What are Siegel modular forms?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The symplectic group $Sp(g,\mathbb{Z})$ is the subgroup of $SL(2g,\mathbb{Z})$ such that all elements satisfy $M=J_g^t M J_g$ with $J_g$ being the canonical almost complex structure. $J_g$ is also known as involution.
$Sp(g,\mathbb{Z})$ acts on $\mathbb{H}_g$ by $M(Z)=(AZ+B)(CZ+D)^{-1}$ where A,B,C and D are the block matrix entries of M.
Let $\rho : GL(g,\mathbb{C}) \to GL(V)$ be a rational representation on a finite dimensional $\mathbb{C}$-vector space then the associated modular forms are the holomorphic functions $f : \mathbb{H}_g \to V$ satisfying $f(M(Z))=\rho(CZ+D)f(Z)$ for all $M \in \Gamma < Sp(g,\mathbb{Z})$.
Since the determinant is also a representation scalar valued modular forms are included in this definition. If $V$'s dimension is 2 or higher then we speak of a vector valued Siegel modular form.
We can spice this definition up by allowing square roots of the determinant $\sqrt{det(CZ+D)}$. Here, we have to solve the ambiguity by a multiplier system $v$. Then we have
$$f(M(Z))=v(M) \cdot \left(\sqrt{det(CZ+D)}\right)^r \cdot \rho(CZ+D)f(Z).$$
Structure theorems
If we fix a representation $\rho_0$ and a subgroup $\Gamma$ but allow arbitrary powers of $\left(\sqrt{det(CZ+D)}\right)^r$ or $det(CZ+D)^k $ , resp., then all functions satisfying
$$ f(M(Z))=v(M) \cdot \left(\sqrt{det(CZ+D)}\right)^r \cdot \rho_0(CZ+D)f(Z)\quad \forall M \in \Gamma $$
form a module over (a subring of) the ring of scalar modular forms of $\Gamma$ (to the multiplier system $v$).
This module is always finitely generated. But it seems hard to find such a finite set of generators and even more all the relations between them. Without these relations it is possible to miss out a nicer description of elements in the module.
For me a structure theorems is a theorem that returns for a group and a representation such a finite set and the relations.
The actual question
Which structure theorems of vector valued Siegel modular forms are known? I'm personally most interested in genus 2.
I would make the question community because I don't expect a uniform theorem but couldn't find the button.
I already got some answers but I'm not sure how to post : all in one answer, one group $\Gamma$ (and one representation $\rho_0$) per answer, one paper per answer ???
Comments on this issue are very welcome. Afterwards I'm very happy to type this half of a dozen papers.
p.s. I hope the post is not too chaotic.
Best Answer
Ibukiyama proved the following results around the year 2000. They are also on the full modular group $Sp(2,\mathbb{Z})$. He covers odd weight of $Sym^2$, even weight of $Sym^6$, and all of $Sym^4$
In this edit, I want to comment on odd weight of $Sym^2$.
$A$ denotes again the ring $\mathbb{C}[\phi_4,\phi_6,\chi_{10},\chi_{12}]$ and abbreviate these 4 forms by $X_1, \dots , X_4$. Let's call the module of vector valued modular forms $M$. We have $$M=\sum_{1\leq i < j < k \leq 4} A \cdot [X_i,X_j,X_k]$$ where $[X_i,X_j,X_k]$ is an other Rankin Cohen bracket. I feel that the best reference for these RC brackets is van Dorp's thesis. Let $f \in M_k(\Gamma)$, $g \in M_l(\Gamma)$ and $h \in M_m(\Gamma)$ then $[f,g,h]$ equals (up to a constant) $$k \cdot f \cdot \nabla g \wedge \nabla h \quad - l \cdot g \cdot \nabla f \wedge \nabla h \quad + m \cdot h\cdot \nabla f \wedge \nabla g . $$ It is a vector valued Seigel modular form of weight $k+l+1$ w.r.t. the representation $Sym^2$. Then the subsequent properties are obvious
$-m\cdot h[f,g,j]+l \cdot g[f,h,j]-k \cdot f[g,h,j]+n \cdot j [f,g,h]=0$
where $f \in M_k(\Gamma)$, $g \in M_l(\Gamma)$,$h \in M_m(\Gamma)$, and $j \in M_n(\Gamma)$.
Proof :
Ibukiyama proves that the above relations are generating ones with the help of Fourier Jacobi expansions. But, he states that there are many ways to prove this result. This gives him the Hilbert function of the RHS which coincides with the LHS's one. This one can be calculated by using Tshushima's famous dimension formula.