§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:47:27Z

Lemma 6.4 · Convolution rules for $\mathrm{IC}s\ast_I\mathrm{IC}{i,n}$ and $\mathrm{IC}_\omega$

proved ★ key step

Decomposes the convolution of each simple affine reflection and of ω with each non-closed IC: only non-closed ICs occur, never the closed orbits A±; proved by a fibre count plus the decomposition theorem, with no weight bookkeeping.

Lemma 6.4.

With \(\mathrm{IC}_s=k_{\overline{IsI}/I}[1]\), \(\mathrm{IC}_\omega=\delta_\omega\), the objects of Lemma 6.3, and \(\mathrm{IC}_{i,-1}:=0\), for all \(n\ge0\) (over \(\overline{\mathbb F}_q\) up to Tate twists, and over \(\mathbb C\)): $$\begin{aligned} &\text{(a)}\quad \mathrm{IC}_{s_0}\ast_I\mathrm{IC}_{0,n}\cong\mathrm{IC}_{1,n+1}\oplus\mathrm{IC}_{1,n-1},\qquad \mathrm{IC}_{s_1}\ast_I\mathrm{IC}_{1,n}\cong\mathrm{IC}_{0,n+1}\oplus\mathrm{IC}_{0,n-1};\\ &\text{(b)}\quad \mathrm{IC}_{s_1}\ast_I\mathrm{IC}_{0,n}\cong\mathrm{IC}_{0,n}[1]\oplus\mathrm{IC}_{0,n}[-1],\qquad \mathrm{IC}_{s_0}\ast_I\mathrm{IC}_{1,n}\cong\mathrm{IC}_{1,n}[1]\oplus\mathrm{IC}_{1,n}[-1];\\ &\text{(c)}\quad \mathrm{IC}_\omega\ast_I\mathrm{IC}_{i,n}\cong\mathrm{IC}_{1-i,n}. \end{aligned}$$ Function shadow (any residue field \(\mathbb F_q\)): \((T_{s_0}+1)f_n=f^{(1)}_{n+1}+q\,f^{(1)}_{n-1}\), \((T_{s_1}+1)f_n=(q+1)f_n\), \(T_\omega f_n=f^{(1)}_n\); in \(\mathcal M\): \(\underline H_{s_0}C_{0,n}=C_{1,n+1}+C_{1,n-1}\), \(\underline H_{s_1}C_{0,n}=(v+v^{-1})C_{0,n}\).

  1. The map. By Lemma 6.3(b), \(\mathrm{IC}_{s_0}\ast_I\mathrm{IC}_{0,n}=\nu'_{*}k[n+2]\) for \(\nu':P_{s_0}\times^I\widetilde Y_n\to LX\) (proper base change), where \(P_{s_0}=\mathrm{Stab}(v_1)\supset I\). The source is a bundle over \(P_{s_0}/I=\mathbb P^1\) with fibre \(\widetilde Y_n\), hence rationally smooth of relative dimension \(n+2\).
  2. Fibres. \(\nu'^{-1}(\gamma)=\{(v',u):\ v'\sim v_1,\ u\in\gamma,\ d(v',u)\le n,\ d(v',u)\equiv n\ (2)\}\). Fix \(u\in\gamma\), \(r=d(v_1,u)\). The neighbours \(v'\) of \(v_1\) are at distance \(r\pm1\) from \(u\) (one at \(r-1\) if \(r\ge1\)); so the condition forces \(r\equiv n+1\), and then holds for all \(v'\in\mathbb P^1\) if \(r\le n-1\), for exactly one \(v'\) if \(r=n+1\), for none if \(r>n+1\). Hence
$$\nu'^{-1}(\gamma)=\bigsqcup_{u\in\gamma,\ r(u)\le n-1,\ r\equiv n-1}\mathbb P^1\ \sqcup\bigsqcup_{u\in\gamma,\ r(u)=n+1}\mathrm{pt},\qquad H^*(\nu'^{-1}(\gamma))=k^{a(\gamma)}\oplus k^{b(\gamma)}[-2],$$

with \(a=\#\{u\in\gamma: d(v_1,u)\le n+1,\equiv n+1\}=\mathrm{rk}\,(\mathrm{IC}_{1,n+1})_\gamma\) and \(b=\#\{u\in\gamma:d(v_1,u)\le n-1,\equiv n-1\}=\mathrm{rk}\,(\mathrm{IC}_{1,n-1})_\gamma\) (Lemma 6.3(b) with \(v_1\)).

  1. Semisimplicity. \(\nu'_*k[n+2]\) is a direct sum of shifted simple perverse sheaves: over \(\overline{\mathbb F}_q\), \(k\) is pure, \(\nu'\) proper, so the pushforward is pure (BBD 5.1.14) and splits (BBD 5.4.5, 5.4.6); over \(\mathbb C\), BBD 6.2.5. Simple constituents are \(I\)-equivariant ICs of orbits with constant coefficients (Setup 6.1(c)).
  2. Peeling. Supports lie in \(\omega Y_{n+1}\). At \(\gamma\in O_{1,n+1}\): stalk \(k[n+2]\), so \(\mathrm{IC}_{1,n+1}\) occurs exactly once, unshifted. The complement \(R\) has stalks \(k^{b(\gamma)}[n]\) by step 2 (dimensions of stalk cohomology add in direct sums). At \(\gamma\in O_{1,n-1}\), \(b=1\): \(\mathrm{IC}_{1,n-1}\) (generic stalk \(k[n]\)) occurs once, unshifted; the remainder \(R'\) has all stalks of rank \(b-b=0\), hence \(R'=0\). For \(n=0\), \(b\equiv0\). This is (a, first half); \(\omega\)-conjugation (\(\omega P_{s_0}\omega^{-1}=P_{s_1}\)) gives the second half.
  3. (b), (c). \(\mathrm{IC}_{0,n}\) is \(K\)-equivariant and \(P_{s_1}\subset K\); then \(P_{s_1}\times^ILX\cong P_{s_1}/I\times LX\), \((p,x)\mapsto(pI,px)\), and \(\mathrm{IC}_{s_1}\ast\mathcal F=R\Gamma(\mathbb P^1,k)[1]\otimes\mathcal F\). \(I\omega I=\omega I\), so \(\mathrm{IC}_\omega\ast=\omega_*\).
  4. Shadow. Taking traces (or counting directly: \((T_{s_0}+1)f_n(\gamma)=\sum_{v'\sim v_1}\#\{u\in\gamma: d(v',u)\le n,\equiv n\}\), and step 2 with \(\#\mathbb P^1(\mathbb F_q)=q+1\)) gives the function identities; with \(C_{i,n}=v^{n+1}f^{(i)}_n\), \(q=v^{-2}\): \(v\cdot v^{n+1}(f^{(1)}_{n+1}+v^{-2}f^{(1)}_{n-1})=C_{1,n+1}+C_{1,n-1}\). The module computation (lv_module.py, lv_checks.py) reproduces this for \(n\le11\) by the Kazhdan–Lusztig reduction against the bar involution, with no geometric input.

Builds on

  • Lemma 6.3 · $\mathrm{IC}(Y_n)=\nu_{n*}k$ with stalk rank $n+1-d(v_0,\gamma)$

Built on by

  • Proposition 6.5 · Sheaves fail for $\mathrm{GL}2/T$: $\mathrm{IC}{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}$
  • Proposition 6.6 · Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two
  • Audit 9.4 · Audit of A6-4: the convolution rules for $\mathrm{IC}s\ast_I\mathrm{IC}{i,n}$
Original record, as the agent wrote it at 18:47 UTC

Convolution rules for $\mathrm{IC}s\ast_I\mathrm{IC}{i,n}$ and $\mathrm{IC}_\omega$

  1. The map. By Lemma 6.3(b), \(\mathrm{IC}_{s_0}\ast_I\mathrm{IC}_{0,n}=\nu'_{*}k[n+2]\) for \(\nu':P_{s_0}\times^I\widetilde Y_n\to LX\) (proper base change), where \(P_{s_0}=\mathrm{Stab}(v_1)\supset I\). The source is a bundle over \(P_{s_0}/I=\mathbb P^1\) with fibre \(\widetilde Y_n\), hence rationally smooth of relative dimension \(n+2\).
  2. Fibres. \(\nu'^{-1}(\gamma)=\{(v',u):\ v'\sim v_1,\ u\in\gamma,\ d(v',u)\le n,\ d(v',u)\equiv n\ (2)\}\). Fix \(u\in\gamma\), \(r=d(v_1,u)\). The neighbours \(v'\) of \(v_1\) are at distance \(r\pm1\) from \(u\) (one at \(r-1\) if \(r\ge1\)); so the condition forces \(r\equiv n+1\), and then holds for all \(v'\in\mathbb P^1\) if \(r\le n-1\), for exactly one \(v'\) if \(r=n+1\), for none if \(r>n+1\). Hence
$$\nu'^{-1}(\gamma)=\bigsqcup_{u\in\gamma,\ r(u)\le n-1,\ r\equiv n-1}\mathbb P^1\ \sqcup\bigsqcup_{u\in\gamma,\ r(u)=n+1}\mathrm{pt},\qquad H^*(\nu'^{-1}(\gamma))=k^{a(\gamma)}\oplus k^{b(\gamma)}[-2],$$

with \(a=\#\{u\in\gamma: d(v_1,u)\le n+1,\equiv n+1\}=\mathrm{rk}\,(\mathrm{IC}_{1,n+1})_\gamma\) and \(b=\#\{u\in\gamma:d(v_1,u)\le n-1,\equiv n-1\}=\mathrm{rk}\,(\mathrm{IC}_{1,n-1})_\gamma\) (Lemma 6.3(b) with \(v_1\)).

  1. Semisimplicity. \(\nu'_*k[n+2]\) is a direct sum of shifted simple perverse sheaves: over \(\overline{\mathbb F}_q\), \(k\) is pure, \(\nu'\) proper, so the pushforward is pure (BBD 5.1.14) and splits (BBD 5.4.5, 5.4.6); over \(\mathbb C\), BBD 6.2.5. Simple constituents are \(I\)-equivariant ICs of orbits with constant coefficients (Setup 6.1(c)).
  2. Peeling. Supports lie in \(\omega Y_{n+1}\). At \(\gamma\in O_{1,n+1}\): stalk \(k[n+2]\), so \(\mathrm{IC}_{1,n+1}\) occurs exactly once, unshifted. The complement \(R\) has stalks \(k^{b(\gamma)}[n]\) by step 2 (dimensions of stalk cohomology add in direct sums). At \(\gamma\in O_{1,n-1}\), \(b=1\): \(\mathrm{IC}_{1,n-1}\) (generic stalk \(k[n]\)) occurs once, unshifted; the remainder \(R'\) has all stalks of rank \(b-b=0\), hence \(R'=0\). For \(n=0\), \(b\equiv0\). This is (a, first half); \(\omega\)-conjugation (\(\omega P_{s_0}\omega^{-1}=P_{s_1}\)) gives the second half.
  3. (b), (c). \(\mathrm{IC}_{0,n}\) is \(K\)-equivariant and \(P_{s_1}\subset K\); then \(P_{s_1}\times^ILX\cong P_{s_1}/I\times LX\), \((p,x)\mapsto(pI,px)\), and \(\mathrm{IC}_{s_1}\ast\mathcal F=R\Gamma(\mathbb P^1,k)[1]\otimes\mathcal F\). \(I\omega I=\omega I\), so \(\mathrm{IC}_\omega\ast=\omega_*\).
  4. Shadow. Taking traces (or counting directly: \((T_{s_0}+1)f_n(\gamma)=\sum_{v'\sim v_1}\#\{u\in\gamma: d(v',u)\le n,\equiv n\}\), and step 2 with \(\#\mathbb P^1(\mathbb F_q)=q+1\)) gives the function identities; with \(C_{i,n}=v^{n+1}f^{(i)}_n\), \(q=v^{-2}\): \(v\cdot v^{n+1}(f^{(1)}_{n+1}+v^{-2}f^{(1)}_{n-1})=C_{1,n+1}+C_{1,n-1}\). The module computation (lv_module.py, lv_checks.py) reproduces this for \(n\le11\) by the Kazhdan–Lusztig reduction against the bar involution, with no geometric input.

Sources