# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 5 Invariant Differential Operators on \(Sp(2n,\mathbb{R})/U(n)\)  
The first part of this section is based on the author's paper [30] (pp. 279-281). Throughout this section, we let \(G:=Sp(2n,\mathbb{R})\) and \(K=U(n)\). We let  
\[\mathbb{H}_{n}:=\{\,\Omega\in\mathbb{C}^{(n,n)}\,|\,\,\Omega=\,^{t}\Omega,\, \,\,\text{Im}\,\Omega>0\,\}\]  
be the Siegel upper half plane of degree \(n\). Then, \(G\) acts on \(\mathbb{H}_{n}\) transitively by  
\[M\cdot\Omega=(A\Omega+B)(C\Omega+D)^{-1}, \tag{43}\]  
where \(M= \begin{pmatrix}A&B\\ C&D\end{pmatrix} \in G\) and \(\Omega\in\mathbb{H}_{n}\). The stabilizer of Action (43) at \(iI_{n}\) is  
\[\left\{ \begin{pmatrix}A&B\\ -B&A\end{pmatrix} \right|\,A+iB\in U(n)\,\right\}\cong U(n).\]  
Thus, we obtain the biholomorphic map  
\[G/K\longrightarrow\mathbb{H}_{n},\qquad gK\mapsto g\cdot iI_{n},\quad g\in G.\]  
\(\mathbb{H}_{n}\) is a Hermitian symmetric manifold. In fact, it is known that \(\mathbb{H}_{n}\) is an Einstein-Kahler Hermitian symmetric space. For \(\Omega=(\omega_{ij})\in\mathbb{H}_{n}\), we write \(\Omega=X+i\,Y\) with \(X=(x_{ij})\), \(Y=(y_{ij})\) real. We put \(d\Omega=(d\omega_{ij})\) and \(d\overline{\Omega}=(d\overline{\omega}_{ij})\). We also put  
\[\frac{\partial}{\partial\Omega}\,=\,\left(\,\frac{1+\delta_{ij}}{2}\,\frac{ \partial}{\partial\omega_{ij}}\,\right)\qquad\text{and}\qquad\frac{\partial} {\partial\overline{\Omega}_{ij}}\,=\,\left(\,\frac{1+\delta_{ij}}{2}\,\frac{ \partial}{\partial\overline{\omega}_{ij}}\,\right).\][PERSON] [31] introduced the symplectic metric \(ds^{2}_{n;A}\) on \(\mathbb{H}_{n}\) invariant under the action (43) of \(Sp(2n,\mathbb{R})\) that is given by  
\[ds^{2}_{n;A}=A\cdot\operatorname{Tr}(Y^{-1}d\Omega\,Y^{-1}d\overline{\Omega}), \qquad A>0. \tag{44}\]  
It is known that the metric \(ds^{2}_{n;A}\) is a Kahler-Einstein metric. [PERSON] [32] proved that its Laplace operator \(\Delta_{n;A}\) is given by  
\[\Delta_{n;A}=\ \frac{4}{A}\cdot\operatorname{Tr}\left(Y^{t}\!\left(Y\frac{ \partial}{\partial\overline{\Omega}}\right)\frac{\partial}{\partial\overline{ \Omega}}\right) \tag{45}\]  
and  
\[dv_{n}(\Omega)=(\det Y)^{-(n+1)}\prod_{1\leq i\leq j\leq n}dx_{ij}\prod_{1 \leq i\leq j\leq n}dy_{ij} \tag{46}\]  
is a \(Sp(2n,\mathbb{R})\)-invariant volume element on \(\mathbb{H}_{n}\) (cf. [33], (p. 130)). We let \(\mathbb{D}(\mathbb{H}_{n})\) be the algebra of all differential operators on \(\mathbb{H}_{n}\) invariant under Action (43). Then, according to [PERSON] [10; 11],  
\[\mathbb{D}(\mathbb{H}_{n})=\mathbb{C}[D_{1},\cdots,D_{n}],\]  
where \(D_{1},\cdots,D_{n}\) are algebraically independent invariant differential operators on \(\mathbb{H}_{n}\). That is, \(\mathbb{D}(\mathbb{H}_{n})\) is a commutative algebra that is finitely generated by \(n\) algebraically independent invariant differential operators on \(\mathbb{H}_{n}\). [PERSON] [2] found the explicit \(D_{1},\cdots,D_{n}\). We let \(\mathfrak{g}_{\mathbb{C}}\) be the complexification of the Lie algebra of \(G\). It is known that \(\mathbb{D}(\mathbb{H}_{n})\) is isomorphic to the center of the universal enveloping algebra of \(\mathfrak{g}_{\mathbb{C}}\) (cf. [4]). Now, we review differential operators on the Siegel upper half plane \(\mathbb{H}_{n}\) invariant under Action (43). The isotropy subgroup \(K\) at \(il_{n}\) for Action (43) is a maximal compact subgroup given by  
\[K=\left\{ \begin{pmatrix}A&-B\\ B&A\end{pmatrix} \right|\ A^{\,t}A+B^{\,t}B=I_{n},\ A^{\,t}B=B^{\,t}A,\ A,B\in \mathbb{R}^{(n,n)}\,\right\}\cong U(n).\]  
We let \(\mathfrak{k}\) be the Lie algebra of \(K\). Then, the Lie algebra \(\mathfrak{g}\) of \(G\) has a Cartan decomposition \(\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}\), where  
\[\mathfrak{g}=\left\{ \begin{pmatrix}X_{1}&X_{2}\\ X_{3}&-^{\,t}X_{1}\end{pmatrix} \Big{|}\ X_{1},X_{2},X_{3}\in\mathbb{R}^{(n,n )},\ X_{2}=\,^{t}X_{2},\ X_{3}=\,^{t}X_{3}\,\right\},\]  
\[\mathfrak{k}=\left\{ \begin{pmatrix}X&-Y\\ Y&X\end{pmatrix} \in\mathbb{R}^{(2n,2n)}\ \Big{|}\ ^{t}X+X=0,\ Y=\,^{t}Y\,\right\},\]  
\[\mathfrak{m}=\left\{