Finding the action of the symplectic group on the Siegel-half plane

Question

Let $S$ be the Siegel-half plane of dimension $n$, i.e. the set of complex $n \times n$ matrices $Z$ which are symmetric and whose imaginary part is positive-definite. In dimension 1 we can identify $S$ with the Poincaré half-plane.

In dimension 1 there is an action of the symplectic group $Sp_2(\mathbb R)$ on $S$, given by fractional linear transformations. One can extend this action to higher dimensions; we write an element $M$ of the symplectic group $Sp_{2n}(\mathbb R)$ as a block matrix with blocks $A$, $B$, $C$, $D$, and then set

$$ M \cdot Z = (AZ + B)(CZ + D)^{-1}. $$

Now, in dimension 1 one sees that this action is induced by the action of the general linear group on the projective line and the embedding $GL_2(\mathbb R) \hookrightarrow GL_2(\mathbb C)$. However, this does not seem to be true in higher dimensions, as $GL_{2n}(\mathbb C)$ acts on the projective space of dimension $2n$, and not on the space of dimension $n$.

The definition of the action of $Sp_{2n}(\mathbb R)$ on $S$ in higher dimensions thus seems very ad-hoc, which motivates:

Question: How can one find this action naturally in higher dimensions?

[Edit:] Question 2: David's answer is very nice, but I'd like to iterate the question before accepting it. Instead of the Siegel half-plane above, consider the space $U$ of complex $(1,1)$-forms whose imaginary part is positive-definite. In a basis we can identify this with the set of $n \times n$ matrices $Z$ whose antihermitian part is positive-definite. The action of the symplectic group now makes sense on this space as well.

In a basis the Siegel half-plane is a closed subspace of $U$ if $n > 1$. It is easiest to define the action of the symplectic group in a basis, but we can define it without reference to a basis by choosing a hermitian inner product on our vector space. Now, can we find the action of the symplectic group on $U$ in the same fashion as on the Siegel half-plane?

Answer 1

The Siegel half-plane is just one example of a very general construction: if one lets $G$ be a semisimple Lie group, and $K$ a maximal compact subgroup, then $G$ acts on the quotient $G / K$, and the quotient is known as a "symmetric space". The Siegel upper half-plane is just a symmetric space for $Sp_{2n}(\mathbb{R})$, with its natural action; and from this perspective it's a completely natural thing to consider. It just takes a little while to find explicit coordinates in which you can write down the action, and when you do so, its origins become slightly obscured.

You can already see this for $n = 1$, where $G$ is just $SL_2$. Then you can easily check that the action of $G$ on the upper half-plane $\mathcal{H}$ is transitive and the stabilizer of $i$ is $K = SO_2(\mathbb{R}) \cong S^1$, a maximal compact in $G$; so we can identify $\mathcal{H}$ with $G / K$.

In the higher-dimensional symplectic cases, $G = Sp_{2n}$ has a maximal compact subgroup $K$ isomorphic to the unitary group $U(n)$, and you can check that the quotient $G / K$ is the Siegel upper half-plane.

(The miracle with symplectic groups, which doesn't happen with most real Lie groups, is that the Siegel half-space has a $G$-invariant complex structure. This is very important in the theory of automorphic forms.)

Answer 2

A nice way to think about the action of $SP_{2g}(\mathbb{R})$ on $S$ is that $S$ is an $SP_{2g}(\mathbb{R})$-invariant open set in the Lagrangian grassmannian.

Let $V$ be a $2g$-dimensional complex vector space equipped with a symplectic form. Let $L$ be the space of isotropic $g$-dimensional subspaces of $V$. Such a subspace is called a Lagrangian. Clearly, $L$ is a complex manifold equipped with a holomorphic action of $SP_{2g}(\mathbb{C})$.

Choose a splitting of $V$ as $U \oplus U^{\vee}$, where $U$ and $U^{\vee}$ are complementary isotropic subspaces of $V$ and the symplectic pairing between $U$ and $U^{\vee}$ makes $U$ and $U^{\vee}$ into dual spaces. Let $\Omega \subset L$ be the open set consisting of those Lagrangians which are the graph of a map $\phi: U \to U^{\vee}$. The condition that the graph of $\phi$ be Lagrangian is equivalent to the condition that $\phi$ be self adjoint, so $\Omega$ is identified with the space of symmetric $g \times g$ complex matrices. The action of $Sp_{2g}(\mathbb{C})$ on $\Omega$ isn't fully defined, because $\Omega$ isn't $Sp_{2g}$-invariant, but when it is defined you can check that it is given by the formula $Z \mapsto (AZ+B)(CZ+D)^{-1}$.

Choose a real $2g$-dimensional symplectic vector space $V_0$ and identify $V$ with $V_0 \otimes_{\mathbb{R}} \mathbb{C}$. Let $\Omega' \subset L$ be the open set of those Lagrangian's $X$ such that $X \cap V_0 = \{ 0 \}$. Clearly, $\Omega'$ has a holmorphic action of $Sp_{2g}(\mathbb{R})$. If we identify $\Omega$ with $g \times g$ symmetric complex matrices, then $\Omega'$ is those matrices $Z$ for which $\det \Im(Z) \neq 0$.

Obviously, one connected component of $\Omega \cap \Omega'$ is $S$: Symmetric matrices for which $\Im(Z)$ is positive definite. And it turns out that this is actually a connected component of $\Omega'$, so it inherits an action of $Sp_{2g}(\mathbb{R})$. This is the action you asked about.

Answer 3

If you can get hold of a copy of Mumford's Lectures on Theta I, you will find a useful discussion of the Siegel upper half plane $H_{g}$ and the action of $Sp_{2g}$ on it in section 4 of chapter II. He gives several coordinate free interpretations. Here is one: Fix the standard symplectic form $\omega$ on $\mathbb{R}^{2g}$. Then $H_{g}$ can be identified with the space of complex structures on $\mathbb{R}^{2g}$ such that $\omega= Im(H)$ for some (uniquely determined) positive definite Hermitean form $H$. It is clear that $Sp_{2g}(\mathbb{R})= Sp(\omega)$ acts the space of such complex structures.

In case you like Hodge structures, let me supplement his Lemma 4.1 with one more (well known) equivalent. (He proves this without quite saying it.) $H_g$ can be identified with the space of pure Hodge structures of type $\lbrace (1,0),(0,1)\rbrace$ polarized by $\omega$. Griffiths generalized this to show that the space of arbitrary polarized pure Hodge structures of fixed type is a homogeneous complex manifold. Unlike the Siegel case, however, it is usually not Hermitean symmetric, so lots of things don't generalize.