Complexification (Lie group)

Let GC be the simply connected complex Lie group with Lie algebra 𝖌C = 𝖌 ⊗ C, let Φ: G → GC be the natural homomorphism (the unique morphism such that Φ*: 𝖌 ↪ 𝖌 ⊗ C is the canonical inclusion) and suppose π: G → G is the universal covering map, so that ker π is the fundamental group of G. We have the inclusion Φ(ker π) ⊂ Z(GC), which follows from the fact that the kernel of the adjoint representation of GC equals its centre, combined with the equality which holds for any k ∈ ker π. Denoting by Φ(ker π)* the smallest closed normal Lie subgroup of GC that contains Φ(ker π), we must now also have the inclusion Φ(ker π)* ⊂ Z(GC).

is simply connected, Lie's second fundamental theorem now provides us with a unique complex analytic morphism

[3] Onishchik & Vinberg (1994) give an example of a connected real Lie group for which the homomorphism is not injective even at the Lie algebra level: they take the product of T by the universal covering group of SL(2,R) and quotient out by the discrete cyclic subgroup generated by an irrational rotation in the first factor and a generator of the center in the second.

If G is a compact Lie group, the *-algebra A of matrix coefficients of finite-dimensional unitary representations is a uniformly dense *-subalgebra of C(G), the *-algebra of complex-valued continuous functions on G. It is naturally a Hopf algebra with comultiplication given by The characters of A are the *-homomorphisms of A into C. They can be identified with the point evaluations f ↦ f(g) for g in G and the comultiplication allows the group structure on G to be recovered.

It is a complex Lie group and can be identified with the complexification GC of G. The *-algebra A is generated by the matrix coefficients of any faithful representation σ of G. It follows that σ defines a faithful complex analytic representation of GC.

[4] The original approach of Chevalley (1946) to the complexification of a compact Lie group can be concisely stated within the language of classical invariant theory, described in Weyl (1946).

Let G be a closed subgroup of the unitary group U(V) where V is a finite-dimensional complex inner product space.

By the functional calculus for polynomial functions it follows that h⊗N lies in the commutant of AN if h = exp z T with z in C. In particular taking z purely imaginary, T must have the form iX with X in the Lie algebra of G. Since every finite-dimensional representation of G occurs as a direct summand of W⊗N, it is left invariant by GC and thus every finite-dimensional representation of G extends uniquely to GC.

[7] The Gauss decomposition can be extended to complexifications of other closed connected subgroups G of U(V) by using the root decomposition to write the complexified Lie algebra as[8] where 𝖙 is the Lie algebra of a maximal torus T of G and 𝖓± are the direct sum of the corresponding positive and negative root spaces.

If N± and TC are the complex Lie groups corresponding to 𝖓+ and 𝖙C, then the Gauss decomposition states that the subset is a direct product and consists of the elements in GC for which the principal minors are non-vanishing.

[9] If W = NG(T) / T denotes the Weyl group of T and B denotes the Borel subgroup TC N+, the Gauss decomposition is also a consequence of the more precise Bruhat decomposition decomposing GC into a disjoint union of double cosets of B.

The complex dimension of a double coset BσB is determined by the length of σ as an element of W. The dimension is maximized at the Coxeter element and gives the unique open dense double coset.

Its inverse conjugates B into the Borel subgroup of lower triangular matrices in GC.

For g in SL(n,C), take b in B so that bg maximizes the number of zeros appearing at the beginning of its rows.

Chevalley (1955) showed that the expression of an element g as g = b1σb2 becomes unique if b1 is restricted to lie in the upper unitriangular subgroup Nσ = N+ ∩ σ N− σ−1.

In fact, if Mσ = N+ ∩ σ N+ σ−1, this follows from the identity The group N+ has a natural filtration by normal subgroups N+(k) with zeros in the first k − 1 superdiagonals and the successive quotients are Abelian.

Indeed, Nσ(k)N+(k + 1) and Mσ(k)N+(k + 1) are specified in N+(k) by the vanishing of complementary entries (i, j) on the kth superdiagonal according to whether σ preserves the order i < j or not.

Analogues of B, TC and N± are defined by intersection with Sp(n,C), i.e. as fixed points of θ.

From the properties of the Iwasawa decomposition for GL(V), the map G × A × N is a diffeomorphism onto its image in GC, which is closed.

It states that each irreducible representation of G can be obtained by holomorphic induction from a character of T, or equivalently that it is realized in the space of sections of a holomorphic line bundle on G / T. The closed connected subgroups of G containing T are described by Borel–de Siebenthal theory.

Given an irreducible finite-dimensional representation Vλ with highest weight vector v of weight λ, the stabilizer of C v in G is a closed subgroup H. Since v is an eigenvector of T, H contains T. The complexification GC also acts on V and the stabilizer is a closed complex subgroup P containing TC.

Since v is annihilated by every raising operator corresponding to a positive root α, P contains the Borel subgroup B.

Since AN fixes C v, the G-orbit of v in the complex projective space of Vλ coincides with the GC orbit and In particular Using the identification of the Lie algebra of T with its dual, H equals the centralizer of λ in G, and hence is connected.

The Borel–Weil theorem and its generalizations are discussed in this context in Serre (1954), Helgason (1994), Duistermaat & Kolk (2000) and Sepanski (2007).

In that case σ is inner and implemented by an element in a one-parameter subgroup exp tT contained in the center of Gσ.

[24] The starting point is the Abelian version of the result: if T is a maximal torus of a simply connected group G and σ is an involution leaving invariant T and a choice of positive roots (or equivalently a Weyl chamber), then the fixed point subgroup Tσ is connected.

Then T is a maximal torus in G containing x and S. It is invariant under σ and the identity component of Tσ is S. In fact since x and S commute, they are contained in a maximal torus which, because it is connected, must lie in T. By construction T is invariant under σ.

In this case t is a regular element of T—the identity component of its centralizer in G equals T. There is a unique Weyl alcove A in

The map c(g) = (g*)−1 defines an automorphism of GC as a real Lie group with G as fixed point subgroup.

On the Lie algebra level it defines a self-adjoint operator for the complex inner product where B is the Killing form on