), (L2), (D1)–(D3)).

(i) For a simple $S=\mathrm{IC}(P)$, $P\in\mathcal P_{N,M}$: $\mathrm{tr}\,S=(-1)^{\delta(P)}\mathbf 1_{P(\mathbb F_q)}+\sum_{P'<P}c_{P'P}\mathbf 1_{P'(\mathbb F_q)}$ with $c_{P'P}=\mathrm{tr}(S)(x_{P'})$. For each level, $\{\mathrm{tr}\,\mathrm{IC}(P):P\in\mathcal P_{N,M}\}$ is a basis of $C_I^{(N,M)}\otimes\overline{\mathbb Q}\ell$; hence $\{\mathrm{tr}\,S\}{S\in\mathrm{Irr}\,\mathcal P_I}$ is a basis of $C_I\otimes\overline{\mathbb Q}_\ell$.

(ii) Put $E_{S,i}(w):=\mathrm{Hom}(S,{}^p\mathcal H^i(\mathrm{IC}w*_I\mathrm{IC}_0))$ with Frobenius $A{S,i}(w)\colon f\mapsto\Phi_i\circ\mathrm{Fr}^*f\circ\Phi_S^{-1}$. Then $$\mathrm{tr}(\mathrm{IC}_w*_I\mathrm{IC}_0)=\sum_Sm_{S,w}\,\mathrm{tr}\,S,\qquad m_{S,w}=\sum_i(-1)^i\,\mathrm{tr}\,A_{S,i}(w),$$ a finite sum, and $m_{S,w}=0$ whenever $S$ is not a shifted direct summand of $\mathrm{IC}_w*_I\mathrm{IC}_0$.

(iii) (IG) over $F=\mathbb F_q(\!(t)\!)$ holds iff the columns $(m_{S,w})S$, $w\in\widetilde W$, span $\bigoplus{S\in\mathrm{Irr}\,\mathcal P_I}\overline{\mathbb Q}_\ell$.

  1. (i), one level. $\mathrm{tr}\,S$ lies in $C_I^{(N,M)}\otimes\overline{\mathbb Q}\ell$: it is supported on $\overline P(\mathbb F_q)\subseteq Y_N(\mathbb F_q)$, constant on $\pi{N,M}$-fibres ($S$ and $\Phi_S$ are pulled back), and $I(\mathcal O)$-invariant ($\Phi_S$ is equivariant, Setup 3.1(iii)). By Lemma 3.2(c) it is constant on each $P'(\mathbb F_q)$, and $\overline P=\pi_{N,M}^{-1}(\bar o)$ is a union of pieces $P'\le P$; on $P$ the value is $(-1)^{\delta(P)}$. So the matrix from $\{\mathrm{tr}\,\mathrm{IC}(P)\}$ to the basis $\{\mathbf 1_{P(\mathbb F_q)}\}$ of Lemma 3.2(c) is triangular for the closure order with diagonal $\pm1$, hence invertible (every piece has rational points, Lemma 3.2(c), and is the support of exactly one simple object by (L2)).
  2. (i), all levels. Spanning: $C_I=\bigcup C_I^{(N,M)}$ (Lemma 3.2(a)). Independence: finitely many simple objects lie at a common level (directedness and Setup 3.1(ii)), where step 1 applies; distinct simple objects there are $\mathrm{IC}$'s of distinct pieces.
  3. (ii), Frobenius on isotypic parts. By (D3), over $k$, $\mathrm{IC}w*_I\mathrm{IC}_0\simeq\bigoplus_iP_i[-i]$, $P_i={}^p\mathcal H^i$, each $P_i$ semisimple, and $P_i\simeq\bigoplus_SS\otimes E{S,i}(w)$ via evaluation (End $S=\overline{\mathbb Q}\ell$, no maps between distinct simples). $\mathrm{Fr}^*$ is t-exact, so $P_i$ inherits $\Phi_i$; under evaluation $\Phi_i=\bigoplus_S\Phi_S\otimes A{S,i}(w)$ (trace.tex, Lemma (Frobenius on the summands), verbatim). On stalks $\mathrm{tr}(\Phi_S\otimes A)=\mathrm{tr}\,\Phi_S\cdot\mathrm{tr}\,A$, so $\mathrm{tr}\,P_i=\sum_S\mathrm{tr}A_{S,i}(w)\,\mathrm{tr}\,S$.
  4. (ii), signs. $\mathrm{tr}$ is additive on the $\mathrm{Fr}$-compatible perverse truncation triangles and $\mathrm{tr}(P[-i])=(-1)^i\mathrm{tr}P$ [Laumon (1.1.1.1)]. If $S$ is not a shifted summand, every $E_{S,i}(w)=0$, so $m_{S,w}=0$.
  5. (ii), without (L2). If some constituent $S$ had $\mathrm{Fr}^*S\not\simeq S$, $\Phi_i$ would map the isotypic part of $\mathrm{Fr}^*S$ onto that of $S$, cyclically along the $\mathrm{Fr}$-orbit of $S$; a block-permutation matrix without fixed blocks has trace $0$ at every rational stalk, so such constituents contribute nothing to $\mathrm{tr}$. Under (L2) this case does not occur.
  6. (iii). The coordinate map $\kappa\colon C_I\otimes\overline{\mathbb Q}\ell\to\bigoplus_S\overline{\mathbb Q}\ell$ in the basis (i) is an isomorphism sending $\mathrm{tr}(\mathrm{IC}w*_I\mathrm{IC}_0)$ to column $w$ (by (ii)); with Lemma 3.3(c), $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}\ell=C_I\otimes\overline{\mathbb Q}\ell$ iff the columns span (trace.tex, Lemma (Spans in coordinates)). Finally $\mathcal H_I*\varphi_0$ is spanned by $\mathbb Z$-valued functions and $C_I$ has the $\mathbb Z$-basis of piece indicators (Lemma 3.2(c)); expressing one indicator through finitely many $T_w*\varphi_0$ is an integral linear system, solvable over a field of characteristic $0$ iff two ranks agree, independently of the field. So (IG) over $\mathbb C$ is equivalent to its version over $\overline{\mathbb Q}\ell$ (trace.tex, Lemma (Columns), verbatim). $\square$

Not claimed. The converse of the last clause of (ii) fails in general: eigenvalues $\alpha,-\alpha$ on one $E_{S,i}(w)$, or equal eigenvalues in degrees of opposite parity, cancel. It is never used in Theorem 3.6.

Builds on

  • Lemma 3.2 · Functions at a finite level: bases $\{\mathbf 1_{P(\mathbb F_q)}\}$ by Lang
  • Lemma 3.3 · Trace of $*_I$ is convolution (vol $I=1$); IC traces span $\mathcal H_I$

Built on by

  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$
  • Proposition 3.7 · Disconnected stabilizers: (IG)$_n$ for all $n$ gives $\mathcal M_I=\mathrm{Irr}$
  • Proposition 3.9 · Group: no $\mathrm{IC}_\omega$ is a summand, so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Proposition 3.10 · Sheaves to functions: non-zero rows only; (WT) gives (IG)
  • Audit 7.7 · Audit of A3-4, A3-5, A3-6: unitriangular IC-trace basis, $m_{S,w}=\sum_i(-1)^i\mathrm{tr}\,A_{S,i}(w)$, and (IG) $\Rightarrow\mathcal M_I=\mathrm{Irr}\,\mathcal P_I$
  • 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 18:40 UTC

Unitriangular IC-trace basis of $C_I$ and the decomposition matrix $m_{S,w}$

  1. (i), one level. $\mathrm{tr}\,S$ lies in $C_I^{(N,M)}\otimes\overline{\mathbb Q}\ell$: it is supported on $\overline P(\mathbb F_q)\subseteq Y_N(\mathbb F_q)$, constant on $\pi{N,M}$-fibres ($S$ and $\Phi_S$ are pulled back), and $I(\mathcal O)$-invariant ($\Phi_S$ is equivariant, Setup 3.1(iii)). By Lemma 3.2(c) it is constant on each $P'(\mathbb F_q)$, and $\overline P=\pi_{N,M}^{-1}(\bar o)$ is a union of pieces $P'\le P$; on $P$ the value is $(-1)^{\delta(P)}$. So the matrix from $\{\mathrm{tr}\,\mathrm{IC}(P)\}$ to the basis $\{\mathbf 1_{P(\mathbb F_q)}\}$ of Lemma 3.2(c) is triangular for the closure order with diagonal $\pm1$, hence invertible (every piece has rational points, Lemma 3.2(c), and is the support of exactly one simple object by (L2)).
  2. (i), all levels. Spanning: $C_I=\bigcup C_I^{(N,M)}$ (Lemma 3.2(a)). Independence: finitely many simple objects lie at a common level (directedness and Setup 3.1(ii)), where step 1 applies; distinct simple objects there are $\mathrm{IC}$'s of distinct pieces.
  3. (ii), Frobenius on isotypic parts. By (D3), over $k$, $\mathrm{IC}w*_I\mathrm{IC}_0\simeq\bigoplus_iP_i[-i]$, $P_i={}^p\mathcal H^i$, each $P_i$ semisimple, and $P_i\simeq\bigoplus_SS\otimes E{S,i}(w)$ via evaluation (End $S=\overline{\mathbb Q}\ell$, no maps between distinct simples). $\mathrm{Fr}^*$ is t-exact, so $P_i$ inherits $\Phi_i$; under evaluation $\Phi_i=\bigoplus_S\Phi_S\otimes A{S,i}(w)$ (trace.tex, Lemma (Frobenius on the summands), verbatim). On stalks $\mathrm{tr}(\Phi_S\otimes A)=\mathrm{tr}\,\Phi_S\cdot\mathrm{tr}\,A$, so $\mathrm{tr}\,P_i=\sum_S\mathrm{tr}A_{S,i}(w)\,\mathrm{tr}\,S$.
  4. (ii), signs. $\mathrm{tr}$ is additive on the $\mathrm{Fr}$-compatible perverse truncation triangles and $\mathrm{tr}(P[-i])=(-1)^i\mathrm{tr}P$ [Laumon (1.1.1.1)]. If $S$ is not a shifted summand, every $E_{S,i}(w)=0$, so $m_{S,w}=0$.
  5. (ii), without (L2). If some constituent $S$ had $\mathrm{Fr}^*S\not\simeq S$, $\Phi_i$ would map the isotypic part of $\mathrm{Fr}^*S$ onto that of $S$, cyclically along the $\mathrm{Fr}$-orbit of $S$; a block-permutation matrix without fixed blocks has trace $0$ at every rational stalk, so such constituents contribute nothing to $\mathrm{tr}$. Under (L2) this case does not occur.
  6. (iii). The coordinate map $\kappa\colon C_I\otimes\overline{\mathbb Q}\ell\to\bigoplus_S\overline{\mathbb Q}\ell$ in the basis (i) is an isomorphism sending $\mathrm{tr}(\mathrm{IC}w*_I\mathrm{IC}_0)$ to column $w$ (by (ii)); with Lemma 3.3(c), $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}\ell=C_I\otimes\overline{\mathbb Q}\ell$ iff the columns span (trace.tex, Lemma (Spans in coordinates)). Finally $\mathcal H_I*\varphi_0$ is spanned by $\mathbb Z$-valued functions and $C_I$ has the $\mathbb Z$-basis of piece indicators (Lemma 3.2(c)); expressing one indicator through finitely many $T_w*\varphi_0$ is an integral linear system, solvable over a field of characteristic $0$ iff two ranks agree, independently of the field. So (IG) over $\mathbb C$ is equivalent to its version over $\overline{\mathbb Q}\ell$ (trace.tex, Lemma (Columns), verbatim). $\square$

Not claimed. The converse of the last clause of (ii) fails in general: eigenvalues $\alpha,-\alpha$ on one $E_{S,i}(w)$, or equal eigenvalues in degrees of opposite parity, cancel. It is never used in Theorem 3.6.

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