# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 5 Invariant Differential Operators on \(Sp(2n,\mathbb{R})/U(n)\)
According to the work of [PERSON] [10; 11], the algebra \(\mathbb{D}(\mathbb{H}_{n})\) is generated by \(n\) algebraically independent generators and is isomorphic to the commutative algebra \(\mathbb{C}[x_{1},\cdots,x_{n}]\) with \(n\) indeterminates. We note that \(n\) is the real rank of \(G\). We let \(\mathfrak{g}_{\mathbb{C}}\) be the complexification of \(\mathfrak{g}\). It is known that \(\mathbb{D}(\mathbb{H}_{n})\) is isomorphic to the center of the universal enveloping algebra of \(\mathfrak{g}_{\mathbb{C}}\). [PERSON] found algebraically independent generators \(D_{1},D_{2},\cdots,D_{n}\) of \(\mathbb{D}(\mathbb{H}_{n})\) ([2], (pp. 112-118)). In fact, we see that  
\[-D_{1}=\Delta_{n;1}=\,4\,\mathrm{Tr}\bigg{(}Y^{\prime}\bigg{(}Y\frac{\partial} {\partial\overline{\Omega}}\bigg{)}\frac{\partial}{\partial\overline{\Omega}} \bigg{)} \tag{55}\]  
is the Laplace operator for the invariant metric \(ds^{2}_{n;1}\) on \(\mathbb{H}_{n}\). [PERSON] [8] found another algebraically independent generators of \(\mathbb{D}(\mathbb{H}_{n})\). **Example 1**. : _We consider the case when \(n=1\). The algebra \(\mathrm{Pol}(T_{1})^{U(1)}\) is generated by the polynomial_  
\[q(\omega)=\omega\,\overline{\omega},\quad\omega=x+\,iy\in\mathbb{C}\,\text{ with }x,y\,\text{ real}.\]  
_Using Formula (54), we obtain_  
\[\Theta_{1}(q)=\,4\,y^{2}\bigg{(}\frac{\partial^{2}}{\partial x^{2}}+\frac{ \partial^{2}}{\partial y^{2}}\,\bigg{)}.\]  
_Therefore, \(\mathbb{D}(\mathbb{H}_{1})=\mathbb{C}\big{[}\Theta_{1}(q)\big{]}=\,\mathbb{C}[ D_{1}]\)._  
For two complex numbers \(\alpha,\beta\in\mathbb{C}\), [PERSON] considered the following matrix-valued differential operator given by  
\[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]0  
where  
\[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]1  
and  
\[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]2  
That is,  
\[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]3 \[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]4  
We refer to [32] (p. 49), [34] (p. 176), and [2], (p. 119). Then,  
\[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]5  
where \(\Delta_{n;1}\) is the Laplace operator of \((\mathbb{H}_{n},ds^{2}_{n;1})\) (see Formulas (44), (45) and (55)). **Definition 6**. : _The differential operator_  
\[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]6  
_is called the Siegel-Maass Laplacian of weight \((\alpha,\beta)\)._  
**Remark 11**. : _We note that \(\mathcal{L}_{0,0}=\Delta_{n;1}\in\mathbb{D}(\mathbb{H}_{n})\) but \(\mathcal{L}_{\alpha,\beta}\
otin\mathbb{D}(\mathbb{H}_{n})\) if \((\alpha,\beta)\
eq(0,0)\)._  
The following definition is given by [PERSON] and [PERSON] (cf. [9], (Definition 4.7)). **Definition 7**. : _We set \(\Gamma_{n}^{\flat}=Sp(2n,\mathbb{Z})\). We let \(\Gamma\subset Sp(2n,\mathbb{R})\) be a subgroup of \(Sp(2n,\mathbb{R})\) commensurable with \(\Gamma_{n}^{\flat}\), i.e., the intersection \(\Gamma\cap\Gamma_{n}^{\flat}\) is a finite index subgroup of \(\Gamma\) as well as of \(\Gamma_{n}^{\flat}\). We let \(\gamma_{j}\in\Gamma_{n}^{\flat}\)\((1\leq j\leq h)\) denote a set of representatives for the left cosets of \(\Gamma\cap\Gamma_{n}^{\flat}\) in \(\Gamma_{n}^{\flat}\). For two complex numbers \(\alpha,\beta\in\mathbb{C}\), we then let \(\mathfrak{P}_{\alpha,\beta}^{n}(\Gamma)\) denote the space of all functions \(\varphi:\mathbb{H}_{n}\longrightarrow\mathbb{C}\) satisfying the following conditions (KM1)-(KM3):_  
* \(\varphi\) _is real analytic;_
* \(\varphi(\gamma\cdot\Omega)=\det(C\Omega+D)^{\alpha}\det(C\overline{\Omega}+D )^{\beta}\varphi(\Omega)\) _for all_ \(\gamma= \begin{pmatrix}A&B\\ C&D\end{pmatrix}