Fibres over the Lie orbit space

Companion to the note · rank two · the Lie algebra \(M_3(\mathbb R)\) and its unitary forms \(\mathrm{Herm}(J_3)\)

The Lie algebra version of the companion page Fibres over the orbit space. Here the orbit space is a cone over the Whitney umbrella, and through the Cayley charts it is the local picture of the group at every full reduced origin.

\(\mathrm U(3)\)
chamber

\(\mathrm U(1,2)\)
chamber

\(\mathrm U(2,1)\)

(a)

(b)

(c)

(d) split corner:
\(y\) has real eigenvalues

(e) nonsplit corners:
\(y\) has eigenvalues \(\pm is\)

full reduced origin:
\(y=0\)

Figure 1 of the note, schematic. It is the projection of a sheet through the umbrella of the Lie orbit space; the section below draws it to scale. 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. In the labels, \(y\) is the traceless part of \(Y|_W\) on the closed orbit over the point, so \(\det y=q_1\).

semisimple (closed orbit), not regular regular (trivial stabilizer), not semisimple neither regular semisimple

Conventions

\(Y=\left(\begin{smallmatrix}y+uI_2&v\\w&d\end{smallmatrix}\right)\in M_3(\mathbb R)\) with \(y\in\mathfrak{sl}_2(\mathbb R)\), \(v\in\mathbb R^2\), \(w\in(\mathbb R^2)^*\) and \(u,d\in\mathbb R\). An element \(h\in\mathrm{GL}_2(\mathbb R)\) acts by conjugation by \(\mathrm{diag}(h,1)\), that is \((y,v,w)\mapsto(hyh^{-1},hv,wh^{-1})\), and \(u,d\) do not change. The vector \(e=e_3\) is the last basis vector, and \(W=\mathrm{span}(e_1,e_2)\). All examples have \(u=d=0\).
The invariants are \(u,d\) and \(q=(q_1,q_2,q_3)=(\det y,\ wv,\ wyv)\). Elements with the same invariants form one fibre, and every value of \((u,d,q)\in\mathbb R^5\) occurs.
An element is semisimple if its orbit is closed and regular if its stabilizer is trivial. Both properties belong to the whole orbit. Regular semisimple means both, which is \(\Delta\ne0\).
\(K(Y)=[v,yv]\) and \(R(Y)=\left(\begin{smallmatrix}w\\wy\end{smallmatrix}\right)\). Since \(y^2=-q_1\), one has \(R(Y)K(Y)=\left(\begin{smallmatrix}q_2&q_3\\q_3&-q_1q_2\end{smallmatrix}\right)\), so \(\det K(Y)\det R(Y)=-\Delta\) with \(\Delta=q_1q_2^2+q_3^2\). Also \(\det[e,Ye,Y^2e]=\det K(Y)\).
An arrow \(A\to B\) means that \(B\) lies in the closure of \(A\). Conjugating by \(\mathrm{diag}(h_1,h_2,1)\) multiplies \(Y_{ij}\) by \(h_i/h_j\) (with \(h_3=1\)).
Every matrix was checked numerically: the invariants, the stabilizer dimension, \(\det K(Y)\), \(\det R(Y)\), and the limit along each arrow.

Stabilizers of the closed orbits

Kind of point	Stabilizer \(S\) of the closed orbit	Elements fixed by \(S\)	Generic condition	Example
(a), (b) regular semisimple	\(\{1\}\)	every \(Y\)	\(\Delta\ne0\), that is \(\det K(Y)\det R(Y)\ne0\)	\(q=(-1,2,0)\), \((-1,0,2)\), \((-1,-2,0)\)
(c) wall	\(T_L=\{\mathrm{diag}(t,1,1)\}\cong\mathbb R^\times\), after moving the hidden line \(L\) to \(\mathbb Re_1\)	\(\mathrm{diag}(\mu,Y_\Pi)\) with \(\mu\in\mathbb R\) and \(Y_\Pi=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\) on \(\Pi=\mathrm{span}(e_2,e)\)	\(bc\ne0\). The wall is next to \(\mathrm U(3)\) if \(bc>0\), next to \(\mathrm U(1,2)\) if \(bc<0\), since \(bc=q_2\)	\(Y_\Pi=\left(\begin{smallmatrix}-1&1\\1&0\end{smallmatrix}\right)\) and \(\left(\begin{smallmatrix}-1&1\\-1&0\end{smallmatrix}\right)\)
(d) split corner	\(\{\mathrm{diag}(t_1,t_2,1)\}\cong(\mathbb R^\times)^2\)	\(\mathrm{diag}(\mu_1,\mu_2,\mu_3)\), all real	\(\mu_1\ne\mu_2\)	\(\mathrm{diag}(1,-1,0)\)
(e) nonsplit corner	\(\{\mathrm{diag}(a+bJ_0,1)\}\cong\mathbb C^\times\), \(J_0=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\)	\(\mathrm{diag}(y_0,\mu)\) with \(y_0J_0=J_0y_0\), that is \(y_0=\alpha+\beta J_0\): eigenvalues \(\alpha\pm i\beta\) on \(\mathbb C(1,\mp i)\)	\(\beta\ne0\)	\(y_0=J_0\)
full reduced origin	\(\mathrm{GL}_2(\mathbb R)\)	\(\mathrm{diag}(\lambda,\lambda,\mu)\)	none	\(Y=0\)

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.

Conventions on the unitary side

\(V=W\oplus\mathbb Ce\) with \(\langle x,x'\rangle=x'^*J_3x\) and \(J_3=\mathrm{diag}(J,1)\), so \(\langle e,e\rangle=1\) and \(e\perp W\). The unitary Lie side is \(\mathrm{Herm}(J_3)=\{Y:Y^*J_3=J_3Y\}=i\,\mathfrak u(V)\). There \(Y=\left(\begin{smallmatrix}y+uI_2&v\\v^*J&d\end{smallmatrix}\right)\) with \(y\) traceless and \(J\)-self-adjoint, \(v\in\mathbb C^2\) and \(u,d\in\mathbb R\). Every representative comes with its own \(J\).
\(J\) of signature \((2,0)\), \((1,1)\), \((0,2)\) gives \(\mathrm U(V)=\mathrm U(3)\), \(\mathrm U(2,1)\), \(\mathrm U(1,2)\). The group \(\mathrm U(W)=\{k:k^*Jk=J\}\) acts by conjugation by \(\mathrm{diag}(k,1)\). It is \(\mathrm U(2)\), which is compact, for \(\mathrm U(3)\) and \(\mathrm U(1,2)\), and \(\mathrm U(1,1)\) for \(\mathrm U(2,1)\).
The invariants are the same: \(u\), \(d\), \(q_1=\det y\), \(q_2=\langle v,v\rangle\) and \(q_3=\langle yv,v\rangle\), all real. An element of \(M_3(\mathbb R)\) and one of \(\mathrm{Herm}(J_3)\) match when their \((u,d,q)\) agree.
The Gram matrix \(K(Y)^*JK(Y)\) of \(v,yv\) is \(\left(\begin{smallmatrix}q_2&q_3\\q_3&-q_1q_2\end{smallmatrix}\right)\), so \(\Delta=-\det J\,|\det K(Y)|^2\). Hence \(Y\) is regular semisimple exactly when \(e\) is cyclic for \(Y\), and the sign of \(\Delta\) is \(-\operatorname{sign}\det J\): positive for \(\mathrm U(2,1)\), negative for \(\mathrm U(3)\) and \(\mathrm U(1,2)\), where \(q_2=\langle v,v\rangle\) has the sign of \(J\).
Weights. If \(q_1=-s^2<0\), then \(y\) has orthogonal eigenlines \(f_1,f_2\) for \(s,-s\), with \(\langle f_j,f_j\rangle=\epsilon_j=\pm1\). For \(v=v_1f_1+v_2f_2\) the weights \(a_j=\epsilon_j|v_j|^2\) satisfy \(q_2=a_1+a_2\) and \(q_3=s(a_1-a_2)\). Their signs are the signs of the eigenlines, and they determine the closed orbit. On the source side the same numbers are \(a_j=w_jv_j\) in an eigenbasis of \(y\).
If \(q_1=s^2>0\), the eigenlines of \(y\) for \(\pm is\) are isotropic, so \(J\) has signature \((1,1)\). In a hyperbolic eigenbasis, \(z=\bar v_1v_2\) gives \(q_2=2\,\mathrm{Re}\,z\), \(q_3=2s\,\mathrm{Im}\,z\) and \(\Delta=4s^2|z|^2\).
Since \(\mathrm U(W)\) is compact for \(\mathrm U(3)\) and \(\mathrm U(1,2)\), every fibre there is one closed orbit. Non-closed orbits exist only for \(\mathrm U(2,1)\), and only over nonsplit corners and the full reduced origin: these are the points where \(y\) can have an isotropic eigenline, or be nilpotent and nonzero.
When \(J\) is hyperbolic, \(k_t=\mathrm{diag}(t,\bar t^{-1},1)\) lies in \(\mathrm U(W)\) for \(t\in\mathbb C^\times\). Conjugating by \(k_t\) multiplies \(Y_{12},Y_{13},Y_{32}\) by \(|t|^2,t,\bar t\), and \(Y_{21},Y_{23},Y_{31}\) by \(|t|^{-2},\bar t^{-1},t^{-1}\).
Every matrix was checked numerically: \(Y^*J_3=J_3Y\), the common invariants, the stabilizer dimension in \(\mathrm U(W)\), and the limit along each arrow.

Stabilizers, and the two sides compared

Kind of point	Stabilizer of a closed orbit in \(\mathrm U(W)\)	Source side	Orbits in \(M_3(\mathbb R)\)	\(\mathrm U(3)\)	\(\mathrm U(2,1)\)	\(\mathrm U(1,2)\)
(a), (b) regular semisimple	\(\{1\}\)	\(\{1\}\)	1	1 for the group of its chamber, none for 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

The counts are the same as on the group page, because the Cayley charts of the last section identify the fibres. The stabilizers are again real forms of one 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 the section through it.

The orbit space in three dimensions

The orbit space \(\mathbb O(M_3(\mathbb R))=\mathbb R^2\times\mathbb R^3\) has the coordinates \((u,d,q)\). The passive scalars change nothing: translating \(Y\) by \(\mathrm{diag}(uI_2,d)\) commutes with \(\mathrm{GL}_2(\mathbb R)\) and \(\mathrm U(W)\), and keeps \(q\), the chambers and the fibres. So it is enough to draw \(\mathbb R^3\) with the coordinates \(q=(q_1,q_2,q_3)\). There the discriminant is the Whitney umbrella

\[\Delta=q_1q_2^2+q_3^2=0,\qquad\text{that is}\qquad q_3^2=(-q_1)\,q_2^2.\]

For \(q_1=-s^2<0\), \(y\) has the eigenvalues \(\pm s\), and the umbrella is the pair of planes \(q_3=\pm s\,q_2\). Away from their crossing these are walls: \(\mathbb C[Y]e\) is a plane. The crossing \(q_2=q_3=0\) is the line of split corners: \(e\) is an eigenvector of \(Y\).
For \(q_1=s^2>0\), \(y\) has the eigenvalues \(\pm is\), and only the handle \(q_2=q_3=0\) is left. It consists of nonsplit corners. Around it \(\Delta\ge0\), so only \(\mathrm U(2,1)\) is adjacent.
At \(q_1=0\), \(y\) is nilpotent. The two sheets meet tangentially along the fold \(q_3=0\), which consists of walls, and \(q=0\) is the full reduced origin.
Chambers: write \(a_{1,2}=\tfrac12(q_2\pm q_3/s)\). These are the weights of \(e\) on the two eigenlines of \(y\): \(q_2=a_1+a_2\), \(q_3=s(a_1-a_2)\) and \(\Delta=-4s^2a_1a_2\). The number of negative \(a_j\) is \(0\), \(1\) or \(2\) in the \(\mathrm U(3)\), \(\mathrm U(2,1)\) or \(\mathrm U(1,2)\) chamber. So \(\mathrm U(3)\) is the trough \(\{|q_3|<s\,q_2\}\), \(\mathrm U(1,2)\) is the trough \(\{|q_3|<-s\,q_2\}\), and \(\mathrm U(2,1)\) is the rest, including every point with \(q_1\ge0\) off the discriminant.
The closure of the \(\mathrm U(3)\) chamber is the image of the real symmetric matrices: for them \(a_j=v_j^2\ge0\). Its section at a fixed \(q_1<0\) is an infinite wedge, while on the group page it is a triangle.
The picture is a cone. \(Y\mapsto tY\) sends \(q\) to \((t^2q_1,t^2q_2,t^3q_3)\) and \(\Delta\) to \(t^6\Delta\), and it preserves the chambers. So all the sections at \(q_1<0\) look alike after rescaling, and so do all those at \(q_1>0\).

colour the chambers walls next to \(\mathrm U(1,2)\) the sheet \(S\) the section

click one to show it alone split corners nonsplit corners full reduced origin \(q=0\) the fibres above (click one) projection along \(q_2\)

The \(q_2\)-axis always points up, and \(-q_1\) grows to the right. Drag sideways to turn the picture about the \(q_2\)-axis, 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 \(\mathrm U(3)\) trough opens upward from the black line of split corners, and the \(\mathrm U(1,2)\) trough hangs below it. The dashed handle on the left consists of nonsplit corners. Under the umbrella hangs a copy of the sheet \(S\), moved straight down along \(q_2\), and under that its projection, which is Figure 1. The dashed vertical lines join the place of \(S\), the copy and the projection, and the thin teal and blue curves show where \(S\) cuts the walls. The first button lifts the copy back into place.

The section at \(q_1=-1\)

\(q_1\)-1.00

Click a point of the section to read off the weights and the closed unitary orbit over it.

The sheet \(S\) seen along the \(q_2\)-axis: Figure 1, drawn to scale

The floor of the picture on the left, seen from straight above: \(-q_1\) to the right and \(q_3\) up. Teal and blue curves are walls next to \(\mathrm U(3)\) and next to \(\mathrm U(1,2)\).

How Figure 1 arises. Let \(S=\{q_2=\beta(q_1)\}\) be a curved sheet. Here \(\beta\) vanishes for \(q_1\ge0\), is positive on \((-1,0)\), vanishes at \(-1\) and is negative after; in the picture \(\beta(q_1)=5\,q_1^2(1+q_1)\) on \([-1,0]\). In the coordinates \((-q_1,q_3)\) on \(S\):

For \(q_1>0\), \(S\) is the plane \(q_2=0\). It lies in \(\mathrm U(2,1)\), except for the axis \(q_3=0\): this is the dashed line (e).
At \(q_1=0\), \(S\) passes through the full reduced origin.
For \(-1<q_1<0\), \(S\) cuts the \(\mathrm U(3)\) trough in \(|q_3|<w(q_1)\), where \(w(q_1)=\sqrt{-q_1}\,\beta(q_1)\). This is the lens, bounded by walls (c). Since \(w\sim5\,(-q_1)^{5/2}\), the lens has a cusp at the origin.
At \(q_1=-1\), \(S\) crosses the axis at the split corner (d), the point of tab (d).
For \(q_1<-1\), \(S\) cuts the \(\mathrm U(1,2)\) trough in \(|q_3|<\sqrt{-q_1}\,|\beta(q_1)|\). This is the wedge.

So the projection \((q_1,q_2,q_3)\mapsto(-q_1,q_3)\) along the \(q_2\)-axis maps \(S\) with its partition onto Figure 1. On the group page the same figure comes from a sheet through the full reduced origin \(P_-\); the next section explains why the two pictures agree.

Figure 1	Where it is in \(\mathbb R^3\)	The elements	Fibre above
(a) the lens, \(\mathrm U(3)\)	the open trough \(\{q_1<0,\ \|q_3\|<s\,q_2\}\), \(s=\sqrt{-q_1}\)	both weights positive	\(q=(-1,2,0)\)
(a) the wedge, \(\mathrm U(1,2)\)	the open trough \(\{q_1<0,\ \|q_3\|<-s\,q_2\}\)	both weights negative	\(q=(-1,-2,0)\)
(b) \(\mathrm U(2,1)\)	the rest: \(\Delta>0\)	weights of opposite signs, or \(q_1\ge0\)	\(q=(-1,0,2)\)
(c) the boundary of the lens	the half-planes \(q_3=\pm s\,q_2\) with \(q_2>0\), and the half of the fold with \(q_2>0\)	\(\mathbb C[Y]e\) is a positive definite plane	\(q=(-1,1,-1)\)
(c) the edges of the wedge	the same with \(q_2<0\)	\(\mathbb C[Y]e\) is a plane of signature \((1,1)\)	\(q=(-1,-1,1)\)
(d) the right tip	the line \(q_2=q_3=0\), \(q_1<0\)	\(e\) is an eigenvector of \(Y\), and \(y\) has the real eigenvalues \(\pm s\)	\(q=(-1,0,0)\)
(e) the dashed line	the handle \(q_2=q_3=0\), \(q_1>0\)	\(e\) is an eigenvector of \(Y\), and \(y\) has the eigenvalues \(\pm is\)	\(q=(1,0,0)\)
the left tip	\(q=0\)	\(y=0\) on the closed orbit	\(Y=0\)

The Lie picture inside the group picture

For \(\xi\in\mathrm U(1)\) the Cayley chart \(c_\xi(Y)=\xi(1+iY)(1-iY)^{-1}\) is an equivariant isomorphism from \(\{P>0\}\), where \(P(Y)=|\det(1-iY)|^2\), onto \(U_\xi=\{\gamma:\det(\xi+\gamma)\ne0\}\), and likewise from \(\mathrm{Herm}(J_3)\) to \(\mathrm U(J_3)\) (the note's (C1)). It preserves matching (C3). So it identifies the part of the Lie orbit space where \(P>0\) with the part of \(\mathbb O(S_3(\mathbb R))\) where \(\det(\xi+\gamma)\ne0\), chambers with chambers, strata with strata and fibres with fibres. On eigenvalues it is \(\mu\mapsto\xi\frac{1+i\mu}{1-i\mu}\), which sends the real line to the unit circle.

Lie algebra	The elements	Group	The elements
regular semisimple, \(\Delta\ne0\)	\(e\) cyclic for \(Y\)	regular semisimple, \(\Delta_S\ne0\)	\(e\) cyclic for \(g\)
wall	\(\mathbb C[Y]e\) is a plane	wall	\(\mathbb C[g]e\) is a plane
split corner, \(q_1<0\)	\(y\) has real eigenvalues \(\pm s\)	split corner	\(g\|_W\) elliptic: eigenvalues \(\xi\frac{1\pm is}{1\mp is}\) on the unit circle
nonsplit corner, \(q_1>0\)	\(y\) has eigenvalues \(\pm is\)	nonsplit corner	\(g\|_W\) hyperbolic: eigenvalues \(\xi\frac{1\mp s}{1\pm s}\), off the circle
full reduced origin, \(q=0\)	closed orbit \(Y=\mathrm{diag}(u,u,d)\)	full reduced origin	\(g\|_W\) central: \(\mathrm{diag}(\lambda,\lambda,\mu)\)

The table is written for the passive scalars \(u=d=0\); in general the eigenvalues of \(Y|_W\) are \(u\pm s\) or \(u\pm is\). The split corner of the group page, \(\mathrm{diag}(i,-i,1)\), is \(c_1\) of the split corner \(\mathrm{diag}(1,-1,0)\) here, and \(\gamma=1\) is \(c_1(0)\).

One umbrella at every full reduced origin. The plane \(q=0\) of the Lie orbit space, with \(u,d\) free, goes to the torus of full reduced origins \(\mathrm{diag}(\lambda,\lambda,\mu)\) of the group. Near each of them the group picture is this cone. This is why Figure 1 of the note reads the same in both settings.
Infinity. Since \(|\det(\xi+c_\xi(Y))|^2=64/P(Y)\), the infinity of the Lie algebra goes to the divisor \(D_\xi=\{\det(\xi+\gamma)=0\}\), and the locus \(P=0\), where \(Y\) has an eigenvalue \(\pm i\), goes to the infinity of the group. So a Schwartz function on the group that vanishes near \(D_\xi\) is a Schwartz function on the Lie algebra that is flat along \(\{P=0\}\). This is where the note needs its flat factor (Lemma 7.52), and four centres \(\xi\in\{\pm1,\pm i\}\) cover the group.
The compact chamber. The \(\mathrm U(3)\) trough is not bounded, but \(c_\xi\) sends it to a relatively compact part of the compact \(\mathrm U(3)\) chamber of the group, whose closure meets \(D_\xi\). On the Lie algebra the Gaussian thickening of this chamber is not Schwartz; on the group it is. That is the point of Step 5 and Figure 12 of the note, and why \(f_T\) lives on the group.