Question

In a joint paper with Yifan Yang we constructed an "exotic" embedding of $SL_2(\mathbb R)$ in $Sp_4(\mathbb R)$ (in fact, of $PSL_2(\mathbb R)$ in $PSp_4(\mathbb R)$), namely, $$ \iota\colon\begin{pmatrix} a & b \cr c & d \end{pmatrix} \mapsto\begin{pmatrix} a^2d+2abc & -3a^2c & abd+\frac12b^2c & \frac12b^2d \cr -a^2b & a^3 & -\frac12ab^2 & -\frac16b^3 \cr 4acd+2bc^2 & -6ac^2 & ad^2+2bcd & bd^2 \cr 6c^2d & -6c^3 & 3cd^2 & d^3 \end{pmatrix}. $$ An equivalent form of the embedding was independently discovered by Don Zagier, and we could not find it in the literature.

Although the properties of the embedding (discussed in the preprint above) are nice by themselves, I am interested in an exhaustive list of possibilities to embed other matrix groups and their direct products in $Sp_4(\mathbb R)$ (or $PSp_4(\mathbb R)$). For example, can the direct product of two copies of $SL_2(\mathbb R)$ be embedded?

As I am not a specialist in Lie groups, I would appreciate plainer sources. Thank you for any help in advance!

Answer 1

You can embed the direct product of two copies of $SL_2(\mathbb{R})$. One embedding sends the entries to the center $2\times2$ block, the other sends the entries to the corners with the $2\times2$ identity matrix in the center block. Here, the $J$ matrix is the standard anti-diagonal matrix

For embeddings of other groups, you could look at the Bruhat decomposition of Sp(4) and write a decomposition of each cell. Some explicit information is given on the decomposition for GSp(4) in a book by Ralf Schmidt and Brooks Roberts, which is available on Ralf's website.