3 · Background · Part II of II

Before the theorem II: the symmetric variety and its orbits

This part is about \(X\) and the orbits of \(K\) on \(LX\). Notation from Part I: \(F\), \(\mathfrak o\), \(K=L^+G\), \(\Lambda_T^+\), \(\le\), \(D_{\le\lambda}\). Instances are in \(X=\mathrm{SL}_2/\mathrm{SO}_2\), in blue boxes.

1 The symmetric variety and the Cartan embedding

Definition \(\theta\) is an involution of \(G\), and \(X=G/G^\theta\) is a symmetric variety with base point \(x_0\). The Cartan embedding \(\tau:X\hookrightarrow G\), \(\tau(gx_0)=g\,\theta(g)^{-1}\), satisfies \[ \tau(g\cdot x)=g\,\tau(x)\,\theta(g)^{-1}. \]

Instance

\(\theta(g)=(g^t)^{-1}\), so \(G^\theta=\mathrm{SO}_2\) and \(\tau(g)=gg^t\). Hence \[ X\cong\{A\ \text{symmetric }2\times2,\ \det A=1\},\qquad g\cdot A=gAg^t,\qquad x_0=I . \] Pictures of \(X\) are on the example, Part I.

2 The loop space \(LX\)

Definition \(LX=X(F)\) with the action of \(K\). Its \(K\)-orbits play the role that Schubert cells \(\mathrm{Gr}_\lambda\) play for \(G\). The closed orbit is \(L^+X=K\cdot x_0\).

Instance

\(LX\) is the set of symmetric \(2\times2\) matrices of Laurent series with determinant 1, and \(L^+X\) the set of those without poles.

3 Orbits

Proved (cited) [Chen–Yi, Prop. 17] \(LX=\bigsqcup_{\nu\in\Lambda_S^+}\mathcal O_\nu\), with \(\mathcal O_\nu=K\cdot x_\nu\) and \(\tau(x_\nu)=t^\nu\). Here \(S\subset T\) is a maximal \(\theta\)-split torus and \(\Lambda_S^+=\Lambda_S\cap\Lambda_T^+\).

Instance

\(S=T\) and \(\Lambda_S^+=\{(a,-a):a\ge0\}\). Write \(\mathcal O_a:=\mathcal O_{(a,-a)}\); then \(\operatorname{diag}(t^a,t^{-a})\in\mathcal O_a\).

Orbits O0, O1, O2 drawn as lines at distance 0, 1, 2 from the base vertex of the Bruhat–Tits tree — Each orbit \(\mathcal O_a\) is the set of lines in the tree at distance \(a\) from \(v_0\) (proved in the example, Part III).

4 One number, five names

In the instance, the orbit index \(a\) of \(A\in LX\) appears in the literature under five names. (Earlier versions of these pages called it \(d(A)\).)

Table 1. name in the literature and value for \(A\in\mathcal O_a\) and where
name in the literature	value for \(A\in\mathcal O_a\)	where
pole order	\(a=-\min_{i,j}\operatorname{val}(A_{ij})\)	here
elementary divisors (Smith normal form)	\(A={\kappa}\,\operatorname{diag}(t^{-a},t^{a})\,{\kappa}'\), \({\kappa},{\kappa}'\in\mathrm{GL}_2(\mathfrak o)\)	standard
Chen–Yi index	\(\nu=(a,-a)\)	step 3
norm (Sakellaridis–Venkatesh)	\(N_\chi(A)=a\)	norm balls
tree distance	\(a=\operatorname{dist}(v_0,\ell_A)\)	example, Part III

Proved hereThe pole order is \(K\)-invariant (as in Part I, step 3) and equals \(a\) on \(\operatorname{diag}(t^a,t^{-a})\), so the first three agree. The last two are proved on their pages.

5 Orbit closures

Proved (cited) [Chen–Yi, Thm. 18] \(\overline{\mathcal O}_\nu=\bigsqcup_{\mu\in\Lambda_S^+,\ \mu\le\nu}\mathcal O_\mu\), for the dominance order \(\le\) of Part I. The draft quotes this in a footnote and asks whether it holds for every spherical \(X\).

Instance

\[ \overline{\mathcal O}_a=\{A:\ \text{pole order}\le a\}=\mathcal O_0\sqcup\mathcal O_1\sqcup\dots\sqcup\mathcal O_a . \] Heuristic A family can lose poles in the limit but not gain them: \[ A(s)=u_su_s^t=\begin{pmatrix}1& s\,t^{-1}\\ s\,t^{-1} & 1+s^2t^{-2}\end{pmatrix}\in\mathcal O_2\ (s\neq0),\qquad u_s=\begin{pmatrix}1&0\\ s\,t^{-1}&1\end{pmatrix},\qquad A(0)=I\in\mathcal O_0 . \]

6 Hecke translates: where Parts I and II meet

Definition The Hecke translate of an orbit closure is the set \[ D_{\le\lambda}\cdot\overline{\mathcal O}_\nu:=\{g\cdot x:\ g\in D_{\le\lambda},\ x\in\overline{\mathcal O}_\nu\}\ \subset\ LX, \] with \(D_{\le\lambda}\subset LG\) the set of loops of size \(\le\lambda\). By Part I, step 6, it contains \(\operatorname{Supp}(V_\lambda*\mathcal F)\) for every \(\mathcal F\) supported on \(\overline{\mathcal O}_\nu\).

The question, in the instance

\[ \{gAg^t:\ g\in D_{\le m},\ A\in\overline{\mathcal O}_a\}\ \subset\ ? \]

Each entry of \(gAg^t\) is \(\sum_{k,l}g_{ik}A_{kl}g_{jl}\), with pole order at most \(m+a+m\). So the answer is \(\overline{\mathcal O}_{a+2m}\). The example, Part III shows the geometry behind the \(2m\); the theorem page does general symmetric \(X\).