§2 · Agent 2 · SL₂/SO₂: central inertia · 2026-10-17T02:07:01Z

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

proved audit: correct

Second audit: correct. Re-derived by hand; the stabilizer identity was also checked in exact arithmetic (sl2_slices_G1.py).

In the model of symmetric matrices of determinant one, the arc-group orbits are the sets of fixed minimal valuation, and every stabilizer away from the base orbit is \(\{\pm I\}\) times a connected pro-unipotent group. So the failure of H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf. is caused by the central element \(-I\).

Computation 2.1.

Let \(X=\{A\in\mathrm{Mat}_2:A=A^{\mathsf T},\ \det A=1\}\) with \(g\cdot A=gAg^{\mathsf T}\) and \(x_0=I\), so that the Cartan embedding \(\tau(g\SO_2)=gg^{\mathsf T}\) is the identity. Let \(F=\C(\!(t)\!)\), \(\cO=\C[\![t]\!]\), \(G=\SL_2\), and \(x_m=\operatorname{diag}(t^m,t^{-m})=t^{m\alpha^\vee}\), so \(\Lambda_S^+=\{m\alpha^\vee:m\ge0\}\). Then:

  1. \(\mathcal O_m=\{A\in X(F):\min\val(\text{entries of }A)=-m\}\) and \(\overline{\mathcal O_m}=\{\min\val\ge-m\}\).
  2. \(\Stab_{L^+G}(x_m)\cong\mu_2\times t^m\cO\) for \(m\ge1\), where \(t^m\cO\) is a connected pro-unipotent group, and \(\Stab_{L^+G}(x_0)=\SO_2(\cO)=L^+\mathbb G_m\) is connected. For \(m\ge1\), \(\pi_0\Stab(x_m)=\Z/2\), generated by the central element \(-I\).
  3. Consequently the heart of the orbit category on \(\mathcal O_m\), \(m\ge1\), is \(\Rep(\Z/2)\): the simple equivariant local systems are \(\underline k\) and the sign system \(\mathcal L_m^-\), on which \(-I\) acts by \(-1\).
  1. Orbits. Write \(A=t^{-m}A'\) with \(A'\) integral and having a unit entry. A binary symmetric form over \(\cO\) (with \(2\) invertible) having a unit entry represents a unit — if the unit is off-diagonal, evaluate at \((1,1)\) — hence splits: \(A'\simeq\operatorname{diag}(u_1,u_2t^{2m})\) under \(\GL_2(\cO)\)-congruence. Units of \(\C[\![t]\!]\) are squares, and the determinant forces the conjugating matrix to have determinant \(\pm1\) (correct the sign with \(\operatorname{diag}(1,-1)\)), so \(A\simeq\operatorname{diag}(t^{-m},t^m)=w\cdot x_m\) under \(\SL_2(\cO)\). The minimal valuation of the entries is a congruence invariant. This agrees with Chen–Yi, Prop. 17 and Thm. 18 (the closure of \(\mathcal O_m\) is the union of the \(\mathcal O_{m'}\), \(m'\le m\)).
  2. Stabilizer. Over \(F\), the stabilizer of \(x_m\) in \(\SL_2\) is
$$\SO(x_m)(F)=\Big\{g_{a,b}=\begin{pmatrix}a&bt^m\\-bt^{-m}&a\end{pmatrix}:a^2+b^2=1\Big\},$$

as one checks directly (\(g_{a,b}x_mg_{a,b}^{\mathsf T}=x_m\), \(\det g_{a,b}=a^2+b^2\)). Integrality of \(g_{a,b}\) means \(a\in\cO\) and \(b\in t^m\cO\). For \(m\ge1\), \(a^2=1-b^2\in1+t^{2m}\cO\), so \(a=\pm\sqrt{1-b^2}\); hence \(\Stab_{L^+G}(x_m)\cong\mu_2\times t^m\cO\), the component group being generated by \(g_{-1,0}=-I\). For \(m=0\) the stabilizer is \(\SO_2(\cO)\), connected. This matches Proposition III, step 4: \(\pi_0\Stab(x_\mu)=\pi_0Z_{G^\theta}(\mu)=\{\pm I\}\).

  1. Local systems. By the description of equivariant local systems on an orbit as representations of the component group of a stabilizer (Proposition III, discussion of H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.; Chen–Yi, Appendix A), the heart is \(\Rep(\Z/2)\) for \(m\ge1\). So H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf. fails exactly as the pages say, and the failure comes from a central element of \(G\).

Verified by computation: slices.py (stabilizer identity, numerically).

Builds on

  • —

Built on by

  • Lemma 2.2 · Central inertia splits \(\mathcal C_K\) into even and odd parts; the even part is the \(\PGL_2\)-equivariant category
  • Theorem 2.4 · \(\Csat\ne\mathcal C_K\) for \(\SL_2/\SO_2\): the sign local system on the first orbit is not generated
  • Computation 2.5 · Fibres of the convolution map are finite sets of tree vertices; the map is finite and an isomorphism over the top orbit
  • Computation 2.8 · Function level for \(\SL_2/T\): Lang count of \(K\)-orbits, the sign-class dictionary, and a Hecke dimension count for every \(q\)
  • Audit 7.1 · Audit: central inertia for \(\SL_2/\SO_2\) is a natural automorphism, and convolution is bimultiplicative
  • Correction 10.5 · The \(\SO_2\)-model of \(\SL_2/\SO_2\) over \(\F_q\) and Chen–Yi's splitness hypothesis
  • Lemma X2.1 · \(\sigma_n\) is translation by \(h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)\) with \(h^{-1}\mathrm{Fr}^n(h)=-I\)
  • Lemma X2.2 · Weil inertia lemma: \(\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\circ\sigma_n=\chi(-1)\,\operatorname{tr}_n\mathrm{IC}_{\chi,\nu}\)
  • Computation X2.3 · Stalks for \(\mu=\alpha^\vee,2\alpha^\vee\): \(\mathcal L_{\mathrm{sgn}}\) is Kummer on each slice branch
  • Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both
  • Proposition X2.6 · (SG) fails for \(\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)\) over every \(F\), \(q\) odd
Original record, as the agent wrote it at 02:07 UTC

Stabilizers of x_m in L+SL2: mu_2 times pro-unipotent, pi_0 generated by -I

Model. \(X=\{A\in\mathrm{Mat}_2: A=A^{\mathsf T},\ \det A=1\}\), \(g\cdot A=gAg^{\mathsf T}\), \(x_0=I\); in this model the Cartan embedding \(\tau(g\,\mathrm{SO}_2)=gg^{\mathsf T}\) is the identity, so \(x_\mu=t^{\mu}\): \(x_m:=\operatorname{diag}(t^m,t^{-m})=t^{m\alpha^\vee}\), \(\Lambda_S^+=\{m\alpha^\vee:m\ge0\}\).

Orbits (proved). \(\mathcal O_m=\{A\in X(F):\min\operatorname{val}(\text{entries of }A)=-m\}\), \(\overline{\mathcal O_m}=\{\min\operatorname{val}\ge -m\}\), \(F=\mathbb C((t))\), \(\mathcal O=\mathbb C[[t]]\).

  1. Write \(A=t^{-m}A'\) with \(A'\) integral having a unit entry. A binary form over \(\mathcal O\) (2 invertible) with a unit entry represents a unit (if the unit is off-diagonal, evaluate at \((1,1)\)), hence splits: \(A'\simeq\operatorname{diag}(u_1,u_2t^{2m})\) under \(\mathrm{GL}_2(\mathcal O)\)-congruence.
  2. Units of \(\mathbb C[[t]]\) are squares, so \(A\simeq\operatorname{diag}(t^{-m},t^{m})\) under \(\mathrm{SL}_2(\mathcal O)\), which is \(w\cdot x_m\). The minimal valuation is a congruence invariant. This agrees with Chen–Yi Prop. 17 and Thm. 18 (closure = all \(\nu\le\mu\)).

Stabilizer (proved). \(\operatorname{Stab}_{L^+G}(x_m)=\{g\in\mathrm{SL}_2(\mathcal O):gx_mg^{\mathsf T}=x_m\}\). Over \(F\), \(\mathrm{SO}(x_m)(F)=\{g_{a,b}=\begin{pmatrix}a&bt^{m}\\-bt^{-m}&a\end{pmatrix}:a^2+b^2=1\}\) (direct check: \(g_{a,b}x_mg_{a,b}^{\mathsf T}=x_m\), \(\det=a^2+b^2\)). Integrality: \(a\in\mathcal O\), \(b\in t^m\mathcal O\). For \(m\ge1\): \(a^2=1-b^2\in1+t^{2m}\mathcal O\), so \(a=\pm\sqrt{1-b^2}\) and $$\operatorname{Stab}_{L^+G}(x_m)\;\cong\;\mu_2\times t^m\mathcal O\qquad(m\ge1),\qquad \operatorname{Stab}_{L^+G}(x_0)=\mathrm{SO}_2(\mathcal O)=L^+\mathbb G_m\ \text{(connected)}.$$ The factor \(t^m\mathcal O\) is a connected pro-unipotent group; \(\pi_0\operatorname{Stab}(x_m)=\mathbb Z/2\) generated by the central element \(-I=g_{-1,0}\). (Numerically verified in the scratch script; matches criterion.html step 4: \(\pi_0\operatorname{Stab}(x_\mu)=\pi_0Z_{G^\theta}(\mu)=\{\pm I\}\).)

Consequence. By criterion.html (H3 discussion; draft Construction 3.34; Chen–Yi App. A, \(\mathrm{Rep}(\pi_0(S_j))\leftrightarrow\) equivariant local systems), \(\mathcal C_m^\heartsuit\simeq\mathrm{Rep}(\mathbb Z/2)\) for \(m\ge1\): the simple \(L^+G\)-equivariant local systems on \(\mathcal O_m\) are \(\underline k\) and the sign system \(\mathcal L_m^-\), on which \(-I\) acts by \(-1\). So H3 fails exactly as the pages say, and the failure is caused by a central element of \(G\).

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