Norm balls

\(\operatorname{val}(3t^2+t^5)=2\), \(\operatorname{val}(t^{-1}+7)=-1\), \(\operatorname{val}(1+t)=0\). Let \(z(A)=A_{22}\), a polynomial function on \(X\), and \(A=\begin{pmatrix}1&t^{-1}\\t^{-1}&1+t^{-2}\end{pmatrix}\in LX\). Then \(z(A)=1+t^{-2}\) and \(\operatorname{val}z(A)=-2\). For \(A_{11}\) instead, \(\operatorname{val}A_{11}(A)=0\). The matrix has \(\operatorname{val}A=-2\), so pole order 2.

SV work over a \(p\)-adic field with \(|f|=q^{-\operatorname{val}f}\), \(q\) the size of the residue field. Over \(\mathbb C(\!(t)\!)\) use the same formula with any fixed \(q>1\); below we work with \(\operatorname{val}\) directly.

Steps 1–3 hold for any closed \(H\subset\mathrm{SL}_2\): \(V_n\) occurs in \(k[\mathrm{SL}_2/H]\) exactly \(\dim V_n^H\) times. Only step 4 changes.

• \(H=U=\left\{\left(\begin{smallmatrix}1&u\\0&1\end{smallmatrix}\right)\right\}\). \(\mathrm{SL}_2/U\cong\mathbb C^2\smallsetminus\{0\}\) via \(gU\mapsto ge_1\). \(V_n^U=\mathbb C\,e_1^{\,n}\) (the highest weight line), so every \(V_n\) occurs once. Direct check: functions on \(\mathbb C^2\smallsetminus\{0\}\) extend to \(\mathbb C^2\) (Hartogs), so \(k[\mathbb C^2\smallsetminus0]=\mathbb C[v_1,v_2]\), and its degree-\(n\) part is \(\mathrm{Sym}^n((\mathbb C^2)^*)\cong V_n\).
• \(H=N(T)=T\sqcup wT\), \(w=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\). \(V_n^T=\mathbb C\,e_1^{\,n/2}e_2^{\,n/2}\) for \(n\) even, and \(w\) sends \(e_1\mapsto e_2\), \(e_2\mapsto-e_1\), so it multiplies this vector by \((-1)^{n/2}\). Fixed iff \(4\mid n\).
• \(H=\mathrm{SL}_2\): \(V_n^H=0\) unless \(n=0\); a point has only constants.

The first three rows reappear in step 3 as three different lattices \(\mathcal X(X)\subset X^*(T)\).

\(X^*(T)=\mathbb Z\varpi\), and \(\Lambda_T=\{\lambda_m:m\in\mathbb Z\}\) with \(\lambda_m(s)=\operatorname{diag}(s^m,s^{-m})\), written \((m,-m)\). Then \[ (k\varpi)\big(\lambda_m(s)\big)=s^{mk},\qquad \langle(m,-m),k\varpi\rangle=mk . \] Check with \(f=z=A_{22}\), weight \(2\varpi\) (step 2): the \((2,2)\)-entry of \(t^\lambda\cdot A=t^\lambda At^\lambda\) is \(t^{-2m}z\), and \(\langle(m,-m),2\varpi\rangle=2m\).

Example 3.2 (equal). \(X=\mathrm{SL}_2/U=\mathbb C^2\smallsetminus\{0\}\) (Example 2.2). The open \(B\)-orbit is \(\{v_2\ne0\}\); take \(x_0=e_2\). For \(b=\left(\begin{smallmatrix}s&u\\0&s^{-1}\end{smallmatrix}\right)\), \(be_2=(u,s^{-1})^t=e_2\) forces \(b=1\). So \(T_{x_0}=1\) and \[ \mathcal X(X)=X^*(T)=\mathbb Z\varpi,\qquad f_\varpi=v_2,\quad v_2(b^{-1}v)=s\,v_2(v). \]

Example 3.3 (index 2; the running example). \(X=\mathrm{SL}_2/\mathrm{SO}_2\), \(B\) acting by \(A\mapsto bAb^t\). The open \(B\)-orbit is \(\{z\ne0\}\); take \(x_0=I\). Then \(B_I=\{b\in B:bb^t=I\}\): upper triangular and orthogonal means diagonal with entries \(\pm1\), so \(T_{x_0}=\{\pm I\}\). Since \((k\varpi)(-I)=(-1)^k\), \[ \mathcal X(X)=2\mathbb Z\varpi\subsetneq\mathbb Z\varpi=X^*(T),\qquad A_X=T/\{\pm I\}. \] This \(\{\pm I\}\) is the source of every factor 2 on these pages.

Example 3.4 (smaller rank). \(X=\mathrm{SL}_2\) under \(G=\mathrm{SL}_2\times\mathrm{SL}_2\), \((g_1,g_2)\cdot x=g_1xg_2^{-1}\), with \(T\times T\subset B\times B\). The open orbit is the big cell \(BwB\); take \(x_0=w=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\). Then \(b_1wb_2^{-1}=w\) forces \(b_2=w^{-1}b_1w\in B\cap w^{-1}Bw=T\), so \(T_{x_0}=\{(t,t^{-1}):t\in T\}\), a one-dimensional torus. A character \((a\varpi,b\varpi)\) kills it iff \(a=b\): \[ \mathcal X(X)=\mathbb Z(\varpi,\varpi)\subsetneq\mathbb Z\varpi\oplus\mathbb Z\varpi=X^*(T\times T),\qquad f_{(\varpi,\varpi)}=x_{21}. \] Rank 1 inside rank 2. (Contrast page: group \(\mathrm{SL}_2\).)

By step 2 the eigenfunctions in \(k[X]\) are the highest weight vectors, the multiples of \(z^n\), and every eigenfunction in \(k(X)\) is a ratio of these [Br, Prop. 2.8(i)]. So with \(\chi:=2\varpi\), \[ \mathcal X(X)^+=\mathbb N\chi,\quad f_{n\chi}=z^n;\qquad \mathcal X(X)=\mathbb Z\chi;\qquad \mathfrak a_X=\operatorname{Hom}_{\mathrm{Grp}}(\mathbb Z\chi,\mathbb Q)\xrightarrow{\ \sim\ }\mathbb Q,\ \ \mu\mapsto\langle\mu,\chi\rangle . \] So \(\mathfrak a_X\) is a line (\(X\) has rank one). For \(\lambda=(m,-m)\), \(\overline\lambda=\langle(m,-m),2\varpi\rangle=2m\): The factor 2 in \(\overline\lambda\) is the index \([X^*(T):\mathcal X(X)]=2\) of Example 3.3. Equivalently \(\Lambda_T\to\Lambda_X:=X_*(A_X)\cong\mathbb Z\) is \(\lambda\mapsto2m\).

In \(\mathrm{SL}_2/\mathrm{SO}_2\), \(z^n\) is regular for \(n\ge0\), but \(z^{-1}\) is not, since \(z\) vanishes at the point \(\left(\begin{smallmatrix}0&i\\i&0\end{smallmatrix}\right)\in X\). So \[ \mathcal X(X)^+=\mathbb N\chi\subsetneq\mathbb Z\chi=\mathcal X(X),\qquad \mu\succeq\mu'\iff\langle\mu,\chi\rangle\ge\langle\mu',\chi\rangle , \] the usual \(\ge\) on \(\mathbb Q\). Every two elements are comparable.

\(G=\mathrm{SL}_3\), \(X=\mathrm{SL}_3/U\), \(T\) diagonal, \(B\) upper triangular. Let \(\varpi_1,\varpi_2\in X^*(T)\) send \(\operatorname{diag}(s_1,s_2,s_3)\) to \(s_1\), \(s_1s_2\). As in Example 3.2, \(T_{x_0}=1\), and the regular eigenfunctions are generated by \[ f_{\varpi_2}(gU)=g_{31},\qquad f_{\varpi_1}(gU)=\det\begin{pmatrix}g_{21}&g_{22}\\g_{31}&g_{32}\end{pmatrix},\qquad \mathcal X(X)^+=\mathbb N\varpi_1\oplus\mathbb N\varpi_2 . \] Here \(\mathfrak a_X=\Lambda_T\otimes\mathbb Q=\{(a_1,a_2,a_3)\in\mathbb Q^3:\sum a_i=0\}\), and \(\langle\mu,\varpi_1\rangle=a_1\), \(\langle\mu,\varpi_2\rangle=a_1+a_2\). So \[ \mu\succeq0\iff a_1\ge0\ \text{and}\ a_1+a_2\ge0\iff\mu\in\mathbb Q_{\ge0}\alpha_1^\vee+\mathbb Q_{\ge0}\alpha_2^\vee, \] with \(\alpha_1^\vee=(1,-1,0)\), \(\alpha_2^\vee=(0,1,-1)\): on \(G/U\), \(\succeq\) is the dominance order used in step 7.

\(X=\mathbb C^2\smallsetminus\{0\}\), \(\chi=\varpi\), \(f_\varpi=v_2\) (Example 3.2). For \(v\in LX=F^2\smallsetminus\{0\}\), \[ N_\varpi(v)=-\min(\operatorname{val}v_1,\operatorname{val}v_2),\qquad \|v\|_\varpi=\max(|v_1|,|v_2|), \] the standard sup-norm on \(F^2\), which \(K=\mathrm{SL}_2(\mathfrak o)\) preserves.

Proof. \((\kappa^{-1}v)_2=w\cdot v\), where \(w\) is the second row of \(\kappa^{-1}\), a vector in \(\mathfrak o^2\); so \(\operatorname{val}(w\cdot v)\ge\min_i\operatorname{val}v_i\). The rows \(w=e_2\) (\(\kappa=I\)) and \(w=e_1\) (\(\kappa^{-1}=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\)) give \(v_2\) and \(v_1\), so equality is attained. \(\square\)

The last two lie in the same orbit \(\mathcal O_1\). The single function \(z\) cannot see the pole of \(\operatorname{diag}(t^{-1},t)\); after moving by \({\kappa}=w\) it can.

With \(t^\lambda=\operatorname{diag}(t^m,t^{-m})\), \[ t^\lambda\cdot A=\begin{pmatrix}t^{2m}x&y\\ y&t^{-2m}z\end{pmatrix},\qquad N_\chi(t^\lambda\cdot A)\le N_\chi(A)+2m=N_\chi(A)+\langle\lambda,\chi\rangle, \] with equality when \(z\) carries the worst pole, and strict inequality when \(x\) does.

Only \(\chi\) matters, since \(N_{n\chi}=nN_\chi\). With \(s=\langle\mu,\chi\rangle\in\mathbb Q\): \[ X_{\succeq s}=\{A:\ \text{pole order of }A\le-s\},\qquad X_{\succeq-a}=\overline{\mathcal O}_a, \] by step 4 and the orbit closures. The ball is nonempty only for \(s\le0\). In the tree it is the set of lines meeting the ball of radius \(a\) about \(v_0\) (Part III of the example). The ball index \(\mu\) runs opposite to the orbit index: \(\mu=-a\leftrightarrow\nu=(a,-a)\).

With \(\lambda=(m,-m)\), \(\overline\lambda=2m\) and \(\mu=-a\): \[ D_{\le m}\cdot X_{\succeq-a}\subset X_{\succeq-a-2m},\qquad\text{i.e.}\qquad D_{\le m}\cdot\overline{\mathcal O}_a\subset\overline{\mathcal O}_{a+2m}. \] This is step (1) of the Proposition in the example, and the factor 2 is \(\langle\lambda,\chi\rangle\) with \(\chi=2\varpi\).