9 · Result
The orbit closure relation: statement and proof
This is the general version of what the running example showed as a triangle inequality. Each piece of notation is followed by its value in \(\mathrm{SL}_2/\mathrm{SO}_2\).
1 Notation
Definition
- \(G\) is connected reductive over \(\mathbb C\), \(\theta\) is an involution, \(X=G/G^\theta\) with base point \(x_0\), \(K=L^+G\), and \(LG=G(F)\).
- Fix \(S\subset T\subset B\subset P\): a maximal \(\theta\)-split torus \(S\), a \(\theta\)-stable maximal torus \(T\), and a parabolic \(P\) with \(P\cap\theta(P)=Z_G(S)\). (So \(P\) is \(\theta\)-split; the draft says “minimal \(\theta\)-stable”.) Example: \(S=T\) diagonal.
- \(\Lambda_S^+=X_*(S)\cap\Lambda_T^+\), and \(\mu\le\nu\) means that \(\nu-\mu\) is a non-negative integral sum of simple coroots. Example: \(\{(a,-a):a\ge0\}\), the ladder \(0<1<2<\cdots\).
- \(\tau(gx_0)=g\,\theta(g)^{-1}\), so \(\tau(g\cdot x)=g\,\tau(x)\,\theta(g)^{-1}\). Example: \(\tau(A)=A\) and \(g\cdot A=gAg^t\).
- \(D_{\le\lambda}\)\(=\bigsqcup_{\mu\le\lambda}Kt^\mu K\subset LG\), a set of loops (not the Hecke operator). Example: \(D_{\le m}\), the \(g\) with poles of order \(\le m\), i.e. \(\operatorname{dist}(v_0,gv_0)\le2m\).
- \(r(\lambda):=\lambda-\theta\lambda\). Example: \(r(m,-m)=(2m,-2m)\).
- \(\eta^+\) is the dominant \(W\)-conjugate of \(\eta\), and \(\lambda^\theta:=(-\theta\lambda)^+\). Example: \(\theta\lambda=-\lambda\), so \(\lambda^\theta=\lambda\).
2 Three cited facts
Proved (cited)
(F1) [Chen–Yi, Prop. 17] \(LX=\bigsqcup_{\nu\in\Lambda_S^+}\mathcal O_\nu\), with \(\mathcal O_\nu=K\cdot x_\nu\) and \(\tau(x_\nu)=t^\nu\). Since \(\theta(K)=K\), this gives \(\tau(\mathcal O_\nu)\subset Kt^\nu K\).
(F2) [Chen–Yi, Thm. 18] \(\overline{\mathcal O}_\nu=\bigsqcup_{\mu\in\Lambda_S^+,\ \mu\le\nu}\mathcal O_\mu\).
(F3) [Zhu, (2.1.17)] \(D_{\le\lambda}\cdot D_{\le\mu}\subset D_{\le\lambda+\mu}\) for \(\lambda,\mu\in\Lambda_T^+\). Example: the triangle inequality \(\operatorname{dist}(v_0,ghv_0)\le\operatorname{dist}(v_0,gv_0)+\operatorname{dist}(v_0,hv_0)\).
(F2) [Chen–Yi, Thm. 18] \(\overline{\mathcal O}_\nu=\bigsqcup_{\mu\in\Lambda_S^+,\ \mu\le\nu}\mathcal O_\mu\).
(F3) [Zhu, (2.1.17)] \(D_{\le\lambda}\cdot D_{\le\mu}\subset D_{\le\lambda+\mu}\) for \(\lambda,\mu\in\Lambda_T^+\). Example: the triangle inequality \(\operatorname{dist}(v_0,ghv_0)\le\operatorname{dist}(v_0,gv_0)+\operatorname{dist}(v_0,hv_0)\).
3 Theorem
Proved here (stated without proof in
Satake_full/satake_full.tex, “Symmetric-space support”)
For \(\lambda\in\Lambda_T^+\) and \(\nu\in\Lambda_S^+\):
\[
\text{(i)}\quad D_{\le\lambda}\cdot\overline{\mathcal O}_\nu\ \subset \bigsqcup_{\substack{\eta\in\Lambda_S^+\\ \eta\le \nu+\lambda+\lambda^\theta}}\mathcal O_\eta .
\]
\[
\begin{aligned}
\text{(ii)}\quad \text{if } -\theta\lambda\in\Lambda_T^+:\qquad & r(\lambda):=\lambda-\theta\lambda\in\Lambda_S^+,\\
& D_{\le\lambda}\cdot\overline{\mathcal O}_\nu\subset\overline{\mathcal O}_{\nu+r(\lambda)},\\
& t^\lambda\cdot x_\nu\in\mathcal O_{\nu+r(\lambda)} .
\end{aligned}
\]
In the example
\(r(\lambda)=2\lambda\), so (ii) says \(D_{\le m}\cdot\overline{\mathcal O}_a\subset\overline{\mathcal O}_{a+2m}\), with the top orbit attained. This is (T3) in the example, Part III.4 Proof
Let \(g\in D_{\le\lambda}\) and \(x\in\overline{\mathcal O}_\nu\). The right-hand column says what each step is in the tree.
| step | general | in the tree (\(\mathrm{SL}_2/\mathrm{SO}_2\)) |
|---|---|---|
| 1 | \(\tau(gx)=g\cdot\tau(x)\cdot\theta(g)^{-1}\) | \(\ell_{gA}=g\ell_A\) (T1) |
| 2 | \(\tau(x)\in D_{\le\nu}\), by (F1) and (F2) | \(\ell_A\) meets the ball of radius \(a\) about \(v_0\) |
| 3 | \(\theta(g)^{-1}\in D_{\le\lambda^\theta}\) | \(g^t\) has the same pole order as \(g\) |
| 4 | \(\tau(gx)\in D_{\le\lambda}D_{\le\nu}D_{\le\lambda^\theta}\subset D_{\le\kappa}\), with \(\kappa=\nu+\lambda+\lambda^\theta\), by (F3) twice | \(m+a+m\) |
| 5 | intersect with \(\tau(LX)\), using (F1) | read off the orbit |
| 6 | if \(-\theta\lambda\) is dominant, \(\kappa=\nu+r(\lambda)\); apply (F2) | \(\kappa=a+2m\) |
Details.
- The map \(g\mapsto\theta(g)^{-1}\) is an automorphism of \(LG\) preserving \(K\). It sends \(Kt^\mu K\) to \(Kt^{-\theta\mu}K=Kt^{(-\theta\mu)^+}K\), and hence carries \(D_{\le\lambda}\), the closure of \(Kt^\lambda K\), onto \(D_{\le\lambda^\theta}\).
- \(\tau(LX)=\bigsqcup_\eta\tau(\mathcal O_\eta)\) with \(\tau(\mathcal O_\eta)\subset Kt^\eta K\). Since Cartan cells are disjoint, \(\tau(LX)\cap D_{\le\kappa}=\bigsqcup_{\eta\le\kappa}\tau(\mathcal O_\eta)\). Since \(\tau\) is an embedding, (i) follows.
- \(\theta r(\lambda)=-r(\lambda)\), so \(r(\lambda)\in X_*\big((T^{\theta=-1})^\circ\big)=\Lambda_S\). It is dominant as a sum of dominant coweights. Finally, \(\tau(t^\lambda x_\nu)=t^\lambda t^\nu t^{-\theta\lambda}=t^{\nu+r(\lambda)}\). \(\square\)
5 Corollary (the draft's actual target)
Proved here
If \(-\theta\lambda\) is dominant, then \(\operatorname{Supp}(V_\lambda * \mathrm{IC}_0)\subset\overline{\mathcal O}_{r(\lambda)}\). This combines \(\operatorname{Supp}V_\lambda\subset\overline{\mathrm{Gr}}_{\lambda}\) and \(\operatorname{Supp}\mathrm{IC}_0\subset\mathcal O_0=L^+X\) with the support bound of Part I, step 6 (the draft's step (3), still marked “find reference”).
6 Dictionary to the draft's wording
Heuristic
The draft writes \(X_{\ge\mu}\cdot L^+Gt^\lambda L^+G\subset X_{\ge\mu+\overline\lambda}\) with \(\lambda\in\Lambda_G^-\), following Sakellaridis–Venkatesh, Lemma 5.5.2. Under the norm-balls dictionary, \(X_{\ge\mu}\) corresponds to \(\overline{\mathcal O}_\nu\), the antidominant \(\lambda\) to its dominant conjugate, and \(\overline\lambda\) to \(r(\lambda)\). Part (ii) is then the draft's statement. The identification of norm balls with orbit closures is verified only in the example.