# Problems in Invariant Differential Operators on Homogeneous Manifolds
###### Abstract
We note that the symmetric space (III) is biholomorphic to the so-called Siegel upper half plane \(\mathbb{H}_{n}\) given by  
\[\mathbb{H}_{n}:=\{\,\Omega\in\mathbb{C}^{(n,n)}\,|\,\,\Omega=\,^{t}\Omega,\, \,\,\text{Im}\,\Omega>0\,\,\}. \tag{3}\]  
It is known that \(\mathbb{H}_{n}\) is an Einstein-Kahler Hermitian manifold of complex dimension \(\frac{n(n+1)}{2}\) which is biholomorphic to the generalized unit disk \(\mathbb{D}_{n}\) given by  
\[\mathbb{D}_{n}:=\{\,W\in\mathbb{C}^{(n,n)}\,|\,\,W=\,^{t}W,\,\,\,I_{n}-W\, \overline{W}>0\,\,\}. \tag{4}\]  
The symmetric complex manifold \(\mathbb{H}_{n}\) provides the rich, deep, and beautiful theory in algebraic geometry (e.g., Satake compactification, toroidal compactifications, moduli of abelian varieties, etc.) and number theory (e.g., Siegel modular forms, \(L\)-functions, etc.). We provide the notion of Maass-Siegel functions using the algebra \(\mathbb{D}(\mathbb{H}_{n})\). We also provide the notion of Siegel-Maass forms defined by [PERSON] and [PERSON] (cf. [9]). In Section 6, we study \(GL_{n,m}(\mathbb{R})\)-invariant differential operators on the non-symmetric homogeneous space (IV). We show that the semidirect product \(GL_{n,m}(\mathbb{R})\) of \(GL(n,\mathbb{R})\) and the additive group \(\mathbb{R}^{(m,n)}\) given by  
\[GL_{n,m}(\mathbb{R}):=GL(n,\mathbb{R})\ltimes\mathbb{R}^{(m,n)}\]  
acts naturally and transitively on the following space:  
\[\mathscr{P}_{n,m}:=\mathscr{P}_{n}\times\mathbb{R}^{(m,n)}=\{\,Y\in\mathbb{R} ^{(n,n)}\mid Y=\,^{t}Y>0\,\,\}\times\mathbb{R}^{(m,n)}. \tag{5}\]  
We refer to Formula (61) for the precise action of \(GL_{n,m}(\mathbb{R})\) on \(\mathscr{P}_{n,m}\). We see that the homogeneous space (IV) is diffeomorphic to the homogeneous space \(\mathscr{P}_{n,m}\). It is shown that the algebra \(\mathbb{D}(\text{IV})\) of \(GL_{n,m}(\mathbb{R})\)-invariant differential operators on the homogeneous space (IV) is not commutative. So far, nobody has found a set of generators of \(\mathbb{D}(\text{IV})\). We provide some examples of explicit invariant differential operators on \(\mathscr{P}_{n,m}\) and investigate invariant differential operators on (IV). We provide some open problems that should be solved in the future. In Section 7, we study \(SL_{n,m}(\mathbb{R})\)-invariant differential operators on the non-symmetric homogeneous space (V). The semidirect product \(SL_{n,m}(\mathbb{R})\) of \(SL(n,\mathbb{R})\) and the additive group \(\mathbb{R}^{(m,n)}\) given by  
\[SL_{n,m}(\mathbb{R}):=SL(n,\mathbb{R})\ltimes\mathbb{R}^{(m,n)}\]  
acts naturally and transitively on the following space:  
\[\mathfrak{P}_{n,m}:=\mathfrak{P}_{n}\times\mathbb{R}^{(m,n)}=\{\,Y\in\mathbb{ R}^{(n,n)}\mid Y=\,^{t}Y>0,\,\det(Y)=1\,\,\}\times\mathbb{R}^{(m,n)}. \tag{6}\]  
See Formula (81) for the precise action of \(SL_{n,m}(\mathbb{R})\) on \(\mathfrak{P}_{n,m}\). We see that the homogeneous space (V) is diffeomorphic to the homogeneous space \(\mathfrak{P}_{n,m}\). It is shown that the algebra \(\mathbb{D}(\text{V})\) of \(SL_{n,m}(\mathbb{R})\)-invariant differential operators on the homogeneous space (V) is not commutative. So far, nobody has found a set of generators of \(\mathbb{D}(\text{V})\). We provide some examples of explicit invariant differential operators on \(\mathfrak{P}_{n,m}\) and investigate invariant differential operators on (V). We provide some open problems that should be solved in the future. In the final section, we study \(G^{J}\)-invariant differential operators on the non-symmetric homogeneous space (VI). The homogeneous space (VI) is biholomorphic to the so-called Siegel-Jacobi space \(\mathbb{H}_{n,m}\) given by  
\[\mathbb{H}_{n,m}:=\mathbb{H}_{n}\times\mathbb{C}^{(m,n)}=\{\,\Omega\in \mathbb{C}^{(n,n)}\mid\Omega=\,^{t}\Omega,\,\operatorname{Im}\Omega>0\,\,\} \times\mathbb{C}^{(m,n)}. \tag{7}\]  
The Jacobi group  
\[G^{J}:=Sp(2n,\mathbb{R})\ltimes H_{\mathbb{R}}^{(n,m)}\qquad(\text{semidirect product})\]  
acts naturally and transitively on the Siegel-Jacobi space \(\mathbb{H}_{n,m}\).