Several complex variables and analytic spaces Complex spaces with automorphism groups Theorem

Harish-Chandra embedding

Builds on Hermitian symmetric space Bounded symmetric domain Lie group Complexification of a Lie group

Statement

Let $G$ be a connected noncompact simple real Lie group, contained in its simply connected complexification $G_\C$ . Let $K \subset G$ be a maximal compact subgroup such that $X = G/K$ is a Hermitian symmetric space. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition, and let $\mathfrak{p}_\C = \mathfrak{p}^+ \oplus \mathfrak{p}^-$ be the splitting into the $\pm i$ eigenspaces of the complex structure. Put $P^\pm = \exp \mathfrak{p}^\pm$ , and let $K_\C$ be the complexification of $K$ in $G_\C$ . Then:

Every $g \in G$ can be written uniquely as $g = p^+ k p^-$ with $p^\pm \in P^\pm$ and $k \in K_\C$ .
The map $\zeta \colon X \to \mathfrak{p}^+$ , $gK \mapsto \log p^+$ , is a holomorphic embedding onto a bounded domain in the complex vector space $\mathfrak{p}^+ \cong \C^n$ , where $n = \dim_\C X$ .

Remarks

The image is a bounded symmetric domain, the Harish-Chandra realization of $X$ , and it is convex and circled. The map extends to the Borel embedding of $X$ as an open subset of the compact dual $G_\C / K_\C P^-$ . In this picture $\zeta$ is the restriction of an affine chart $P^+ \cong \mathfrak{p}^+$ of the compact dual. The same holds for semisimple $G$ by taking products.

Example: for $G = SU(1, 1)$ and $K = U(1)$ the Harish-Chandra realization of $G/K$ is the unit disc. For $G = Sp(2n, \R)$ and $K = U(n)$ , the space $G/K$ is the Siegel upper half-space $\mathbb{H}_n$ , and the realization is the domain $\set{W : W^T = W,\ I - W^* W > 0}$ , reached by the Cayley transform $Z \mapsto (Z - iI)(Z + iI)^{-1}$ .

Going deeper

For a student

Together with the fact that bounded symmetric domains with the Bergman metric are Hermitian symmetric, this gives an equivalence between the two notions. The embedding is the tool behind holomorphic discrete series representations and the analysis of automorphic forms on $G/K$ . It also explains why $X$ sits as an open subset inside its compact dual, as the disc sits inside the Riemann sphere.

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Tag	`4RXXE3BB6W`
URI	`m2://4RXXE3BB6W`
Kind	statement, theorem
Writing	W0 stub
MSC	32M15, 22E46
Status	This copy of the site has no build data. Run `m2 status 4RXXE3BB6W`.
About	Hermitian symmetric space, Bounded symmetric domain
Uses	Hermitian symmetric space, Bounded symmetric domain, Lie group, Complexification of a Lie group

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