# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 5 Invariant Differential Operators on \(Sp(2n,\mathbb{R})/U(n)\)
\in\Gamma_{n}^{\flat}\,(1\leq j\leq h)\)_._  
_We denote by \(M_{n}(\Gamma,k)\) the vector space of all Siegel modular form of weight \(k\) and degree \(n\) for \(\Gamma\). It is known that \(M_{n}(\Gamma,k)\) is finite dimensional. Moreover, a Siegel modular form \(f\in M_{n}(\Gamma,k)\) is called a _Siegel cusp form_ of weight \(k\) and degree \(n\) for \(\Gamma\) if the condition (SI3) is strengthened to the following condition \((\mathrm{SI3})^{*}\):  
1. [label=(SI3)]
2. _given_ \(Y_{0}\in\mathbb{R}^{(n,n)}\) _with_ \(Y_{0}=\,^{t}Y_{0}>0\)_, the quantities_ \(\det(C_{j}\Omega+D_{j})^{-k}f(\gamma_{j}\cdot\Omega)\) _become arbitrarily small in the region_ \(\{\Omega=X+i\,Y\in\mathbb{H}_{n}\,|\,\,Y\geq Y_{0}\,\}\) _for the set of representatives_ \(\gamma_{j}= \begin{pmatrix}A_{j}&B_{j}\\ C_{j}&D_{j}\end{pmatrix} \in\Gamma_{n}^{\flat}\,(1\leq j\leq h)\)_._  
_We denote by \(\mathfrak{C}_{k}^{n}(\Gamma)\) the vector space of Siegel cusp form of weight \(k\) and degree \(n\) for \(\Gamma\). The vector space \(\mathfrak{C}_{k}^{n}(\Gamma)\) is a Hermitian inner product space equipped with the Petersson inner product given by_  
\[\langle f,g\rangle:=\int_{\Gamma\backslash\mathbb{H}_{n}}\det(Y)^{k}\,f( \Omega)\,\overline{g(\Omega)}\,dv_{n}(\Omega)\qquad(\,f,g\in\mathfrak{C}_{k}^{ n}(\Gamma)\,).\]  
We have  
\[\Omega_{\frac{k}{2},-\frac{k}{2}}=-4\cdot Y^{t}\!\!\left(Y\frac{\partial}{ \partial\overline{\Omega}}\right)\frac{\partial}{\partial\Omega}+i\,k\,Y \frac{\partial}{\partial X}\qquad\text{and}\qquad\mathcal{L}_{\frac{k}{2},- \frac{k}{2}}=\,\mathrm{Tr}(\Omega_{\frac{k}{2},-\frac{k}{2}}). \tag{58}\]  
Theorem 11: We let \(\Gamma\subset Sp(2n,\mathbb{R})\) be a subgroup of \(Sp(2n,\mathbb{R})\) commensurable with \(\Gamma_{n}^{\flat}\) and let \(\varphi\in\mathcal{H}_{\frac{k}{2},-\frac{k}{2}}^{n}(\Gamma)\) be a Siegel-Maass form of weight \((\frac{k}{2},-\frac{k}{2})\) for \(\Gamma\). Then, if \(\mathcal{L}_{\frac{k}{2},-\frac{k}{2}}\varphi=\lambda\,\varphi\), then \(\lambda\in\mathbb{R}\) and  
\[\lambda\geq\frac{nk}{4}\,(n+1-k).\]  
The equality holds if and only if \(\varphi(\Omega)=\det(Y)^{k/2}f(\Omega)\) for some Siegel cusp form \(f\in\mathfrak{C}_{k}^{n}(\Gamma)\) of weight \(k\) for \(\Gamma\). In other words,  
\[\mathfrak{C}_{k}^{n}(\Gamma)\cong\mathrm{Ker}\!\left(\mathcal{L}_{\frac{k}{2},- \frac{k}{2}}+\frac{nk}{4}\,(n+1-k)\cdot I_{n}\right) \tag{59}\]  
of complex vector spaces induced by the assignment  
\[f(\Omega)\longmapsto\det(Y)^{k/2}f(\Omega),\qquad(\,\Omega=X+i\,Y\in\mathbb{H }_{n},\,f\in\mathfrak{C}_{k}^{n}(\Gamma)\,).\]  
Proof: The proof can be found in [9] (pp. 15-19). Using the commutative algebra \(\mathbb{D}(\mathbb{H}_{n})\), we introduce the notion of Maass-Siegel function for \(\Gamma_{n}\). **Definition 10**. : _Function \(f:\mathbb{H}_{n}\longrightarrow\mathbb{C}\) is said to be a Maass-Siegel function for \(\Gamma_{n}^{\flat}\) if it satisfies the conditions_ (MS1)-(MS4)_:_  
* \(f\) _is real analytic;_
* \(f(\gamma\cdot\Omega)=f(\Omega)\quad\text{for all }\gamma\in\Gamma_{n}^{\flat}\) _and_ \(\Omega\in\mathbb{H}_{n}\)_;_
* \(f\) _is an eigenfunction of all invariant differential operators in_ \(\mathbb{D}(\mathbb{H}_{n})\)_;_
* _given_ \(Y_{0}\in\mathbb{R}^{(n,n)}\) _with_ \(Y_{0}=\,^{t}Y_{0}>0\)_, the quantities_ \(f(\Omega)\) _are bounded in the region_ \(\{\Omega=X+i\,Y\in\mathbb{H}_{n}\mid Y\geq Y_{0}\,\}\)_._  
_Function \(f:\mathbb{H}_{n}\longrightarrow\mathbb{C}\) is said to be a weak Maass-Siegel function for \(\Gamma_{n}^{\flat}\) if it satisfies the above conditions_ (MS1)_,_ (MS2) _and_ (MS4) _together with_ (MS3)\({}^{*}\)_:_  
* \(f\) _is an eigenfunction of the Laplace operator_ \(\Delta_{n;A}\) _of_ \((\mathbb{H}_{n},ds^{2}_{n;A})\)_._  
**Problem 1**. : _Develop the theory of harmonic analysis of \(L^{2}(\Gamma_{n}^{\flat}\backslash\mathbb{H}_{n})\). Develop the spectral theory of the Laplace operator \(\Delta_{n;A}\) (see Formula (45)) on \(L^{2}(\Gamma_{n}^{\flat}\backslash\mathbb{H}_{n})\)._