Companion to the note · rank two · source side \(S_3(\mathbb R)\) and unitary side
Figure 1 of the note, schematic. Click anywhere, or drag the small ring, and the explorer shows the fibre over the point under the ring, on the side chosen above the tabs. A chamber gives a regular semisimple point, the boundary of the lens a wall next to \(\mathrm U(3)\), an edge of the blue wedge a wall next to \(\mathrm U(1,2)\), the dashed line a nonsplit corner, and the right and left tips of the lens the split corner and the full reduced origin. Near a wall, the dashed line or a tip, the ring snaps onto it. The last section draws this figure to scale.
| Kind of point | Stabilizer \(S\) of the closed orbit | Elements fixed by \(S\) | Generic condition | Example |
|---|---|---|---|---|
| (a), (b) regular semisimple | \(\{1\}\) | every \(\gamma\) | \(\Delta_S(\gamma)\ne0\), that is \(\det K_\gamma\det L_\gamma\ne0\) | \(b=\tfrac13,\ -1,\ 2\) |
| (c) wall | \(T_L=\{\mathrm{diag}(t,1,1)\}\cong\mathbb R^\times\), after moving the hidden line \(L\) to \(\mathbb Re_1\) | \(\mathrm{diag}(z,\gamma_\Pi)\) with \(|z|=1\) and \(\gamma_\Pi=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in S_2(\mathbb R)\) | \(bc\ne0\). The wall is next to \(\mathrm U(3)\) if \(|d|<1\), next to \(\mathrm U(1,2)\) if \(|d|>1\) | \(b=0\) and \(b=2i\) |
| (d) split corner | \(\{\mathrm{diag}(t_1,t_2,1)\}\cong(\mathbb R^\times)^2\) | \(\mathrm{diag}(z_1,z_2,\mu)\), all of absolute value one | \(z_1\ne z_2\) | \(\mathrm{diag}(i,-i,1)\) |
| (e) nonsplit corner | \(\{\mathrm{diag}(a+bJ_0,1)\}\cong\mathbb C^\times\), \(J_0=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\) | \(\mathrm{diag}(\gamma_0,\mu)\) with \(\gamma_0J_0=J_0\gamma_0\): eigenvalues \(\alpha,\bar\alpha^{-1}\) on \(\mathbb C(1,\pm i)\) | \(|\alpha|\ne1\) | \(\alpha=2\) |
| full reduced origin | \(\mathrm{GL}_2(\mathbb R)\) | \(\mathrm{diag}(\lambda,\lambda,\mu)\) | none | \(\gamma=1\) |
In every case the semisimple representatives are exactly the elements fixed by \(S\) that are generic for what \(S\) leaves free. When the generic condition fails, the element lies over a deeper point.
| Kind of point | Stabilizer of a closed orbit in \(\mathrm U(W)\) | Source side | Orbits in \(S_3(\mathbb R)\) | \(\mathrm U(3)\) | \(\mathrm U(2,1)\) | \(\mathrm U(1,2)\) |
|---|---|---|---|---|---|---|
| (a), (b) regular semisimple | \(\{1\}\) | \(\{1\}\) | 1 | 1 in the group of its chamber, none in the other two | ||
| (c) wall next to \(\mathrm U(3)\) | \(\mathrm U(1)\), rotating the hidden line | \(\mathbb R^\times\) | 3 | 1 | 1 | 0 |
| (c) wall next to \(\mathrm U(1,2)\) | \(\mathrm U(1)\), rotating the hidden line | \(\mathbb R^\times\) | 3 | 0 | 1 | 1 |
| (d) split corner | \(\mathrm U(1)^2\), rotating the two eigenlines in \(W\) | \((\mathbb R^\times)^2\) | 9 | 1 | 2 | 1 |
| (e) nonsplit corner | \(\{\mathrm{diag}(t,\bar t^{-1})\}\cong\mathbb C^\times\) | \(\mathbb C^\times\) | 3 | 0 | 3 | 0 |
| full reduced origin | all of \(\mathrm U(W)\) | \(\mathrm{GL}_2(\mathbb R)\) | 8 and a family | 1 | 3 and a family | 1 |
On the two sides the stabilizers are real forms of the same complex group: \(\mathbb R^\times\) and \(\mathrm U(1)\) of \(\mathbb C^\times\), \((\mathbb R^\times)^2\) and \(\mathrm U(1)^2\) of \((\mathbb C^\times)^2\), and \(\mathrm{GL}_2(\mathbb R)\) and \(\mathrm U(W)\) of \(\mathrm{GL}_2(\mathbb C)\). Over a split corner the four closed orbits match the four quadrants of Figure 1, and \(\mathrm U(2,1)\) has two of them: over a real group one fibre can contain several closed orbits that are conjugate only over \(\mathbb C\).
\(\mathbb O(S_3(\mathbb R))=\mathrm U(1)\times\mathbb C^2\) has real dimension five. To draw it, replace it by its real analogue \(\{1\}\times\mathbb R\times\mathbb C\), the slice
\[\Sigma=\{\det\gamma=1,\ \operatorname{tr}\gamma\in\mathbb R\},\]with coordinates \(h=\tfrac12(\operatorname{tr}\gamma-1)\) and \(b=e_0^t\gamma e_0=x+iy\). Fixing \(\det\gamma=1\) loses nothing. Multiplying \(\gamma\) by a scalar \(\zeta\in\mathrm U(1)\) commutes with \(\mathrm{GL}_2(\mathbb R)\), sends \(\mathrm{inv}=(d,c,b)\) to \((\zeta^3d,\zeta c,\zeta b)\), and preserves \(\Delta_S\) and the chambers. On \(\Sigma\) the characteristic polynomial is \((X-1)(X^2-2hX+1)\). So every \(\gamma\) has the eigenvalues \(1,\lambda,\lambda^{-1}\): \(\lambda=e^{i\theta}\) with \(\cos\theta=h\) if \(|h|\le1\), and \(\lambda=h+\sqrt{h^2-1}\), real, if \(|h|>1\). All eight fibres of the explorer lie in \(\Sigma\). On \(\Sigma\) the discriminant factors as
\[\Delta_S=4\,(x-h)\,\bigl((1-h)\,y^2-(1+h)\,(x-1)^2\bigr).\]The \(x\)-axis always points down. Drag sideways to turn the picture about it, and up or down to tilt it. The chambers are tinted: \(\mathrm U(3)\) teal, \(\mathrm U(2,1)\) orange and \(\mathrm U(1,2)\) blue; each wall is coloured on each side by the chamber that side faces. Click a chamber in the legend to show it alone. The thick edges of the solid are the axis \(b=1\) for \(|h|<1\) and the ellipse \(b=\lambda^{\pm1}\) on the plane; the dashed axis beyond \(P_\pm=(h,b)=(\pm1,1)\) consists of nonsplit corners. Under the slice hangs a copy of the sheet \(S\), moved straight down along \(x\), and under that its projection, which is Figure 1. The dashed vertical lines join the place of \(S\) in the slice, the copy and the projection, and the thin teal and blue curves on the walls show where \(S\) cuts them. The first button lifts the copy back into place.
The floor of the picture on the left, seen from straight above: \(h\) to the right and \(y=\operatorname{Im}b\) up. Teal and blue curves are walls next to \(\mathrm U(3)\) and next to \(\mathrm U(1,2)\).
How Figure 1 arises. Let \(S=\{x=1-\beta(h)\}\) be a curved sheet. Here \(\beta\) vanishes for \(h\le-1\), is positive on \((-1,0)\), vanishes at \(0\) and is negative after; in the picture \(\beta(h)=2.6\,(1+h)^2(-h)\) on \([-1,0]\). In the coordinates \((h,y)\) on \(S\):
So the projection \((h,x,y)\mapsto(h,y)\) along the \(x\)-axis maps \(S\) with its partition onto Figure 1, with \(h\) increasing to the right. The picture draws this literally. Under the slice hangs a copy of \(S\), moved straight down along \(x\), and under the copy lies its flat projection; all three sit at the same \((h,y)\).
| Figure 1 | Where it is in \(\Sigma\) | The elements | Fibre above |
|---|---|---|---|
| (a) the lens, \(\mathrm U(3)\) | the open solid; at height \(|h|<1\), the open triangle with vertices \(1,\lambda,\lambda^{-1}\) | no \(\beta_j\) negative: \(g\in\mathrm U(3)\) | \(b=\tfrac13\), \(h=0\) |
| (a) the wedge, \(\mathrm U(1,2)\) | at \(|h|<1\), the three regions at the vertices of the triangle; \(x>h\) if \(h>1\); \(x<h\) if \(h<-1\) | two \(\beta_j\) negative | \(b=2\), \(h=0\) |
| (b) \(\mathrm U(2,1)\) | the rest; at \(|h|<1\), the three regions across the sides of the triangle | one \(\beta_j\) negative | \(b=-1\), \(h=0\) |
| (c) the boundary of the lens | the faces of the solid: the elliptic disc \(\{x=h,\ h^2+y^2<1\}\) and two pieces of the conoid | \(\mathbb C[g]e\) is a positive definite plane | \(b=0\), \(h=0\), on the disc |
| (c) the edges of the wedge | the rest of the plane \(x=h\) and of the conoid | \(\mathbb C[g]e\) is a plane of signature \((1,1)\) | \(b=2i\), \(h=0\), on the plane |
| (d) the right tip | the edges of the solid: the axis \(b=1\) for \(|h|<1\), and the ellipse \(b=\lambda^{\pm1}\), \(x=h\) | \(e\) is an eigenvector of \(g\), and \(g|_W\) is elliptic | \(b=1\), \(h=0\) |
| (e) the dashed line | the axis \(b=1\) for \(|h|>1\) | \(e\) is an eigenvector of \(g\), and \(g|_W\) is hyperbolic, with eigenvalues \(\lambda^{\pm1}\) | \(b=1\), \(h=\tfrac54\) |
| the left tip | \(P_\pm=(h,b)=(\pm1,1)\) | \(g|_W=\pm1\) is central | \(P_+\), which is \(\gamma=1\) |
At \(P_-\), whose eigenvalues are \(1,-1,-1\), the slice is transverse to the torus of full reduced origins. The conoid \((1-h)y^2=(1+h)(x-1)^2\) is then a Whitney umbrella, as in Figure 2 of the note, and its handle is the nonsplit axis below \(P_-\). At \(P_+\), which is \(\gamma=1\), the slice is tangent to that torus, and the plane \(x=h\) passes through \(P_+\) as well. That is why Figure 1 is drawn at \(P_-\). The fibre of the full reduced origin tab lies over \(P_+\). Translating in the Cayley chart of Lemma 7.8 identifies the fibres over all full reduced origins, so that tab also describes the fibre over \(P_-\).