§2 · Agent 2 · SL₂/SO₂: central inertia · 2026-10-17T02:22:38Z

Lemma 2.3 · Hecke convolution preserves the central grading, so every Satake object is even

proved audit: correct

Second audit: correct.

The base object \(\IC_0\) and all Hecke kernels have trivial central inertia, and inertia is multiplicative under convolution. Hence every Satake object, and so all of \(\Csat\), lies in the even part \(\mathcal C_K^+\).

Lemma 2.3.

With \(\varepsilon\) and \(\mathcal C_K^\pm\) as in Lemma 2.2:

  1. \(\IC_0\in\mathcal C_K^+\);
  2. every object of the Hecke category \(\mathcal H\) has \(\varepsilon=\mathrm{id}\) (left \(L^+G\)-inertia of \(-I\) on \(\Gr\));
  3. \(\varepsilon_{V\ast F}=\varepsilon_V\ast\varepsilon_F\) for \(V\in\mathcal H\), \(F\in\mathcal C_K\).

Hence \(\mathcal H\ast\mathcal C_K^\pm\subseteq\mathcal C_K^\pm\) and $$\Csat=\Thick\{V_\lambda\ast\IC_0\}\subseteq\mathcal C_K^+ .$$

  1. The base object. \(\IC_0=a_{0!}\underline k_{\mathcal O_0}\). The constant sheaf is pulled back from a point, so \(\varepsilon=\mathrm{id}\) on it, and \(a_{0!}\) commutes with \(\varepsilon\). (Also, \(\Stab(x_0)=\SO_2(\cO)\) is connected and contains \(-I\), so the equivariant structure is unique and \(-I\) acts trivially on the stalk.)
  2. Hecke kernels. On \(\Gr\), \(-I\) acts trivially (it is central), and \(\Stab_{L^+G}(t^\lambda)=L^+G\cap\operatorname{Ad}_{t^\lambda}L^+G\) is connected: for \(\SL_2\) and \(\lambda=n\alpha^\vee\) it is the group of integral matrices whose lower-left entry lies in \(t^{2n}\cO\), which reduces modulo \(t\) onto a Borel subgroup with pro-unipotent kernel. So every equivariant local system on every \(\Gr_\lambda\) has \(\varepsilon=\mathrm{id}\). The sheaf \(V_\lambda=\IC(\Gr^{\le\lambda})\) is simple with \(\End(V_\lambda)=k\), so \(\varepsilon_{V_\lambda}\) is a scalar of square \(1\), equal to \(+1\) on \(\Gr_\lambda\). More generally, split \(V=V^+\oplus V^-\) by Lemma 2.2(c); all cohomology sheaves of \(V^-\) restricted to orbits are equivariant local systems with \(\varepsilon=-1\), hence zero, so \(V^-=0\).
  3. Bimultiplicativity. On \(LG\times^{L^+G}LX\), \((-I)\cdot[g,x]=[-g,x]=[g(-I),x]=[g,(-I)x]=[g,x]\). The twisted product \(V\mathbin{\widetilde\boxtimes}F\) is obtained by descent from \(V\boxtimes F\) on \(LG\times LX\), the middle copy of \(L^+G\) acting through the right invariance of the pullback of \(V\) and the equivariant structure of \(F\). Tracing the identification \([-g,x]=[g,x]\) through the middle element \(-I\), the inertia of \(-I\) acts on the stalk \(V_{gL^+G}\otimes F_x\) by \(\varepsilon(V)_{gL^+G}\otimes\varepsilon(F)_x\). Since \(m:[g,x]\mapsto gx\) is \(L^+G\)-equivariant, \(m_!\) commutes with \(\varepsilon\), and \(\varepsilon_{V\ast F}=\varepsilon_V\ast\varepsilon_F\).
  4. Conclusion. \(\mathcal C_K^+\) is thick (Lemma 2.2) and contains every \(V_\lambda\ast\IC_0\) by steps 1–3, hence their thick span.

Equivalent formulation. The whole convolution diagram descends to \(\PGL_2\): \(\Gr_{\SL_2}\simeq\Gr^\circ_{\PGL_2}\) (neutral component) and \(L\SL_2\times^{L^+\SL_2}LX\simeq L\PGL_2^\circ\times^{L^+\PGL_2}LX\), because the kernel \(\mu_2\subseteq L^+\SL_2\) acts trivially on pairs; \(V_\lambda\) and \(\IC_0\) are \(L^+\PGL_2\)-equivariant. So \(V_\lambda\ast p^{*}\overline F\simeq p^{*}(V_\lambda\mathbin{\bar\ast}\overline F)\), and the Satake objects lie in \(p^{*}\overline{\mathcal C_K}=\mathcal C_K^+\). The function-level counterpart is Proposition J(2): \(\sigma\) commutes with \(\mathcal H_K\) and fixes \(\varphi_0\).

Builds on

  • Lemma 2.2 · Central inertia splits \(\mathcal C_K\) into even and odd parts; the even part is the \(\PGL_2\)-equivariant category

Built on by

  • Theorem 2.4 · \(\Csat\ne\mathcal C_K\) for \(\SL_2/\SO_2\): the sign local system on the first orbit is not generated
  • Proposition 2.7 · The Satake span is exactly the even-label, constant-coefficient part, strictly inside the even part
  • Audit 7.1 · Audit: central inertia for \(\SL_2/\SO_2\) is a natural automorphism, and convolution is bimultiplicative
  • Lemma X2.2 · Weil inertia lemma: \(\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\)
Original record, as the agent wrote it at 02:22 UTC

Every Satake object is even: Hecke convolution preserves the central grading

Claim. \(\mathrm{IC}_0\in\mathcal C^+\); every object of \(\mathcal H\) has \(\varepsilon=\mathrm{id}\) (left \(L^+G\)-inertia of \(-I\) on \(\mathrm{Gr}\)); and \(\varepsilon_{V\ast F}=\varepsilon_V\otimes\varepsilon_F\). Hence \(\mathcal H\ast\mathcal C^\pm\subset\mathcal C^\pm\) and $$\mathcal C^{\mathrm{Sat}}=\operatorname{Thick}\{V_\lambda\ast\mathrm{IC}_0\}\subseteq\mathcal C^+ .$$ Proof.

  1. \(\mathrm{IC}_0=a_{0!}\underline k_{\mathcal O_0}\); \(\underline k_{\mathcal O_0}\) is pulled back from a point (A2-2.4), and \(a_{0!}\) commutes with \(\varepsilon\) (A2-2.3). Also \(\operatorname{Stab}(x_0)=\mathrm{SO}_2(\mathcal O)\) is connected and contains \(-I\), so the equivariant structure is unique and \(-I\) acts trivially on the stalk.
  2. On \(\mathrm{Gr}\), \(-I\) acts trivially (central) and \(\operatorname{Stab}_{L^+G}(t^\lambda)=J_\lambda=L^+G\cap\mathrm{Ad}_{t^\lambda}L^+G\) is connected (for \(\mathrm{SL}_2\): \(J_{n\alpha^\vee}=\{c\in t^{2n}\mathcal O\}\), reduction mod \(t\) onto the connected Borel \(B(k)\) with pro-unipotent kernel). By A2-2.5 every equivariant local system on every \(\mathrm{Gr}_\lambda\) has \(\varepsilon=\mathrm{id}\). For \(V_\lambda=\mathrm{IC}(\mathrm{Gr}^{\le\lambda})\): it is simple, \(\operatorname{End}(V_\lambda)=k\), so \(\varepsilon_{V_\lambda}\) is a scalar with square 1 equal to \(+1\) on \(\mathrm{Gr}_\lambda\). (More generally every \(V\in\mathcal H\) lies in \(\mathcal H^+\): split \(V=V^+\oplus V^-\) by A2-2(c); all cohomology sheaves of \(V^-\) restricted to orbits are equivariant local systems with \(\varepsilon=-1\), hence zero, so \(V^-=0\).)
  3. Bimultiplicativity. On \(LG\times^{L^+G}LX\), \((-I)\cdot[g,x]=[-g,x]=[g\,(-I),x]=[g,(-I)x]=[g,x]\). The twisted product \(V\mathbin{\widetilde\boxtimes}F\) is defined by descent from \(p_1^*V\boxtimes p_2^*F\) on \(LG\times LX\), the middle \(L^+G\) acting through the right-invariance of \(p_1^*V\) (pulled back from \(\mathrm{Gr}\)) and the equivariant structure of \(F\). Tracing the identification \([-g,x]=[g,x]\) through the middle element \(h=-I\), the inertia action of \(-I\) on the stalk \(V_{gL^+G}\otimes F_x\) is \(\varepsilon_{-I}(V)_{gL^+G}\otimes\varepsilon_{-I}(F)_x\). Since \(m:[g,x]\mapsto gx\) is \(L^+G\)-equivariant, \(m_!\) commutes with \(\varepsilon\) (A2-2.3). Hence \(\varepsilon_{V\ast F}=\varepsilon_V\otimes\varepsilon_F\).
  4. \(\mathcal C^+\) is thick (A2-2) and contains all \(V_\lambda\ast\mathrm{IC}_0\) (steps 1–3), so it contains their thick span. \(\square\)

Equivalent formulation. The whole convolution diagram descends to \(\mathrm{PGL}_2\): \(\mathrm{Gr}_{\mathrm{SL}_2}\simeq\mathrm{Gr}^\circ_{\mathrm{PGL}_2}\) (neutral component), \(L\mathrm{SL}_2\times^{L^+\mathrm{SL}_2}LX\simeq L\mathrm{PGL}_2^\circ\times^{L^+\mathrm{PGL}_2}LX\) (the kernel \(\mu_2\subset L^+\mathrm{SL}_2\) acts trivially on pairs), \(V_\lambda\) and \(\mathrm{IC}_0\) are \(L^+\mathrm{PGL}_2\)-equivariant; so \(V_\lambda\ast p^*\bar F\simeq p^*(V_\lambda\bar\ast\bar F)\) and the Satake objects lie in \(p^*\bar{\mathcal C}=\mathcal C^+\). The function-level counterpart is J(2): \(\sigma\) commutes with \(\mathcal H_K\) and fixes \(\varphi_0\).

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