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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?

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?

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?

Remark --- The question doesn't actually have much to do with the symplectic group and the Siegel half-plane. Let $V$ be a finite dimensional complex vector space, equipped with a hermitian inner product $g$. The same formula defines an action of $GL(V \oplus V)$ on the open subset $U$ of $End(V)$ which consists of those endomorphisms whose imaginary part is non-degenerate${}^{[1]}$ with respect to $g$, so that might be a more natural setting for the question.

[1]: Recall: an endomorphism $f$ is non-degenerate with respect to $g$ if the induced bilinear form $g \circ f$ is non-degenerate on $V$.

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?

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?

Remark --- The question doesn't actually have much to do with the symplectic group and the Siegel half-plane. Let $V$ be a finite dimensional vector space, equipped with an inner product $g$. The same formula defines an action of $GL(V \oplus V)$ on the open subset $U$ of $End(V)$ which consists of those endomorphisms which are non-degenerate${}^{[1]}$ with respect to $g$, so that might be a more natural setting for the question.

[1]: Recall: an endomorphism $f$ is non-degenerate with respect to $g$ if the induced bilinear form $g \circ f$ is non-degenerate on $V$.

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?

Remark --- The question doesn't actually have much to do with the symplectic group and the Siegel half-plane. Let $V$ be a finite dimensional complex vector space, equipped with a hermitian inner product $g$. The same formula defines an action of $GL(V \oplus V)$ on the open subset $U$ of $End(V)$ which consists of those endomorphisms whose imaginary part is non-degenerate${}^{[1]}$ with respect to $g$, so that might be a more natural setting for the question.

[1]: Recall: an endomorphism $f$ is non-degenerate with respect to $g$ if the induced bilinear form $g \circ f$ is non-degenerate on $V$.