# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 5 Invariant Differential Operators on \(Sp(2n,\mathbb{R})/U(n)\)
\begin{pmatrix}X&Y\\ Y&-X\end{pmatrix} \Big{|}\ X=\,^{t}X,\ Y=\,^{t}Y,\ X,Y\in\mathbb{R}^{(n,n)}\,\right\}.\]  
The subspace \(\mathfrak{m}\) of \(\mathfrak{g}\) may be regarded as the tangent space of \(\mathbb{H}_{n}\) at \(il_{n}\). The adjoint representation of \(G\) on \(\mathfrak{g}\) induces the action of \(K\) on \(\mathfrak{m}\) given by  
\[k\circ Z=\,kZ\,^{t}k,\quad\text{where}\ k\in K\ \text{and}\ Z\in\mathfrak{m}. \tag{47}\]We let \(T_{n}\) be the vector space of \(n\times n\) symmetric complex matrices. We let \(\Psi:\mathfrak{m}\longrightarrow T_{n}\) be the map defined by  
\[\Psi\left( \begin{pmatrix}X&Y\\ Y&-X\end{pmatrix} \right)=\,X\,+\,i\,Y,\quad \begin{pmatrix}X&Y\\ Y&-X\end{pmatrix} \in\mathfrak{m}. \tag{48}\]  
We let \(\delta:K\longrightarrow U(n)\) be the isomorphism defined by  
\[\delta\left( \begin{pmatrix}A&-B\\ B&A\end{pmatrix} \right)=\,A\,+\,i\,B,\quad \begin{pmatrix}A&-B\\ B&A\end{pmatrix} \in K, \tag{49}\]  
where \(U(n)\) denotes the unitary group of degree \(n\). We identify \(\mathfrak{m}\) (resp. \(K\)) with \(T_{n}\) (resp. \(U(n)\)) through the map \(\Psi\) (resp. \(\delta\)). We consider the action of \(U(n)\) on \(T_{n}\) defined by  
\[h\cdot\omega\,=\,h\omega\,^{t}h,\quad h\in U(n),\,\,\omega\in T_{n}. \tag{50}\]  
Then, the adjoint Action (47) of \(K\) on \(\mathfrak{m}\) is compatible with Action (50) of \(U(n)\) on \(T_{n}\) through map \(\Psi\). Precisely for any \(k\in K\) and \(Z\in\mathfrak{m}\), we obtain  
\[\Psi(k\,Z\,^{t}k)=\delta(k)\,\Psi(Z)\,^{t}\delta(k). \tag{51}\]  
Action (50) induces the action of \(U(n)\) on the polynomial algebra \(\operatorname{Pol}(T_{n})\) and the symmetric algebra \(S(T_{n})\), respectively. We denote by \(\operatorname{Pol}(T_{n})^{U(n)}\) (resp. \(S(T_{n})^{U(n)}\)) the subalgebra of \(\operatorname{Pol}(T_{n})\) (resp. \(S(T_{n})\) ) consisting of \(U(n)\)-invariants. The following inner product \((,)\) on \(T_{n}\) defined by  
\[(Z,W)=\,\operatorname{Tr}\bigl{(}Z\,\overline{W}\,\bigr{)},\quad Z,W\in T_{n}\]  
gives an isomorphism as vector spaces  
\[T_{n}\cong T_{n}^{*},\quad Z\mapsto h_{Z},\quad Z\in T_{n}, \tag{52}\]  
where \(T_{n}^{*}\) denotes the dual space of \(T_{n}\) and \(f_{Z}\) is the linear functional on \(T_{n}\) defined by  
\[h_{Z}(W)=(W,Z),\quad W\in T_{n}.\]  
It is known that there is a canonical linear bijection of \(S(T_{n})^{U(n)}\) onto the algebra \(\mathbb{D}(\mathbb{H}_{n})\) of differential operators on \(\mathbb{H}_{n}\) invariant under Action (43) of \(G\). Identifying \(T_{n}\) with \(T_{n}^{*}\) by the above isomorphism (52), we obtain a canonical linear bijection  
\[\Theta_{n}:\operatorname{Pol}(T_{n})^{U(n)}\longrightarrow\mathbb{D}( \mathbb{H}_{n}) \tag{53}\]  
of \(\operatorname{Pol}(T_{n})^{U(n)}\) onto \(\mathbb{D}(\mathbb{H}_{n})\). The map \(\Theta_{n}\) is described explicitly as follows. Similarly, Action (47) induces the action of \(K\) on the polynomial algebra \(\operatorname{Pol}(\mathfrak{m})\) and the symmetric algebra \(S(\mathfrak{m})\), respectively. Through map \(\Psi\), the subalgebra \(\operatorname{Pol}(\mathfrak{m})^{K}\) of \(\operatorname{Pol}(\mathfrak{m})\) consisting of \(K\)-invariants is isomorphic to \(\operatorname{Pol}(T_{n})^{U(n)}\). We put \(N_{*}=n(n+1)\). We let \(\{\xi_{\alpha}\mid 1\leq\alpha\leq N_{*}\,\}\) be a basis of a real vector space \(\mathfrak{m}\). If \(P\in\operatorname{Pol}(\mathfrak{m})^{K}\), then  
\[\Bigl{(}\Theta_{n}(P)f\Bigr{)}(gK)=\left[P\Bigl{(}\frac{\partial}{\partial t_ {\alpha}}\Bigr{)}f\left(g\cdot\exp\left(\sum_{\alpha=1}^{N_{*}}t_{\alpha}\xi_{ \alpha}\right)K\right)\right]_{(t_{\alpha})=0}, \tag{54}\]  
where \(f\in C^{\infty}(\mathbb{H}_{n})\). We refer to [4] (Theorem 4.6, pp. 285-286) for more detail. In general, it is hard to express \(\Theta_{n}(P)\) explicitly for a polynomial \(P\in\operatorname{Pol}(\mathfrak{m})^{K}\).