# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 8 Invariant Differential Operators on \(G^{j}/(U(n)\times S(m,\mathbb{R}))\)
Now, we define the isomorphism \(\theta:K^{J}\longrightarrow U(n)\times S(m,\mathbb{R})\) by  
\[\theta(h,(0,0,\kappa))=(\delta(h),\kappa),\quad h\in K,\,\kappa\in S(m, \mathbb{R}), \tag{105}\]  
where \(\delta:K\longrightarrow U(n)\) is the map defined by Formula (49). Identifying \(\mathbb{R}^{(m,n)}\times\mathbb{R}^{(m,n)}\) with \(\mathbb{C}^{(m,n)}\), we can identify \(\mathfrak{p}^{J}\) with \(T_{n,m}=T_{n}\times\mathbb{C}^{(m,n)}\). **Theorem 15**. : _The adjoint representation of \(K^{J}\) on \(\mathfrak{p}^{J}\) is compatible with the natural action of \(U(n)\times S(m,\mathbb{R})\) on \(T_{n,m}\) defined by_  
\[(h,\kappa)\cdot(\omega,z):=(h\,\omega\,^{\prime}h,\,z\,^{\prime}h),\qquad h \in U(n),\,\,\kappa\in S(m,\mathbb{R}),\,\,(\omega,z)\in T_{n,m} \tag{106}\]  
_through maps \(\Phi\) and \(\theta\). Precisely, if \(k^{J}\in K^{J}\) and \(\alpha\in\mathfrak{p}^{J}\), then we have the following equality:_  
\[\Phi\left(\operatorname{Ad}\left(k^{J}\right)\alpha\right)=\theta\big{(}k^{J }\big{)}\cdot\Phi(\alpha). \tag{107}\]  
_Here, we regard the complex vector space \(T_{n,m}\) as a real vector space._  
Proof. : The proof can be found in [30] (pp. 286-287). We now study algebra \(\mathbb{D}(\mathbb{H}_{n,m})\) of all differential operators on \(\mathbb{H}_{n,m}\) invariant under the _natural Action_ (96) of \(G^{J}\). Action (106) induces the action of \(U(n)\) on the polynomial algebra \(\operatorname{Pol}_{n,m}:=\,\operatorname{Pol}\left(T_{n,m}\right)\). We denote by \(\operatorname{Pol}_{n,m}^{U(n)}\) the subalgebra of \(\operatorname{Pol}_{n,m}\) consisting of all \(U(n)\)-invariants. Similarly, Action (99) of \(K\) induces the action of \(K\) on the polynomial algebra \(\operatorname{Pol}\left(\mathfrak{p}^{J}\right)\). We see that through the identification of \(\mathfrak{p}^{J}\) with \(T_{n,m}\), the algebra \(\operatorname{Pol}\left(\mathfrak{p}^{J}\right)\) is isomorphic to \(\operatorname{Pol}_{n,m}\). The following \(U(n)\)-invariant inner product \((\,,)_{J}\) of the complex vector space \(T_{n,m}\) defined by  
\[\left((\omega,z),(\omega^{\prime},z^{\prime})\right)_{J}=\operatorname{Tr} \left(\omega\overline{\omega^{\prime}}\,\right)+\operatorname{Tr}\left(z\,^{ \prime}\overline{z^{\prime}}\,\right),\quad(\omega,z),\,(\omega^{\prime},z^{ \prime})\in T_{n,m}\]  
gives a canonical isomorphism  
\[T_{n,m}\cong\,T_{n,m}^{*},\quad(\omega,z)\mapsto f_{\omega,z},\quad(\omega,z) \in T_{n,m},\]where \(f_{\omega,z}\) is the linear functional on \(T_{n,m}\) defined by  
\[f_{\omega,z}\big{(}(\omega^{\prime},z^{\prime})\big{)}=\big{(}(\omega^{\prime},z^ {\prime}),(\omega,z)\big{)}_{J^{\prime}}\quad(\omega^{\prime},z^{\prime})\in T_{ n,m}.\]  
According to [PERSON] [4] (p. 287), one obtains a canonical linear bijection of \(S(T_{n,m})^{U(n)}\) onto \(\mathbb{D}(\mathbb{H}_{n,m})\). Here, \(S(T_{n,m})\) denotes the symmetric algebra of \(T_{n,m}\) and \(S(T_{n,m})^{U(n)}\) denotes the subalgebra of all \(U(n)\)-invariants in \(S(T_{n,m})\). Identifying \(T_{n,m}\) with \(T_{n,m}^{*}\) by the above isomorphism, one obtains a natural linear bijection  
\[\Theta_{n,m}:\,\mathrm{Pol}_{n,m}^{U(n)}\longrightarrow\mathbb{D}(\mathbb{H}_{ n,m})\]  
of \(\mathrm{Pol}_{n,m}^{U(n)}\) onto \(\mathbb{D}(\mathbb{H}_{n,m})\). The map \(\Theta_{n,m}\) is described explicitly as follows. We put \(N_{*}=n(n+1)+2\,mn\). We let \(\big{\{}\eta_{\alpha}\mid 1\leq\alpha\leq N_{*}\big{\}}\) be a basis of \(\mathfrak{p}^{J}\). If \(P\in\mathrm{Pol}\big{(}\mathfrak{p}^{J}\big{)}^{K}=\mathrm{Pol}_{n,m}^{U(n)}\), then  
\[\Big{(}\Theta_{n,m}(P)f\Big{)}(gK^{J})=\left[P\bigg{(}\frac{\partial}{ \partial t_{\alpha}}\Big{)}f\bigg{(}g\cdot\exp\left(\sum_{\alpha=1}^{N_{*}}t_{ \alpha}\eta_{\alpha}\right)K^{J}\bigg{)}\right]_{(t_{\alpha})=0}, \tag{108}\]  
where \(g\in G^{J}\) and \(f\in C^{\infty}(\mathbb{H}_{n,m})\). In general, it is hard to express \(\Theta_{n,m}(P)\) explicitly for a polynomial \(P\in\mathrm{Pol}_{n,m}^{U(n)}\). We propose the following natural problems. **Problem 19**.