Seminar Sophus Lie Version of July 22, 1995 1 (1991) 135–141

defVersion of July 22, 1995

The hyperboloid as ordered symmetric space

Joachim Hilgert

Abstract: We study the geometry of the one sheeted hyperboloid viewed as an ordered symmetric space. The geometric results are indispensible prerequesites for the harmonic analysis of causal integral equations on such spaces. The present note may be used as an introduction to the geometric parts of the two forthcoming papers [1, 2].

Consider ℝ^{3} with the Lorentzian metric

(1)  (x_{0}, x_{1}, x_{2}) · (y_{0}, y_{1}, y_{2}) = −x_{0}y_{0} + x_{1}y_{1} + x_{2}y_{2}.

This metric induces an ordering on ℝ^{3} via

(2)  x ≥ y ⇔ x_{0} ≥ y_{0} and (x_{0} − y_{0})^{2} ≥ (x_{1} − y_{1})^{2} + (x_{2} − y_{2})^{2}.

In this note we study various properties of the hyperboloid

X = {x ∈ ℝ^{3} | x · x = 1}

Figure 1

which are related to the ordering of X that comes from ≥. We identify ℝ^{3} with 𝔰𝔩(2, ℝ) via

(x_{0}, x_{1}, x_{2}) ↔ ( \begin{matrix}x_{1} & x_{0} + x_{2} \\ −x_{0} + x_{2} & −x_{1} \end{matrix} )

so that the metric (1) gets identified with a multiple of the Cartan-Killing form of 𝔰𝔩(2, ℝ). We let G = Sl(2, ℝ) act on X via conjugation and fix a base point

e = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Then the stabilizer H of e is given by

H = \{( \begin{matrix}c & s \\ s & c \end{matrix} ) | c^{2} − s^{2} = 1\}

and X = G/H.

The homogeneous space X carries the structure of a pseudo Riemannian symmetric space via the involution

\begin{matrix}τ: G & → G \\ ( \begin{matrix}a & b \\ c & d \end{matrix} ) & ↦ ( \begin{matrix}d & c \\ b & a \end{matrix} ) . \end{matrix}

In fact, τ is just conjugation by

w = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )

and H is the group of fixed points G^{τ}. Note that

\begin{matrix}θ: G & → G \\ ( \begin{matrix}a & b \\ c & d \end{matrix} ) & ↦ ( \begin{matrix}d & −c \\ −b & a \end{matrix} ) . \end{matrix}

is a Cartan involution commuting with τ. The group of its fixed elements

K = \{( \begin{matrix}c & s \\ −s & c \end{matrix} ) | c^{2} + s^{2} = 1\}

is a maximal compact subgroup of G. On the level of Lie algebras we have

θ ( \begin{matrix}x & y \\ z & −x \end{matrix} ) = ( \begin{matrix}−x & −z \\ −y & x \end{matrix} )

τ ( \begin{matrix}x & y \\ z & −x \end{matrix} ) = ( \begin{matrix}−x & z \\ y & x \end{matrix} )

and

𝔨 = Lie(K) = \{( \begin{matrix}0 & y \\ −y & 0 \end{matrix} ) | y ∈ ℝ\}

𝔥 = Lie(H) = \{( \begin{matrix}0 & y \\ y & 0 \end{matrix} ) | y ∈ ℝ\}.

The −1-eigenspaces of θ and τ are given by

𝔭 = \{( \begin{matrix}x & y \\ y & −x \end{matrix} ) | x, y ∈ ℝ\}

𝔮 = \{( \begin{matrix}x & y \\ −y & −x \end{matrix} ) | x, y ∈ ℝ\}.

In order to find a suitable Cartan decomposition of G we consider the maximal abelian subspace

𝔞 = \{( \begin{matrix}x & 0 \\ 0 & −x \end{matrix} ) | x ∈ ℝ\}

of 𝔭 ∩ 𝔮 (which is also maximal abelian in 𝔭 and 𝔮!) and set

A = \{( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ) | t > 0\}

A^{+} = \{( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ) | t > 1\}.

Then

(3)  G = KA^{+}K ∪ K

and

(4)  G = HT H ∪ HAH ∪ HwAH ∪ lower dimensional things.

At this point we write T for K to stress the fact that K in our example is at the same time a maximal torus in G. The easiest way to see (4) is to consider the H -orbits on X:

Figure 2

The correct analogon of (3) for X comes from the ordering: Note first that on X we have

(2′)  x ≥ y ⇔ x_{0} ≥ y_{0} and x · y ≥ 1.

Let

X^{+} = {x ∈ X | x ≥ e}.

Figure 3

If π: G → X is the quotient map then S = π^{−1}(X^{+}) is a semigroup containing H. In fact, g, g′ ∈ S implies

π(gg′) = gπ(g′) ≥ ge = π(g) ≥ e.

Let S^{o} be the interior of S then

(5)  S^{o} = HA^{+}H.

Next we consider a suitable Iwasawa decomposition KAN of G and its analogon HAN which will be just an open subset of G containing S. To this end we set

N = \{( \begin{matrix}1 & r \\ 0 & 1 \end{matrix} ) | r ∈ ℝ\}

𝔫 = Lie(N) = \{( \begin{matrix}0 & r \\ 0 & 0 \end{matrix} ) | r ∈ ℝ\}

and note that

\begin{matrix}K × A × N & → KAN = G \\ (( \begin{matrix}c & s \\ −s & c \end{matrix} ) , ( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ) , ( \begin{matrix}1 & r \\ 0 & 1 \end{matrix} )) & ↦ ( \begin{matrix}ct & ctr + st−1 \\ −st & −str + ct^{−1} \end{matrix} ) \end{matrix}

is a diffeomorphism. Similarly

\begin{matrix}H × A × N & → HAN ⊆ G \\ (( \begin{matrix}c & s \\ s & c \end{matrix} ) , ( \begin{matrix}t & 0 \\ 0 & t−1 \end{matrix} ) , ( \begin{matrix}1 & r \\ 0 & 1 \end{matrix} )) & ↦ ( \begin{matrix}ct & ctr + st−1 \\ st & str + ct−1 \end{matrix} ) \end{matrix}

is a diffeomorphism onto an open subset of G. In order to show that S ⊆ HAN we realize S as a semigroup of nonlinear contractions. Let

G^{c} = SU(1, 1) = \{( \begin{matrix}a & b \\ \overline{b} & \overline{a} \end{matrix} ) | |a|^{2} − |b|^{2} = 1\}

K^{c} = (G^{c})^{θ} = \{( \begin{matrix}a & 0 \\ 0 & \overline{a} \end{matrix} ) | |a| = 1\}.

Then

𝔤^{c} = 𝔥 + i𝔮 𝔨^{c} = 𝔥 ∩ 𝔨 + i(𝔮 ∩ 𝔭)

and

G^{c}/K^{c} = D := {z ∈ ℂ | |z| < 1}

if we let G^{c} act on D via

( \begin{matrix}a & b \\ \overline{b} & \overline{a} \end{matrix} ) · z = \frac{az + b}{\overline{b}z + \overline{a}}.

Both G and G^{c} have the same complexification G_{ℂ} = Sl(2, ℂ). The hermitean symmetric space D can be viewed as G-orbit in the flag manifold

(6)  G_{ℂ}/K_{ℂ}^{c}\overline{P_{ℂ}^{c}} ≅ ℙ_{ℂ}^{1},

where

K_{ℂ}^{c} = \{( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) | a ∈ ℂ^{×}\}

\overline{P_{ℂ}^{c}} = \{( \begin{matrix}1 & 0 \\ c & 1 \end{matrix} ) | c ∈ ℂ\}.

The isomorphy (6) is easy to see via fractional linear transformations

Figure 4

Note that τ induces a complex conjugation on D and ℙ_{ℝ}^{1} . Restricting the Borel embedding D → ℙ_{ℝ}^{1} to the real points yields

H/\{±1\} ≅ D_{τ} = \{r ∈ ℝ | |r| < 1\} → ℙ_{ℝ}^{1} ≅ G/A\overline{N},

where \overline{N} = τN. Using D_{τ} = H ·0 = H ·D_{τ} and the doubl e coset decomposition (4) one now finds

(7)  S = {g ∈ G | g^{−1} · D_{τ} ⊆ D_{τ} }

which in turn implies S ⊆ HA\overline{N} and then

(8)  S ⊆ HAN

because of τ(S) = S^{−1} (this follows from (5)) and τ(HA\overline{N}) = HAN. In order to visualize the difference between S and HAN we consider

(8′)  X^{−} = S^{−1}/H ⊆ NAH/H ⊆ X.

The N -orbits on X, described by

( \begin{matrix}1 & r \\ 0 & 1 \end{matrix} ) ( \begin{matrix}x & y \\ z & −x \end{matrix} ) ( \begin{matrix}1 & −r \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}x + rz & −2xr − zr^{2} + y \\ z & −zr − x \end{matrix} )

simply are the parabolas which occur as intersections of X with the plane 𝔞 + 𝔫. In analogy to the Riemannian case we call these parabolas horocycles:

Figure 5

For the harmonic analysis on the unit disk it is important to study Poisson integrals of functions on its boundary, the circle, which is a homogeneous space under the action of the maximal compact subgroup. These integrals can be written in terms of the Iwasawa projection KAN → A. Note here that one should rather write an Iwasawa decomposition for G^{c} since it is not a general fact that

(9)  G/K ≅ G^{c}/K^{c}

is also a bounded symmetric domain—such ordered symmetric spaces are called Cayley type spaces since the isomorphism (9) is effected by a Cayley transform. In our case K acts transitively on the boundary of G/K which gets identified with the homogeneous space G/MAN = K/M = ℙ_{ℝ}^{1} , where M = {±1} (this also can be se en via fractional linear transformations). This is contrasted by the four H -orbits on G/MAN:

Figure 6 The orbit HMAN/MAN may be considered as a boundary of X^{+} :

Figure 7

and the harmonic analysis on X^{+} depends heavily on the “Iwasawa projection” (this is why (8) is so important)

\begin{matrix}HAN & → A \\ ( \begin{matrix}a & b \\ c & d \end{matrix} ) & ↦ ( \begin{matrix}\sqrt{a^{2} − c^{2}} & 0 \\ 0 & \sqrt{a^{2} − c^{2}}^{−1} \end{matrix} ) \end{matrix}

which should be compared to the usual Iwasawa projection

\begin{matrix}KAN & → A \\ ( \begin{matrix}a & b \\ c & d \end{matrix} ) & ↦ ( \begin{matrix}\sqrt{a^{2} + c^{2}} & 0 \\ 0 & \sqrt{a^{2} + c^{2}}^{−1} \end{matrix} ) . \end{matrix}

References

[1] Faraut, J., J. Hilgert and G. ’Olafsson, Harmonic analysis on ordered symmetric spaces, to appear.

[2] Hilgert J. and K.H. Neeb, Wiener-Hopf operators on ordered symmetric spaces, to appear.

Mathematisches Institut Universität Erlangen Bismarckstr. 1\frac{1}{2} 8520 Erlangen, Germany

Received July 31, 1991