Download Basic Lie Theory by Hossein Abbaspour and Martin Moskowitz PDF

By Hossein Abbaspour and Martin Moskowitz

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Note that τ ′ : H × H → G is smooth as the inclusion H × H ֒→ G × G is smooth. Now we are going to introduce a chart for H which makes τ smooth. 3 Correspondence between Lie Subgroups and Subalgebras 47 k +n and k is the dimension of h. By Frobenius theorem, for each x ∈ H there is a chart φ : U → Rk+m U ∩ H = φ−1 ({(x1 , . . , xk+m )|xk+1 = · · · = xk+m = 0}). Consider V = U ∩ H and π : Rk+m → Rk the projection on the first k coordinates, then (V, ψ = π ◦ φ ◦ i) is the desired chart. We have ψ ◦ τ = π ◦ φ ◦ i ◦ τ = π ◦ φ ◦ τ′ which is smooth as τ ′ and π are.

Let F(V ) be the set of all flags on V . By choosing a basis for V1 , extending this to a basis of V2 , and eventually to V , we see that GL(V ) operates transitively and continuously on F(V ). The isotropy subgroup is easily seen to be the group of upper triangular matrices B in GL(V ). Arguing, as in the Grassmann manifold we see that F(V ) is connected and its dimension is n(n−1) . By using the Gram-Schmidt method to get an orthonormal basis 2 compatible with a flag we see that O(n, R) (respectively U(n, C)) acts transitively.

Show this σ-compact result is rather general. For instance it shows that it applies whenever G is locally compact and second countable. To deal with the σ-compact case we recall a version of the Baire Category theorem [32], pp. 110. Baire’s Category Theorem: Let X be a σ-compact space. That is X = ∪Cn , where each Cn is compact. If X is locally compact, then one of the Cn must have a non-void interior. Continuing the proof in the σ-compact case, we have the following: Since the orbit map π is G-equivariant and the action of G on itself as well as the action of G on X are both transitive, to show openness we may consider a neighborhood of 1 in G.

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