# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 5 Invariant Differential Operators on \(Sp(2n,\mathbb{R})/U(n)\)
\in\Gamma\)_;_
* _given_ \(Y_{0}\in\mathbb{R}^{(n,n)}\) _with_ \(Y_{0}=\,^{t}\!Y_{0}>0\)_, there exist a positive real number_ \(M\in\mathbb{R}^{+}\) _and a positive integer_ \(N\in\mathbb{Z}^{+}\) _such that the inequalities_ \[|\det(C_{j}\Omega+D_{j})^{-\alpha}\det(C\overline{\Omega}+D_{j})^{-\beta} \varphi(\gamma_{j}\cdot\Omega)|\leq M\cdot\mathrm{Tr}(\gamma)^{N}\] _holds in the region_ \(\{\Omega=X+i\,Y\in\mathbb{H}_{n}\,|\,\,Y\geq Y_{0}\,\}\) _for the set of representatives_ \[\gamma_{j}= \begin{pmatrix}A_{j}&B_{j}\\ C_{j}&D_{j}\end{pmatrix} \in\Gamma_{n}^{\flat}\,(1\leq j\leq h).\]  
**Remark 12**. : _For \(\varphi\in\mathfrak{P}_{\alpha,\beta}^{n}(\Gamma)\), we set_  
\[\parallel\varphi\parallel_{\alpha,\beta}=\int_{\Gamma\backslash\mathbb{H}_{n} }\det(Y)^{\alpha+\beta}|\varphi(\Omega)|^{2}\,dv_{n}(\Omega),\]  
_where \(dv_{n}(\Omega)\) is a \(Sp(2n,\mathbb{R})\)-invariant volume element on \(\mathbb{H}_{n}\) (see Formula (46)). In this way, we obtain the Hilbert space_  
\[\mathcal{H}_{\alpha,\beta}^{n}(\Gamma):=\{\,\varphi\in\mathfrak{P}_{\alpha, \beta}^{n}(\Gamma)\,|\,\parallel\varphi\parallel_{\alpha,\beta}<\infty\,\}\]  
_equipped with the inner product_  
\[\langle\varphi,\psi\rangle_{\alpha,\beta}:=\int_{\Gamma\backslash\mathbb{H}_{ n}}\det(Y)^{\alpha+\beta}\varphi(\Omega)\,\overline{\psi(\Omega)}\,dv_{n}(\Omega)\]  
_for all \(\varphi,\psi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)\). We note that in order to enable \(\|\varphi\|<\infty\), the exponent \(N\in\mathbb{Z}^{+}\) in part \((\mathrm{KM3})\) of Definition 7 has to be \(0\)._  
**Remark 13**. : _[PERSON] and [PERSON] showed that the Siegel-Maass Laplacian \(\mathcal{L}_{\alpha,\beta}\) acts as a symmetric operator on \(\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)\) (cf. [9], (Theorem 5.1, pp. 11-15). That is,_  
\[\langle\mathcal{L}_{\alpha,\beta}\varphi,\psi\rangle_{\alpha,\beta}=\,\langle \varphi,\mathcal{L}_{\alpha,\beta}\psi\rangle_{\alpha,\beta}\]  
_for all \(\varphi,\psi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)\)._  
**Definition 8**. : _We let \(\Gamma\subset Sp(2n,\mathbb{R})\) be a subgroup of \(Sp(2n,\mathbb{R})\) commensurable with \(\Gamma_{n}^{\flat}\). The elements of \(\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)\) are called automorphic forms of weight \((\alpha,\beta)\) and degree \(n\) for \(\Gamma\). Moreover, if \(\varphi\in\mathcal{H}_{\alpha,\beta}^{n}(\Gamma)\) is an eigenfunction of \(\mathcal{L}_{\alpha,\beta}\), it is called a Siegel-Maass form of weight \((\alpha,\beta)\) and degree \(n\) for \(\Gamma\)._For the present being, we assume that \(\Gamma\subset Sp(2n,\mathbb{R})\) is a subgroup of \(Sp(2n,\mathbb{R})\) commensurable with \(\Gamma_{n}^{\flat}\). The case that \(\alpha=\frac{k}{2}\) and \(\beta=-\frac{k}{2}\) with \(k\in\mathbb{Z}^{+}\) provides an application to the study of Siegel cusp forms of weight \(k\) for \(\Gamma\). We recall the notion of Siegel modular forms. Definition 9: We let \(\Gamma\subset Sp(2n,\mathbb{R})\) be a subgroup of \(Sp(2n,\mathbb{R})\) commensurable with \(\Gamma_{n}^{\flat}\). 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 \(\Gamma_{n}^{\flat}\). Function \(f:\mathbb{H}_{n}\longrightarrow\mathbb{C}\) is called a _Siegel modular form of weight \(k\)_and degree \(n\) for \(\Gamma\) if it satisfies the following conditions_ (SI1)-(SI3)_:_  
1. \(f\) _is holomorphic;_
2. \(f(\gamma\cdot\Omega)=\det(C\Omega+D)^{k}f(\Omega)\quad\text{for all }\gamma= \begin{pmatrix}A&B\\ C&D\end{pmatrix} \in\Gamma\);
3. _given_ \(Y_{0}\in\mathbb{R}^{(n,n)}\) _with_ \(Y_{0}=\,^{t}Y_{0}>0\)_, the quantities_ \(\det(C_{j}\Omega+D_{j})^{-k}f(\gamma_{j}\cdot\Omega)\) _are bounded in the region_ \(\{\Omega=X+i\,Y\in\mathbb{H}_{n}\,|\,\,Y\geq Y_{0}\,\}\) _for the set of representatives_ \(\gamma_{j}= \begin{pmatrix}A_{j}&B_{j}\\ C_{j}&D_{j}\end{pmatrix}