Convexity theorems for semisimple symmetric spaces

Let n∈N and let N be the connected, simply-connected Lie group with Lie algebra n. Let λ1,…,λm be the distinct weights of h in n. Then the map

defines a diffeomorphism

Proof.

We will use induction on dimℂ⁢(𝔫). If dimℂ⁡𝔫=1, then 𝔫 is abelian and the result holds trivially.

Next, assume that m>1 and assume that the result has been established for 𝔫 with dimℂ⁡𝔫<m. Assume that 𝔫∈𝒩 has dimension m.

Denote by 𝔫1 the center of 𝔫, which is non-trivial. If 𝔫1=𝔫, then 𝔫 is abelian and the result is trivially true. Thus, we may as well assume that 0⊊𝔫1⊊𝔫. In particular, this implies that both 𝔫1 and 𝔫/𝔫1 have dimensions at most m-1. Put l:=dim⁡𝔫1.

The ideal 𝔫1 is stable under the action of 𝔥 and it is readily verified that 𝔫1 and 𝔫/𝔫2 with the natural 𝔥-representations belong to 𝒩. Furthermore, since all weight spaces are 1-dimensional, we see that

We will first prove that ψ is a diffeomorphism under the assumption that the 𝔥-weights in 𝔫 are numbered in such a way that

Since N is simply-connected, the map exp:𝔫→N is a diffeomorphism; hence, N1:=exp⁡(𝔫1) is the connected subgroup of N with Lie algebra 𝔫1. In particular, N1 is simply connected as well. Since 𝔫1 is an ideal, N/N1 has a unique structure of Lie group for which the natural map N→N/N1 is a Lie group homomorphism. We now observe that N→N/N1 is a principal fiber bundle with fiber N1. By standard homotopy theory, we have a natural exact sequence

Since N is simply-connected, and N1 connected, we conclude that N/N1 is the simply connected group with Lie algebra 𝔫/𝔫1.

By the induction hypothesis, the maps

are diffeomorphisms. For every j∈{l+1,…,m} the canonical projection 𝔫→𝔫/𝔫1 induces the isomorphisms of weight spaces 𝔫λj→(𝔫/𝔫1)λj. Let ψ¯:𝔫λl+1×⋯×𝔫λm→N/N1 be defined by

Then the following diagram commutes:

From this we infer that ψ¯ is a diffeomorphism. We now obtain that the map ψ~:𝔫λl+1×⋯×𝔫λm×N1→N,

is a diffeomorphism onto N. Since

it follows that ψ is a diffeomorphism as well. Clearly, the above proof works for every enumeration of the weights in Λ⁢(𝔫/𝔫1). Since the weight spaces (𝔫1)λ for λ∈Λ⁢(𝔫1) are all central in 𝔫, we conclude that the result holds for any enumeration of the weights in Λ⁢(𝔫). ∎

Let G be a connected complex semisimple Lie group and let nB be the nilpotent radical of a Borel subalgebra b of g. Let h be a Cartan subalgebra contained in b. Let n1,…,nk be linearly independent subalgebras of nB, each of which is a direct sum of h-root spaces, and assume that their direct sum n:=n1⊕⋯⊕nk is again a subalgebra. Put N:=exp⁡n and Nj:=exp⁡(nj) for 1≤j≤k. Then the multiplication map

is a diffeomorphism.

Proof.

This is an immediate consequence of Lemma A.1. ∎

Proof of Lemma 2.11.

We assume that G is a real reductive Lie group of the Harish-Chandra class. Define

the semisimple part of the Lie algebra of G. Let G1 be the analytic subgroup of G with Lie algebra 𝔤1. Since the nilpotent radical NP of P is completely contained in G1, we may assume from the start that G=G1, i.e., G is connected semisimple with finite center.

Since Ad is a finite covering homomorphism from G onto Aut⁢(𝔤)∘, mapping N diffeomorphically onto Ad⁢(N), whereas Aut⁢(𝔤)∘ is a connected real form of Int⁢(𝔤ℂ), we may assume that G is a connected real form of a connected complex semisimple Lie group Gℂ with trivial center. Let τ be the conjugation on Gℂ, such that

Let 𝔤ℂ denote the Lie algebra of Gℂ, then 𝔤ℂ=𝔤⊕i⁢𝔤. Note that the complexification 𝔫P⁢ℂ of 𝔫P equals 𝔫P⊕i⁢𝔫P and that

Take a Cartan subalgebra of 𝔤ℂ, containing 𝔞ℂ=𝔞⊕i⁢𝔞. It is of the form

where 𝔱 is a maximal abelian subspace of 𝔪:=Z𝔨⁢(𝔞). Since 𝔱 centralizes 𝔞, all 𝔞-root spaces are invariant under ad⁢(𝔱). This implies that the subalgebras 𝔫j⁢ℂ:=𝔫j⊕i⁢𝔫j (j∈{1,…,k}) of 𝔫P⁢ℂ are direct sums of 𝔥ℂ-root spaces. Furthermore, their direct sum equals 𝔫ℂ=𝔫⊕i⁢𝔫, hence is a subalgebra. Finally, there exists a Borel subalgebra containing 𝔥ℂ+𝔫ℂ. By Corollary A.2, the multiplication map

is a diffeomorphism. It readily follows that μℂ restricts to a bijection from (N1⁢ℂ)τ×⋯×(Nk⁢ℂ)τ onto (Nℂ)τ. Since

it follows that μ is a bijective embedding from N1×⋯×Nk onto N, hence a diffeomorphism. ∎