# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 8 Invariant Differential Operators on \(G^{j}/(U(n)\times S(m,\mathbb{R}))\)
: _For any two positive real numbers \(A\) and \(B\),_  
\[ds_{n,m;\mu;A,B}^{2} = A\cdot\text{Tr}\Big{(}Y^{-1}d\Omega\,Y^{-1}d\overline{\Omega} \Big{)}\] \[+B\left\{\text{Tr}\Big{(}Y^{-1}\,{}^{t}V\,V\,Y^{-1}d\Omega\,Y^{-1 }d\overline{\Omega}\Big{)}+\,\text{Tr}\Big{(}Y^{-1}\,{}^{t}(dZ)\,d\overline{Z} \Big{)}\right.\] \[\left.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  
\[\mathfrak{e}^{l}=\Big{\{}\big{(}X,(0,0,R)\big{)}\;\big{|}\;X\in\mathfrak{e},\;R=\;^ {l}\!R\in\mathbb{R}^{(m,m)}\;\Big{\}},\]  
\[\mathfrak{p}^{l}=\Big{\{}\big{(}Y,(P,Q,0)\big{)}\;\big{|}\;Y\in\mathfrak{m},\;P,Q\in\mathbb{R}^{(m,n)}\;\Big{\}}.\]  
Here,  
\[\mathfrak{g}=\Bigg{\{} \begin{pmatrix}X_{1}&X_{2}\\ X_{3}&-\,^{t}X_{1}\end{pmatrix} \Big{|}\;X_{1},X_{2},X_{3}\in\mathbb{R}^{(n,n)},\;X_{2}=\;^{t}X_{2},\;X_{3}=\;^{t}X_{3}\;\Bigg{\}},\]  
\[\mathfrak{e}=\Bigg{\{} \begin{pmatrix}X&-Y\\ Y&X\end{pmatrix} \in\mathbb{R}^{(2n,2n)}\;\Big{|}\;^{t}X+X=0,\;Y=\;^{t}Y\Bigg{\}}\]  
and  
\[\mathfrak{m}=\Bigg{\{} \begin{pmatrix}X&Y\\ Y&-X\end{pmatrix} \Big{|}\;X=\;^{t}X,\;Y=\;^{t}Y,\;X,Y\in\mathbb{R}^{(n,n)}\; \Bigg{\}}.\]  
We note that \(\mathfrak{g}\) is the Lie algebra of \(Sp(2n,\mathbb{R})\) and \(\mathfrak{e}\) is the Lie algebra of \(K\cong U(n)\). Thus, the tangent space of the homogeneous space \(\mathbb{H}_{n,m}\) at \((iI_{n},0)\) is identified with \(\mathfrak{p}^{l}\). If \(\alpha=\left( \begin{pmatrix}X_{1}&Y_{1}\\ Z_{1}&-\,^{l}\!X_{1}\end{pmatrix} ,(P_{1},Q_{1},R_{1})\right)\) and \(\beta=\left( \begin{pmatrix}X_{2}&Y_{2}\\ Z_{2}&-\,^{l}\!X_{2}\end{pmatrix} ,(P_{2},Q_{2},R_{2})\right)\) are elements of \(\mathfrak{g}^{l}\), then the Lie bracket \([\alpha,\beta]\) of \(\alpha\) and \(\beta\) is given by  
\[[\alpha,\beta]=\left( \begin{pmatrix}X^{*}&Y^{*}\\ Z^{*}&-\,^{l}\!X^{*}\end{pmatrix} ,(P^{*},Q^{*},R^{*})\right), \tag{98}\]  
where  
\[X^{*} = X_{1}X_{2}-X_{2}X_{1}+Y_{1}Z_{2}-Y_{2}Z_{1},\] \[Y^{*} = X_{1}Y_{2}-X_{2}Y_{1}+Y_{2}\,^{l}\!X_{1}-Y_{1}\,^{l}\!X_{2},\] \[Z^{*} = Z_{1}X_{2}-Z_{2}X_{1}+\,^{l}\!X_{2}Z_{1}-\,^{l}\!X_{1}Z_{2},\] \[P^{*} = P_{1}X_{2}-P_{2}X_{1}+Q_{1}Z_{2}-Q_{2}Z_{1},\] \[Q^{*} = P_{1}Y_{2}-P_{2}Y_{1}+Q_{2}\,^{l}\!X_{1}-Q_{1}\,^{l}\!X_{2},\] \[R^{*} = P_{1}\,^{l}\!Q_{2}-P_{2}\,^{l}\!Q_{1}+Q_{2}\,^{l}\!P_{1}-Q_{1}\, ^{l}\!P_{2}.\]  
**Lemma 3**. : \[[\mathfrak{e}^{l},\mathfrak{e}^{l}]\subset\mathfrak{e}^{l},\quad[\mathfrak{ e}^{l},\mathfrak{p}^{l}]\subset\mathfrak{p}^{l}.\]  
Proof. : The proof follows immediately from Formula (98). **Lemma 4**. : _Let_  
\[k^{l}=\left( \begin{pmatrix}A&-B\\ B&A\end{pmatrix} ,(0,0,\kappa)\right)\in K^{l}\]  
_with \( \begin{pmatrix}A&-B\\ B&A\end{pmatrix} \in K\), \(\kappa=\,^{t}\kappa\in\mathbb{R}^{(m,m)}\) and_  
\[\alpha=\left( \begin{pmatrix}X&Y\\ Y&-X\end{pmatrix} ,(P,Q,0)\right)\in\mathfrak{p}^{l}\]  
_with \(X=\,^{t}X\), \(Y=\,^{t}Y\in\mathbb{R}^{(n,n)},\;P,Q\in\mathbb{R}^{(m,n)}\). Then, the adjoint action of \(K^{l}\) on \(\mathfrak{p}^{l}\) is given by_  
\[\mathrm{Ad}(k^{l})\alpha=\left( \begin{pmatrix}X_{*}&Y_{*}\\ Y_{*}&-X_{*}\end{pmatrix} ,(P_{*},Q_{*},0)\right), \tag{99}\]_where_  
\[X_{*} = AX^{\prime}A-\left(BX^{\prime}B+BY^{\prime}A+AY^{\prime}B\right), \tag{100}\] \[Y_{*} = \left(AX^{\prime}B+AY^{\prime}A+BX^{\prime}A\right)-BY^{\prime}B,\] (101) \[P_{*} = P^{\prime}A-Q^{\prime}B,\] (102) \[Q_{*} = P^{\prime}B+Q^{\prime}A. \tag{103}\]  
Proof. : We leave the proof to the reader. We recall that \(T_{n}\) denotes the vector space of all \(n\times n\) symmetric complex matrices. For brevity, we put \(T_{n,m}:=T_{n}\times\mathbb{C}^{(m,n)}\). We define the real linear isomorphism \(\Phi:\mathfrak{p}^{J}\longrightarrow T_{n,m}\) by  
\[\Phi\left( \begin{pmatrix}X&Y\\ Y&-X\end{pmatrix} ,(P,Q,0)\right)=\left(X\,+\,i\,Y,\,P\,+\,i\,Q\right), \tag{104}\]  
where \( \begin{pmatrix}X&Y\\ Y&-X\end{pmatrix} \in\mathfrak{m}\) and \(P,Q\in\mathbb{R}^{(m,n)}\). We let \(S(m,\mathbb{R})\) denote the additive group consisting of all \(m\times m\) real symmetric matrices.