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?



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


