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

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

proved audit: minor defect

Minor defect (second audit). All claims true. In (a), closure of C⁺ under cofibres does not follow from the cofibre property alone: knowing ε_A = id and ε_B = id determines ε_C only up to adding N = (C → A[1] → B → C), because the homotopy in the naturality square is part of the data. One more line is needed: N² = 0 and ε_C² = id give 2N = 0, so N = 0 (alternatively, deduce (a) from (b) and (c)). Audit 7.1 checks the construction and its compatibility with truncations but does not mention this point.

The central element \(-I\) acts trivially on \(LX\) but nontrivially on equivariant sheaves, through the inertia of the quotient stack; this splits \(\mathcal C_K\) into two orthogonal thick pieces. The even piece is the category for \(\PGL_2\), whose stabilizers are connected.

Lemma 2.2.

Let \(Z=\{\pm I\}\subseteq Z(\SL_2)\cap G^\theta\); it is central in \(L^+G=\SL_2(\cO)\) and acts trivially on \(X\), \(LX\) and \(\Gr=LG/L^+G\).

  1. The element \(-I\) defines a natural automorphism \(\varepsilon\) of the identity functor of \(\mathcal C_K\) with \(\varepsilon^2=\mathrm{id}\). It commutes with shifts, cofibres and with \(f^{*},f_!,f_{*}\) for equivariant \(f\); it is the identity on sheaves pulled back from a point; and on an equivariant local system with monodromy \(\rho\) on an orbit \(H\cdot y\) it is \(\rho(-I)\).
  2. Put \(\mathcal C_K^+=\{F:\varepsilon_F=\mathrm{id}\}\) and \(\mathcal C_K^-=\{F:\varepsilon_F=-\mathrm{id}\}\). Then (a) \(\mathcal C_K^\pm\) are thick; (b) \(\operatorname{Map}(\mathcal C_K^+,\mathcal C_K^-)=0\); (c) \(\mathcal C_K=\mathcal C_K^+\oplus\mathcal C_K^-\).
  3. Let \(\overline G=\PGL_2\) act on \(X\) by \(g\cdot A=gAg^{\mathsf T}/\det g\). The stabilizer of \(I\) is \(\mathrm{PSO}_2\) (connected), the \(L^+\PGL_2\)-orbits on \(LX\) are the \(\mathcal O_m\) with connected stabilizers, and pullback along \(p:L^+\SL_2\backslash LX\to L^+\PGL_2\backslash LX\) is an equivalence
$$p^{*}:\ \overline{\mathcal C_K}:=\Shv_{\mathrm{cons}}(L^+\PGL_2\backslash LX)\xrightarrow{\ \sim\ }\mathcal C_K^+ .$$
  1. Construction. For a quotient stack \(\mathcal Y=H\backslash Y\) and a central subgroup \(Z\subseteq H\) acting trivially on \(Y\), multiplication by \(z\in Z\) is an automorphism of every \(H\)-torsor \(P\to U\) commuting with the equivariant map \(P\to Y\). This is a 2-automorphism of \(\mathrm{id}_{\mathcal Y}\) (a central section of the inertia). Functoriality of sheaves in 2-morphisms gives a natural automorphism \(\varepsilon_z\) of the identity of \(\Shv(\mathcal Y)\); for \(z=-I\) write \(\varepsilon\), so \(\varepsilon^2=\mathrm{id}\). Its properties:
    • naturality: \(\varepsilon_{G}\circ f=f\circ\varepsilon_{F}\) for every \(f:F\to G\);
    • \(\varepsilon_{\Sigma F}=\Sigma\varepsilon_F\), and for a cofibre sequence \(A\to B\to C\), \(\varepsilon_C\) is the map induced on cofibres by \((\varepsilon_A,\varepsilon_B)\) (a natural transformation of \(\infty\)-functors \(\mathrm{id}\Rightarrow\mathrm{id}\) is compatible with every colimit computed in \(\mathcal C_K\));
    • for an \(H\)-equivariant map \(f\), the functors \(f^{*},f_!,f_{*}\) commute with \(\varepsilon\);
    • on a sheaf pulled back from a point, \(\varepsilon=\mathrm{id}\);
    • on an equivariant local system on \(H\cdot y\) with monodromy \(\rho:\pi_0\Stab(y)\to\GL(V)\), \(\varepsilon_z=\rho(z)\), since the inertia acts on the stalk at \(y\) through \(z\in\Stab(y)\).
  2. (a) Thickness. \(\mathcal C_K^\pm\) contain \(0\) and are closed under isomorphisms; shifts and cofibres are handled by the second property. Retracts: if \(F\xrightarrow{i}G\xrightarrow{p}F\) with \(pi=\mathrm{id}\) and \(G\in\mathcal C_K^+\), then \(\varepsilon_F=pi\varepsilon_F=p\varepsilon_Gi=pi=\mathrm{id}\) by naturality; likewise for \(\mathcal C_K^-\).
  3. (b) Orthogonality. For \(f:F\to G\) with \(F\in\mathcal C_K^+\), \(G\in\mathcal C_K^-\): \(-f=\varepsilon_Gf=f\varepsilon_F=f\), so \(2f=0\) and \(f=0\) over \(k=\C\); the same holds after every shift.
  4. (c) Splitting. \(e=(1+\varepsilon_F)/2\) is an idempotent endomorphism of \(F\) in the homotopy category. \(\mathcal C_K\) is idempotent complete (it is defined as such), so \(F\simeq F_1\oplus F_2\) with \(e\) the projector onto \(F_1\). By naturality \(\varepsilon_{F_1}=p\,\varepsilon_F\,i=p(2e-1)i=\mathrm{id}\) and \(\varepsilon_{F_2}=-\mathrm{id}\).
  5. Reformulation. The formula \(g\cdot A=gAg^{\mathsf T}/\det g\) preserves \(\det A=1\), scalars act trivially, and it extends the \(\SL_2\)-action. The stabilizer of \(I\) is \(\{g:gg^{\mathsf T}=\det(g)I\}/\mathbb G_m=\SO_2/\mu_2=\mathrm{PSO}_2\), connected; it is \((\PGL_2^\theta)^\circ\), of index \(2\) in \(\mathrm{PO}_2\). Over \(\C\), \(\cO^\times=\cO^{\times2}\), so \(\PGL_2(\cO)=\SL_2(\cO)/\mu_2\): the orbits on \(LX\) are the same, with stabilizers \((\mu_2\times t^m\cO)/\mu_2\), connected. The map \(p\) is a \(\mu_2\)-gerbe; \(p_{*}p^{*}=(-)\otimes H^{*}(B\mu_2,k)=\mathrm{id}\) in characteristic \(0\), so \(p^{*}\) is fully faithful, and its essential image is exactly the part on which the \(\mu_2\)-inertia acts trivially, i.e. \(\mathcal C_K^+\). Note that \(X\ne\PGL_2/\mathrm{PO}_2\): the latter is the quotient \(X/\{\pm1\}\), the projectivised nondegenerate symmetric matrices (the Summary's row \(\PGL_2/\mathrm{PO}_2\)); here \(X=\PGL_2/\mathrm{PSO}_2\) is its double cover.

Builds on

  • Computation 2.1 · Stabilizers for \(\SL_2/\SO_2\): \(\mu_2\) times a pro-unipotent group

Built on by

  • Lemma 2.3 · Hecke convolution preserves the central grading, so every Satake object is even
  • Theorem 2.4 · \(\Csat\ne\mathcal C_K\) for \(\SL_2/\SO_2\): the sign local system on the first orbit is not generated
  • 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

Central inertia splits C into even and odd parts; even part is the PGL2-equivariant category

Setting. \(Z:=\{\pm I\}\subset Z(\mathrm{SL}_2)\cap G^\theta\). It is central in \(L^+G=\mathrm{SL}_2(\mathcal O)\) and acts trivially on \(X\), \(LX\) and on \(\mathrm{Gr}=LG/L^+G\).

Construction (proved, formal). For a quotient stack \(\mathcal Y=H\backslash Y\) and central \(Z\subset H\) acting trivially on \(Y\), multiplication by \(z\in Z\) is an automorphism of every \(H\)-torsor \(P\to U\) commuting with the equivariant map \(P\to Y\); this is a 2-automorphism of \(\mathrm{id}_{\mathcal Y}\) (a central section of the inertia). Functoriality of \(\mathrm{Shv}\) in 2-morphisms gives a natural automorphism \(\varepsilon_z\) of \(\mathrm{id}_{\mathrm{Shv}(\mathcal Y)}\); for \(z=-I\), \(\varepsilon:=\varepsilon_{-I}\), \(\varepsilon^2=\mathrm{id}\). Properties:

  1. \(\varepsilon_G\circ f=f\circ\varepsilon_F\) for every \(f:F\to G\).
  2. \(\varepsilon_{\Sigma F}\simeq\Sigma\varepsilon_F\) and, for a cofibre sequence \(A\to B\to C\), \(\varepsilon_C\) is the map induced on cofibres by \((\varepsilon_A,\varepsilon_B)\): a natural transformation of \(\infty\)-functors \(\mathrm{id}\Rightarrow\mathrm{id}\) is compatible with every colimit computed in \(\mathcal C\) (apply it to the pushout square).
  3. For an \(H\)-equivariant map \(f\), the functors \(f^*,f_!,f_*\) commute with \(\varepsilon\) (\(f\) intertwines the 2-automorphisms).
  4. On a pullback from a point (constant sheaf), \(\varepsilon=\mathrm{id}\).
  5. On an equivariant local system \(\mathcal L\) on an orbit \(H\cdot y\) with monodromy \(\rho:\pi_0\operatorname{Stab}(y)\to GL\), \(\varepsilon_z=\rho(z)\) (inertia acts on the stalk at \(y\) through \(z\in\operatorname{Stab}(y)\)).

Lemma. \(\mathcal C^+:=\{F:\varepsilon_F=\mathrm{id}\}\), \(\mathcal C^-:=\{F:\varepsilon_F=-\mathrm{id}\}\). Then (a) \(\mathcal C^\pm\) are thick; (b) \(\operatorname{Map}(\mathcal C^+,\mathcal C^-)=0\); (c) \(\mathcal C=\mathcal C^+\oplus\mathcal C^-\). Proof. (a) Contains 0, closed under isomorphisms; shifts and cofibres by 2; retracts: if \(F\xrightarrow{i}G\xrightarrow{p}F\), \(pi=\mathrm{id}\), \(G\in\mathcal C^+\), then \(\varepsilon_F=p\,i\,\varepsilon_F=p\,\varepsilon_G\,i=pi=\mathrm{id}\) by 1. (b) For \(f:F\to G\), \(F\in\mathcal C^+\), \(G\in\mathcal C^-\): \(-f=\varepsilon_Gf=f\varepsilon_F=f\), so \(2f=0\), \(f=0\) (\(k=\mathbb C\)); same for all shifts. (c) \(e=(1+\varepsilon_F)/2\) is an idempotent of \(F\) in \(h\mathcal C\); \(\mathcal C\) is idempotent complete (definition of \(\mathcal C\), criterion.html Categories; thick-spans.html §1), so \(F\simeq F_1\oplus F_2\) with \(e\) the projector onto \(F_1\); by 1, \(\varepsilon_{F_1}=p\varepsilon_Fi=p(2e-1)i=\mathrm{id}\), \(\varepsilon_{F_2}=-\mathrm{id}\). \(\square\)

Reformulation (proved). Let \(\bar G=\mathrm{PGL}_2\) act on \(X\) by \(g\cdot A=gAg^{\mathsf T}/\det g\) (preserves \(\det A=1\); scalars act trivially; extends the \(\mathrm{SL}_2\)-action). Stabilizer of \(I\): \(\{gg^{\mathsf T}=\det(g)I\}/\mathbb G_m=\mathrm{SO}_2/\mu_2=\mathrm{PSO}_2\) (connected; it is \((\mathrm{PGL}_2^\theta)^\circ\), index 2 in \(\mathrm{PO}_2\)). Over \(\mathbb C\), \(\mathcal O^\times=\mathcal O^{\times2}\), so \(\mathrm{PGL}_2(\mathcal O)=\mathrm{SL}_2(\mathcal O)/\mu_2\): same orbits on \(LX\), stabilizers \((\mu_2\times t^m\mathcal O)/\mu_2\) connected. The map \(p:L^+\mathrm{SL}_2\backslash LX\to L^+\mathrm{PGL}_2\backslash LX\) is a \(\mu_2\)-gerbe; \(p_*p^*=(-)\otimes H^*(B\mu_2,k)=\mathrm{id}\) in characteristic 0, so \(p^*\) is fully faithful, and its essential image is exactly the \(\mu_2\)-inertia-trivial part: \(\bar{\mathcal C}:=\mathrm{Shv}_{\mathrm{cons}}(L^+\mathrm{PGL}_2\backslash LX)\xrightarrow{\ p^*,\ \sim\ }\mathcal C^+\). (Note \(X\neq\mathrm{PGL}_2/\mathrm{PO}_2\): the latter is the quotient \(X/\{\pm1\}=\mathbb P(\mathrm{Sym}_2)^{\mathrm{nondeg}}\), the row 'PGL2/PO2' of the Summary; here \(X=\mathrm{PGL}_2/\mathrm{PSO}_2\), its double cover.)

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