# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 8 Invariant Differential Operators on \(G^{j}/(U(n)\times S(m,\mathbb{R}))\)
We can show that algebra \(\mathbb{D}(\mathbb{H}_{1,1})\) is generated by the following differential operators:  
\[(h,\kappa)\cdot(\omega,z):=(h\,\omega\,^{\prime}h,\,z\,^{\prime}h),\qquad h \in U(n),\,\,\kappa\in S(m,\mathbb{R}),\,\,(\omega,z)\in T_{n,m} \tag{106}\]9 \[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]0 \[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]1 \[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]2 \[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]3  
and  
\[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]4 \[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]5where \(\tau=x+iy\) and \(z=u+iv\) with real variables \(x,y,u,v\). Moreover, we have  
\[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]6 \[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]7  
In particular, algebra \(\mathbb{D}(\mathbb{H}_{1,1})\) is not commutative. We refer to [45] for more detail. [PERSON] [46] proved the following results. **Lemma 5**. : _We have the following relation:_  
\[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]8  
_This relation exhausts all the relations among the generators \(q\), \(\alpha\), \(\phi\), and \(\psi\) of \(\mathrm{Pol}^{U(1)}_{1,1}\)._  
Proof. : This follows from a direct computation. **Theorem 18**. : _We have the following relations:_  
1. \([D_{1},D_{2}]=2D_{3}\)_,_
2. \([D_{1},D_{3}]=2D_{1}D_{2}-2D_{3}\)_,_
3. \([D_{2},D_{3}]=-D_{2}^{2}\)_,_
4. \([D_{4},D_{1}]=0\)_,_
5. \([D_{4},D_{2}]=0\)_,_
6. \([D_{4},D_{3}]=0\)_,_
7. \(D_{3}^{2}+D_{4}^{2}=D_{2}D_{1}D_{2}\)_._  
_These seven relations exhaust all the relations among generators_ \(D_{1},\,D_{2},\,D_{3}\)_, and_ \(D_{4}\) _of_ \(\mathbb{D}(\mathbb{H}_{1,1})\)_._  
Proof. : The proof can be found in [46]. Finally, we see that for the case when \(n=m=1\), the above eight problems are completely solved. **Remark 19**. : _According to Theorem 18, we see that \(D_{4}\) is a generator of the center of \(\mathbb{D}(\mathbb{H}_{1,1})\). We observe that the Lapalcian_  
\[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]9  
_of \((\mathbb{H}_{1,1},ds^{2}_{1,1;A,B})\) does not belong to the center of \(\mathbb{D}(\mathbb{H}_{1,1})\)._  
**Remark 20**. : _When \(n=1\) and \(m\) is an arbitrary integer, [PERSON] and [PERSON] [47] found the \(2m^{2}+m+1\) explicit generators of \(\mathbb{D}(\mathbb{H}_{1,m})\) and the explicit one generator of the center of \(\mathbb{D}(\mathbb{H}_{1,m})\). They also found the generators of the center of the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g}^{I})\) of the Jacobi Lie algebra \(\mathfrak{g}^{I}\). The number of generators of the center of \(\mathfrak{U}(\mathfrak{g}^{I})\) is \(1+\frac{m(m+1)}{2}\)._  
We set \(\Gamma_{n}^{\gamma}:=Sp(2n,\mathbb{Z})\) and \(\Gamma^{J}:=\Gamma_{n}^{\gamma}\ltimes H_{\mathbb{Z}}^{(n,m)}\) (see notations). **Definition 11**. : _Function \(f:\mathbb{H}_{n,m}\,\longrightarrow\,\mathbb{C}\) is called a Maass-Jacobi function for \(\Gamma^{J}\) if it satisfies the following conditions (MJ1)-(MJ4):_  
1.