CONTENTS

Characteristic Functions of LSA . . . . . . . . . . . . . .1

(1) LSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 (2) The Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 (3) Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 (4) The Characteristic Polynomial . . . . . . . . . . . . . .8 (5) Developing Images . . . . . . . . . . . . . . . . . . . . . . . . . . 16 (6) The Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

CHARACTERISTIC FUNCTIONS of LSA

August. 13. 1999

Abstract. The purpose of this paper is to introduce the well-known properties of characteristic functions on a Lie group with a left-invariant affinely flat structures.

Theorem 1. ([Hel],[Kim2]) Let p(x) = det(1 + ρ_{x}) :

A → ℝ. Then p(g · x) = ϕ(g)p(x), g ∈ G, where ϕ : G →

ℝ_{+} is a Lie group homomorphism given by ϕ(g) =

det(1 + ρ_{g·0}).

Theorem 2. ([Kim2]) Let us denote the developing image of G by Ω = ev_{0}(G). Then Ω is the connected component of {x ∈ A | p(x) = det(1 + ρ_{x}) ≠ 0} containing the origin 0 ∈ A.

Definition. A real function f : Ω → ℝ will be said to be a characteristic function if

(i) f is real analytic and positive on Ω ; (ii) There exists a Lie group homomorphism

ϕ : G → ℝ_{+} such that for all x ∈ Ω and g ∈ G,

(1-1)  f(g · x) = ϕ(g)f(x).

(1-1)

Ω ⟶^{f} ℝ_{+}
↓^{g-action} ↓^{ϕ(g)}
Ω ⟶^{f} ℝ_{+}

(1) LSA

A ⟶^{dΦ=(λ,Id)} 𝔞𝔣𝔣(A) = 𝔤𝔩(A) + A ⟶ A
↓^{exp} ↓^{exp}
G ⟶^{Φ=(f, q)} Aff(A) = GL(A) ⋉ A ⟶^{Ev_{0}} A^{−} = 𝔤
∥  ∥
G ⟶^{ev_{0} =D}  Ω = G · 0 ⊂ ℝ^{n}  ℝ^{n}
↓^{l_{g}}  ↓^{∃ Φ(g)∈Aff(ℝ^{n})}
G ⟶^{D}  Ω = D(G) ⊂ ℝ^{n}

(i) D = ev_{0} is covering, where 0 := D(e).

(ii) d(ev_{0})|_{e} = d(q) = q = id. (iii) λ = f = (df). A = (𝔤, λ) : LSA with λ_{x}(y) := xy =: ρ_{y}(x), and A^{−} = 𝔤.

(2) The Existence Assume that G acts simply transitively on Ω.

(2-1)  \begin{matrix}A^{−} = 𝔤 & ⟶^{s} & ℝ \\ ↓^{exp} & & ↓^{exp} \\ G & ⟶^{ϕ} & ℝ_{+} \\ ≅↓^{ev_{0}} & & ∥ \\ Ω & ⟶^{f} & ℝ_{+} \end{matrix}

Define f : Ω → ℝ_{+} by

(2-1)  f ∘ ev_{0} := ϕ,

f ∘ ev_{0}(h) = f(h · 0) = ϕ(h) for h ∈ G.

Claim. f(g · x) = ϕ(g)f(x) for g ∈ G, x ∈ Ω.

Proof. (2-2)

G ⟶_{≅}^{ev_{0}} Ω ⟶^{f} ℝ_{+}
↓^{l_{g}} ↓^{g-action} ↓^{ϕ(g)}
G ⟶_{≅}^{ev_{0}} Ω ⟶^{f} ℝ_{+}

Note that

f ∘ ev_{0} = ϕ,
ϕ ∘ l_{g} = ϕ(g) · ϕ.

That is, ϕ ∘ l_{g}(h) = ϕ(gh) = ϕ(g)ϕ(h) for all h ∈ G. □

(3) Examples

A is called left-symmetric if

(3-1)  (a, b, c) = (b, a, c).

A^{−} : the Lie algebra with [a, b] = ab − ba.

λ_{a}(b) := ab =: ρ_{b}(a), where λ, ρ ∈ gl(A). Note ad_{a}(b) := [a, b] = ab − ba = λ_{a}(b) − ρ_{b}(a).

(3-2)  λ_{[a,b]} = [λ_{a}, λ_{b}].
(3-3)  ρ_{[a,b]} = λ_{[a,b]} − ad_{[a,b]}
= [λ_{a}, λ_{b}] − [ad_{a}, ad_{b}].

Hence s = trρ (trλ) : A^{−} → ℝ Lie algebra homomorphism.

Example. Let A be a LSA given by

A = < e_{1}, e_{2} | e_{1}e_{2} = e_{2}, e_{2}e_{1} = e_{2} > .

Let X = xe_{1} + ye_{2} ∈ A. Then Ω = {(x, y) | −1 < x}, and

(i)  \{ \begin{matrix}s(X) & = trρ_{X} = 2x, \\ f_{s}(X) & = (1 + x)^{2}. \end{matrix}

(ii) \{ \begin{matrix}β(X) & = y, \\ f_{β}(X) & = \{ \begin{matrix}e^{y}e^{\{− \frac{2(e^{x}−1)}{x} +1\}} & , for x ≠ 0 \\ e^{y} & , for x = 0. □ \end{matrix} \end{matrix}

(4) The Characteristic Polynomial Consider s = trρ : A^{−} = 𝔤 → ℝ, and its characteristic function p. We will express this explicitly.

(4-1)

A ⟶^{s=trρ} ℝ
↓^{exp} ↓^{exp}
G ⟶^{ϕ} ℝ_{+}
≅↓^{ev_{0}} ∥
Ω ⟶^{p} ℝ_{+}

Lemma. For x = exp(a) · 0 ∈ Ω, a ∈ A, p(x) := e^{trρ_{a}} = det e^{ρ_{a}} = det(1 + ρ_{x}).

(4-1)  \{ \begin{matrix}ϕ(exp a) & = e^{trρ_{a}}, a ∈ A, \\ ϕ(exp a) & = p ∘ ev_{0}(exp a) = p(exp a · 0). \end{matrix}

A ∋ a  ⟶^{trρ}  trρ_{a} ∈ ℝ
↓^{exp}  ↓^{exp}
G ∋ g = exp(a) ⟶^{ϕ} e^{trρ_{a}} = ϕ(g) ∈ ℝ_{+}
↓^{ev_{0}}  ∥
Ω ∋ g · 0 = x ⟶^{p}  p(x) ∈ ℝ_{+}

Claim. For x = exp(a) · 0 ∈ Ω, a ∈ A,

e^{ρ_{a}} = (1 + ρ_{x}).

Step 1. For g = exp a ∈ G, a ∈ A and x ∈ A,

ev_{x}(g) = e^{λ_{a}}(x) + “e^{a} − 1”.

Proof. For a ∈ A,

exp(λ_{a}, a) = I + ( \begin{matrix}λ_{a} & a \\ 0 & 0 \end{matrix} ) + \frac{1}{2!}( \begin{matrix}λ_{a} & a \\ 0 & 0 \end{matrix} )^{2} + ⋯

= ( \begin{matrix}e^{λ_{a}} & “e^{a} − 1” \\ 0 & 1 \end{matrix} ) ∈ Gl(ℝ^{n+1}),

where “e^{a} − 1” := a + \frac{1}{2!}λ_{a}(a) + \frac{1}{3!}λ_{a}^{2}(a) + · · ·.

Thus

ev_{x}(g) = g · x = f(g)(x) + q(g)

= exp(λ, a) ( \begin{matrix}x \\ 1 \end{matrix} )

= e^{λ_{a}}(x) + “e^{a} − 1”. □

Step 2. d(ev_{x})|_{e} = I + ρ_{x}.

Proof. For every a ∈ A, exp(ta) is the integral curve in G. Then for any x ∈ A,

A ∋ a ⟶^{d(ev_{x})|_{e}} d(ev_{x})|_{e}(a) = λ_{a}(x) + a = (I + ρ_{x})(a)
↓^{exp} ↓
G ∋ exp(ta) ⟶^{ev_{x}} ev_{x}(exp(ta)) = e^{λ_{ta}}(x) + “e^{ta} − 1”. □

Remark. λ_{a}(x) + a = ax + a = ρ_{x}(a) + a = (I + ρ_{x})(a). λ_{ta}(x) = (ta)x = t(ax) = tλ_{a}(x), and e^{λ_{ta}}(x) + “e^{ta} − 1” = {x + tλ_{a}(x) + ⋯ } + {ta + ⋯ }.

Step 3. I + ρ_{x} = e^{trρ_{a}}, for x = exp a · 0 ∈ Ω.

Proof.

Let x = g · 0 ∈ Ω for g = exp(a) ∈ G, a ∈ A.

Let r_{g} : G → G, for h ∈ G,

r_{g}(h) = hg = g(g^{−1}hg) = l_{g} ∘ C_{g^{−1}}(h).

Claim1. ev_{x} = ev_{g·0} = ev_{0} ∘ r_{g} = ev_{0} ∘ l_{g} ∘ C_{g^{−1}}.

\begin{matrix}G & ⟶^{ev_{g·0}} & Ω \\ ↓^{r_{g}} & ∥ & ∴ \\ G & ⟶^{ev_{0}} & Ω \end{matrix}  \begin{matrix}h & ⟶^{ev_{g·0}} & h · (g · 0) \\ ↓^{r_{g}} & & ∥ \\ hg & ⟶^{ev_{0}} & (hg) · 0 \end{matrix}

Claim2. I + ρ_{x} = e^{trρ_{a}}.

G ⟶^{ev_{x}=ev_{g·0}} Ω
↓^{C_{g^{−1}}}  ↓^{Φ(g^{−1})}
G ⟶^{ev_{0}} Ω
↓^{l_{g}}  ↓^{Φ(g)=(f(g),q(g))}
G ⟶^{ev_{0}} Ω

h ⟶^{ev_{g·0}}  h · (g · 0)
↓^{C_{g^{−1}}}  ↓^{∃ Φ(g^{−1})}
g^{−1}hg ⟶^{ev_{0}} g^{−1} · (hg · 0) □

General case, i.e., G acts transitively on Ω.

Theorem 1. ([Hel],[Kim2]) Let p(x) = det(1 + ρ_{x}) :

A → ℝ. Then p(g · x) = ϕ(g)p(x), g ∈ G, where ϕ : G →

ℝ_{+} is a Lie group homomorphism given by ϕ(g) =

det(1 + ρ_{g·0}).

Proof.

(4-2)

A^{−} = 𝔤  ⟶^{trρ}  ℝ
↓^{exp}  ↓^{exp}
G  ⟶^{ϕ}  ℝ_{+}
locally at e ≅↓^{ev_{0}}  ∥
Ω ⟶^{p(x):=det(I+ρ_{x})} ℝ_{+}

Step 1. For near e ∈ G and 0 ∈ A,

p ∘ ev_{0} ∘ exp = e^{trρ},

ϕ = p ∘ ev_{0}, that is, ϕ(g) = p(g · 0).

Step 2. For near e ∈ G, 0 ∈ A, and also for all x ∈ A,

p(g · x) = ϕ(g)p(x).

Step 3. For all g ∈ G,

p(g · x) = ϕ(g)p(x). □

(5) Developing Images

Theorem 2. ([Kim2]) Let us denote the developing image of G by Ω = ev_{0}(G). Then Ω is the connected component of {x ∈ A | p(x) = det(1 + ρ_{x}) ≠ 0} containing the origin 0 ∈ A.

Proof. Ω_{0} := the 0-component of {x ∈ A | p(x) ≠ 0}. Claim. Ω_{0} = Ω :

Step 1. G acts on Ω_{0} :

For x ∈ Ω_{0}, g · x ∈ Ω : det(1 + ρ_{g·x}) = p(g · x) = ϕ(g) det(1 + ρ_{x}) ≠ 0. 0 = Id · 0 ∈ G · Ω_{0} = connected ⊂ {x ∈ A | p(x) ≠ 0}.

Hence, by maximality G · Ω_{0} ⊂ Ω_{0}.

Step 2. Ω ⊂ Ω_{0} :

Ω = G · 0 ⊂ G · Ω_{0} ⊂ Ω_{0}, because 0 ∈ Ω_{0}.

Step 3. G · y = ev_{y}(G) ⊂ Ω_{0} is open in A, for y ∈ Ω_{0} :

For y ∈ Ω_{0}, det(1 + ρ_{y}) ≠ 0.

d(ev_{y})|_{e} = 1 + ρ_{y} is an isomorphism. ev_{y} is a covering map. G · y = ev_{y}(G) ⊂ Ω_{0} is an open set in A.

Step 4. Ω = G · 0 is open and closed in Ω_{0} :

Ω_{0} = (G · 0) ∪ (G · y).

Since Ω_{0} is connected, we have Ω = Ω_{0}. □

(6) The Hessian

Lemma. Let s : A^{−} = 𝔤 → ℝ be a Lie algebra homomorphism with its characteristic function p, and let

f(x) = log p(x). Then for a, b ∈ A,

\{ \begin{matrix}df|_{0}(a) = s(a), \\ d^{2}f|_{0}(a, b) = −s(ab). \end{matrix}

(4-3)

A^{−} = 𝔤 ⟶^{s} ℝ ⟶^{id} ℝ
↓^{exp} ↓^{exp} ↓^{id}
G ⟶^{ϕ} ℝ_{+} ⟶^{log} ℝ
↓^{ev_{0}} ∥
Ω ⟶^{p} ℝ_{+}

Step 1. df|_{0}(a) = s(a). d(ev_{0})|_{e} = id.

Diagram (4-3).

Step 2. H(f)(a, b) = H(φ)(a, b) = −s(ab),

for a, b ∈ 𝔤 = A^{−}, where φ := log ∘ϕ.

(a) ((l_{g})^{∗}φ) = φ ∘ l_{g} : G ⟶^{l_{g}} G ⟶^{φ} ℝ.

(b) dφ is left-invariant :

(l_{g})^{∗}(dφ) = d((l_{g})^{∗}φ) = dφ.

(c) H(φ)(a, b) = −s(ab) :

H(φ)(a, b) = (∇(dφ))(a, b)

= (∇_{b}dφ)(a) = ∇_{b}(dφ(a)) − dφ(∇_{b}a) = −dφ(∇_{b}a) = −s(ba) = −s(ab). □

Remark. ([Hel]) s = trρ, f = log ∘ϕ. Then

∇^{k+1}f(a_{1}, ⋯, a_{k}, b) = (−1)^{k} ∑_{τ∈S_{k}} s(ρ_{a_{τ(1)}} ⋯ ρ_{a_{τ(k)}}(b)),

where S_{k} is the set of permutations τ of {1, ⋯, k}.

References

[Hel] J. Helmstetter, Radical d’une algebre symmetrique a gauche, Ann. Inst. Fourier (Grenoble) 29 (1979), 17 – 35.

[Kim1] H. Kim, The geometry of left-symmetric algebra, J. Korean math. Soc. 33 (1996), 1047 – 1067.

[Kim2] H. Kim, Developing maps of affinely flat Lie groups, Bull. Korean Math. Soc. 34, no. 4 (1997), 509 - 518.

[FK] J. Faraut, A. Koranyi, Analysis on Symmetric Cones, Oxford University Press, New York, 1994.

[Vin] E. B. Vinberg, Convex homogeneous cones, Transl. Moscow Math. Soc. No.12 (1963), 340 – 403.