§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:46:10Z

Computation 6.2 · Affine Lusztig–Vogan module of $\mathrm{GL}_2/T$: Hecke action on orbit indicators

proved

Computes, by counting chambers in a panel, the action of the Iwahori–Hecke algebra on the basis of orbit indicators; with q formal this is both the function module C_I and the Grothendieck group of C_I (Chen–Yi's affine Lusztig–Vogan module, formula (29), types IIa, IIb, IIIa, IIIb).

Computation 6.2.

Conventions: \(T_w=\mathbf 1_{IwI}\), \(\mathrm{vol}(I)=1\), \((T_s-q)(T_s+1)=0\), \((f\ast\varphi)(\gamma)=\int f(g)\varphi(g^{-1}\gamma)\,dg\). Write \(\mathbf 1_O\) for the indicator of an orbit of Setup 6.1, \(s_1\) for the finite reflection (panel at \(v_0\)), \(s_0\) for the affine one (panel at \(v_1\)). Then $$\begin{aligned} T_{s_1}\mathbf 1_{A_\pm}&=\mathbf 1_{A_\mp}+\mathbf 1_{O_{0,0}}, & T_{s_1}\mathbf 1_{O_{0,0}}&=(q-1)(\mathbf 1_{A_+}+\mathbf 1_{A_-})+(q-2)\mathbf 1_{O_{0,0}},\\ T_{s_1}\mathbf 1_{O_{1,m-1}}&=\mathbf 1_{O_{0,m}}, & T_{s_1}\mathbf 1_{O_{0,m}}&=q\,\mathbf 1_{O_{1,m-1}}+(q-1)\mathbf 1_{O_{0,m}}\quad(m\ge1), \end{aligned}$$ the formulas for \(T_{s_0}\) are obtained by exchanging the indices \(0\leftrightarrow1\) of the orbits (keeping \(A_\pm\)), and \(T_\omega\mathbf 1_O=\mathbf 1_{\omega O}\); \(T_{\omega^2}\) acts trivially. \(\varphi_0=\mathbf 1_{L^+X}=\mathbf 1_{A_+}+\mathbf 1_{A_-}+\mathbf 1_{O_{0,0}}\) satisfies \(T_{s_1}\varphi_0=q\varphi_0\).

Sheaf version: in the Grothendieck group \(\mathcal M\) of Weil objects of \(\mathcal C_I\) over \(\overline{\mathbb F}_q\) with Frobenius eigenvalues in \(q^{\mathbb Z/2}\) (Chen–Yi §8.2), put \(v=-q^{-1/2}\), \(M_O:=[\Delta_O]\), \(\Delta_O=j_{O!}k[\delta(O)](\delta(O)/2)\), so that trace gives \(M_O\mapsto v^{\delta(O)}\mathbf 1_O\), \(H_s:=vT_s=[\Delta_s]\), \(\underline H_s:=H_s+v=[\mathrm{IC}_s]\) with \(\mathrm{IC}_s=k_{\overline{IsI}/I}[1](1/2)\). The same formulas, read in \(\mathcal M\otimes\mathbb Z[v^{\pm1}]\) with \(q=v^{-2}\), give \([\mathcal G\ast_I\mathcal F]=[\mathcal G]\cdot[\mathcal F]\).

  1. Fibre count. \((T_s\ast\mathbf 1_O)(\gamma)=\#\{gI\subset IsI:\ g^{-1}\gamma\in O\}=\#\{C'\in\mathcal P_s(C_0)\smallsetminus\{C_0\}:\ \mathrm{relpos}(C',\gamma)=O\}\), where \(\mathcal P_s(C_0)\cong\mathbb P^1(\mathbb F_q)\) is the \(s\)-panel: chambers \((v_0,v')\), \(v'\sim v_0\), for \(s_1\); chambers \((v',v_1)\), \(v'\sim v_1\), for \(s_0\). So the coefficient of \(\mathbf 1_{O'}\) in \(T_s\mathbf 1_O\) is \(\#(\mathcal P_s(C_0)\cap O)-[O=O']\) evaluated at any \(\gamma\in O'\).
  2. Panel at \(v_0\). If \(\gamma\ni v_0\) (\(\gamma\in A_\pm\cup O_{0,0}\)), the panel splits as: the neighbour of \(v_0\) towards \(b\) (relative position \(A_+\)), towards \(a\) (\(A_-\)), and the other \(q-1\) (\(O_{0,0}\)). If \(d(v_0,\gamma)=m\ge1\): the neighbour towards \(\gamma\) gives \(O_{1,m-1}\), the other \(q\) give \(O_{0,m}\). Step 1 gives the \(T_{s_1}\) formulas; \(s_0\) is the same with \(v_1\), and \(\omega\) swaps the panels. \(\omega^2=t\cdot\mathrm{Id}\) is central, and the centre acts trivially on \(G/T\).
  3. Checks. \(T_s=S-1\) with \(S\) the full panel sum and \(S^2=(q+1)S\) gives \(T_s^2=(q-1)T_s+q\) on every block; \(T_\omega T_{s_0}T_\omega^{-1}=T_{s_1}\); brute force on finite balls of the tree for \(q=2,3\) reproduces every coefficient (script tree_check.py).
  4. Sheaves. The stalk of \(\mathrm{IC}_s\ast\mathcal F\) at \(\gamma\) is \(R\Gamma(\mathcal P_s(C_0),\mathcal F|_{\mathrm{relpos}(\cdot,\gamma)})[1](1/2)\) (proper base change for \(m\)), and each panel decomposition in step 2 is into points, \(\mathbb A^1\) and \(\mathbb G_m\), whose compactly supported cohomology is Tate with classes \(1,q,q-1\). Hence the trace computation is an identity in \(\mathcal M\) (classes of Tate objects are determined by stalks; Laumon 1987, Théorème 1.1.2, for the trace map), which is the content of Chen–Yi, Proposition 74, formula (29): \(A_\pm\) are of type IIIa and \(O_{0,0}\) of type IIIb (with \(a_{v'}(\xi)=0\)) for \(s_1\); \(O_{1,m-1}\), \(O_{0,m}\) are of types IIa, IIb.
  5. Normalization. \(\mathrm{tr}(\mathrm{Frob},k[1](1/2))=-q^{-1/2}=v\); \((H_s+v)(H_s-v^{-1})=0\), \(\overline{H_s}=H_s^{-1}\), \(\overline v=v^{-1}\) (Soergel's normalization).

Builds on

  • Setup 6.1 · The $I$-orbits on $L(\mathrm{GL}_2/T)$ as relative positions of an edge and a geodesic

Built on by

  • Lemma 6.3 · $\mathrm{IC}(Y_n)=\nu_{n*}k$ with stalk rank $n+1-d(v_0,\gamma)$
  • Proposition 6.6 · Functions: $\mathcal H_I\ast\varphi_0=C_K+C_{K'}$ has codimension two
  • Lemma 6.7 · Sheaf-level Corollary D: positive Steinberg certificates exclude summands
  • Proposition 6.8 · Second proof for $\mathrm{GL}_2/T$: $\psi$ is a positive certificate, $\lambda$ is not
  • Computation 9.2 · Recomputing the Hecke action on $C_I$ for $\mathrm{GL}_2/T$
Original record, as the agent wrote it at 18:46 UTC

Affine Lusztig–Vogan module of $\mathrm{GL}_2/T$: Hecke action on orbit indicators

  1. Fibre count. \((T_s\ast\mathbf 1_O)(\gamma)=\#\{gI\subset IsI:\ g^{-1}\gamma\in O\}=\#\{C'\in\mathcal P_s(C_0)\smallsetminus\{C_0\}:\ \mathrm{relpos}(C',\gamma)=O\}\), where \(\mathcal P_s(C_0)\cong\mathbb P^1(\mathbb F_q)\) is the \(s\)-panel: chambers \((v_0,v')\), \(v'\sim v_0\), for \(s_1\); chambers \((v',v_1)\), \(v'\sim v_1\), for \(s_0\). So the coefficient of \(\mathbf 1_{O'}\) in \(T_s\mathbf 1_O\) is \(\#(\mathcal P_s(C_0)\cap O)-[O=O']\) evaluated at any \(\gamma\in O'\).
  2. Panel at \(v_0\). If \(\gamma\ni v_0\) (\(\gamma\in A_\pm\cup O_{0,0}\)), the panel splits as: the neighbour of \(v_0\) towards \(b\) (relative position \(A_+\)), towards \(a\) (\(A_-\)), and the other \(q-1\) (\(O_{0,0}\)). If \(d(v_0,\gamma)=m\ge1\): the neighbour towards \(\gamma\) gives \(O_{1,m-1}\), the other \(q\) give \(O_{0,m}\). Step 1 gives the \(T_{s_1}\) formulas; \(s_0\) is the same with \(v_1\), and \(\omega\) swaps the panels. \(\omega^2=t\cdot\mathrm{Id}\) is central, and the centre acts trivially on \(G/T\).
  3. Checks. \(T_s=S-1\) with \(S\) the full panel sum and \(S^2=(q+1)S\) gives \(T_s^2=(q-1)T_s+q\) on every block; \(T_\omega T_{s_0}T_\omega^{-1}=T_{s_1}\); brute force on finite balls of the tree for \(q=2,3\) reproduces every coefficient (script tree_check.py).
  4. Sheaves. The stalk of \(\mathrm{IC}_s\ast\mathcal F\) at \(\gamma\) is \(R\Gamma(\mathcal P_s(C_0),\mathcal F|_{\mathrm{relpos}(\cdot,\gamma)})[1](1/2)\) (proper base change for \(m\)), and each panel decomposition in step 2 is into points, \(\mathbb A^1\) and \(\mathbb G_m\), whose compactly supported cohomology is Tate with classes \(1,q,q-1\). Hence the trace computation is an identity in \(\mathcal M\) (classes of Tate objects are determined by stalks; Laumon 1987, Théorème 1.1.2, for the trace map), which is the content of Chen–Yi, Proposition 74, formula (29): \(A_\pm\) are of type IIIa and \(O_{0,0}\) of type IIIb (with \(a_{v'}(\xi)=0\)) for \(s_1\); \(O_{1,m-1}\), \(O_{0,m}\) are of types IIa, IIb.
  5. Normalization. \(\mathrm{tr}(\mathrm{Frob},k[1](1/2))=-q^{-1/2}=v\); \((H_s+v)(H_s-v^{-1})=0\), \(\overline{H_s}=H_s^{-1}\), \(\overline v=v^{-1}\) (Soergel's normalization).

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