# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 7 Invariant Differential Operators on \(SL_{n,m}(\mathbb{R})/SO(n,\mathbb{R})\)
Since \(\text{Ad}_{\circ}(k)\mathfrak{p}_{\circ}\subset\mathfrak{p}_{\circ}\) for any \(k\in K_{\circ}\), \(K_{\circ}\) acts on \(\mathfrak{p}_{\circ}\) via the adjoint representation of \(K_{\circ}\) on \(\mathfrak{p}_{\circ}\) by  
\[k_{\circ}\cdot(X,Z)=\big{(}kX^{t}k,Z^{t}k\big{)}, \tag{91}\]  
where \(k_{\circ}=(k,0)\in K_{\circ}\) with \(k\in SO(n,\mathbb{R})\) and \((X,Z)\in\mathfrak{p}_{\circ}\). We put \(K_{\
atural}=SO(n,\mathbb{R})\). Action (91) induces the action of \(K_{\
atural}\) on the polynomial algebra \(\text{Pol}(\mathfrak{p}_{\circ})\) of \(\mathfrak{p}_{\circ}\) and the symmetric algebra \(S(\mathfrak{p}_{\circ})\). We denote by \(\text{Pol}(\mathfrak{p}_{\circ})^{K_{\
atural}}\) (resp. \(S(\mathfrak{p}_{\circ})^{K_{\
atural}}\)) the subalgebra of \(\text{Pol}(\mathfrak{p}_{\circ})\) (resp. \(S(\mathfrak{p}_{\circ})\)) consisting of all \(K_{\
atural}\)-invariants. The following inner product \((,)_{\circ}\) on \(\mathfrak{p}_{\circ}\) defined by  
\[\big{(}(X_{1},Z_{1}),(X_{2},Z_{2})\big{)}_{\circ}=\text{Tr}(X_{1}X_{2})+ \text{Tr}(Z_{1}\,^{t}Z_{2}),\quad(X_{1},Z_{1}),(X_{2},Y_{2})\in\mathfrak{p}_{\circ}\]  
gives an isomorphism as vector spaces  
\[\mathfrak{p}_{\circ}\cong\mathfrak{p}_{\circ}^{*}\quad(X,Z)\mapsto f_{X,Z}, \quad(X,Z)\in\mathfrak{p}_{\circ}, \tag{92}\]where \(\mathfrak{p}_{\circ}^{*}\) denotes the dual space of \(\mathfrak{p}_{\circ}\) and \(f_{X,Z}\) is the linear functional on \(\mathfrak{p}_{\circ}\) defined by  
\[f_{X,Z}\big{(}(X_{1},Z_{1})\big{)}=\big{(}(X,Z),(X_{1},Z_{1})\big{)}_{\circ}, \quad(X_{1},Z_{1})\in\mathfrak{p}_{\circ}.\]  
We let \(\mathbb{D}(\mathfrak{p}_{n,m})\) be the algebra of all differential operators on \(\mathfrak{P}_{n,m}\) that are invariant under Action (81) of \(GL_{n,m}(\mathbb{R})\). It is known that there is a canonical linear bijection of \(S(\mathfrak{p}_{\circ})^{K}\) onto \(\mathbb{D}(\mathfrak{P}_{n,m})\). Identifying \(\mathfrak{p}_{\circ}\) with \(\mathfrak{p}_{\circ}^{*}\) by the above Isomorphism (92), we obtain a canonical linear bijection  
\[\Psi_{n,m}:\operatorname{Pol}(\mathfrak{p}_{\circ})^{K_{\
atural}}\longrightarrow \mathbb{D}(\mathfrak{P}_{n,m}) \tag{93}\]  
of \(\operatorname{Pol}(\mathfrak{p}_{\circ})^{K_{\
atural}}\) onto \(\mathbb{D}(\mathfrak{P}_{n,m})\). The map \(\Psi_{n,m}\) is described explicitly as follows. We put \(N_{\circ}=n(n+1)/2+mn-1\). We let \(\{\xi_{\alpha}\mid 1\leq\alpha\leq N_{\circ}\,\}\) be a basis of \(\mathfrak{p}_{\circ}\). If \(P\in\operatorname{Pol}(\mathfrak{p}_{\circ})^{K_{\
atural}}\); then,  
\[\Big{(}\Psi_{n,m}(P)f\Big{)}(gK_{\circ})=\left[P\Big{(}\frac{\partial}{ \partial t_{\alpha}}\Big{)}f\Bigg{(}g\cdot\exp\left(\sum_{\alpha=1}^{N_{\circ }}t_{\alpha}\xi_{\alpha}\right)K_{\circ}\Bigg{)}\right]_{(t_{\alpha})=0}, \tag{94}\]  
where \(f\in C^{\infty}(\mathfrak{P}_{n,m})\). We refer to [4] (pp. 280-289). In general, it is very hard to express \(\Psi_{n,m}(P)\) explicitly for a polynomial \(P\in\operatorname{Pol}(\mathfrak{p}_{\circ})^{K_{\
atural}}\). We take a coordinate \((X,Z)\) in \(\mathfrak{p}_{\circ}\) such that  
\[X= \begin{pmatrix}x_{11}&\frac{1}{2}x_{12}&\dots&\frac{1}{2}x_{1n}\\ \frac{1}{2}x_{12}&x_{22}&\dots&\frac{1}{2}x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{1}{2}x_{1n}&\frac{1}{2}x_{2n}&\dots&x_{nn}\end{pmatrix} \in\mathfrak{p}_ {\flat}\quad\text{and}\quad Z=(z_{kl})\in\mathbb{R}^{(m,n)},\]  
where  
\[\mathfrak{p}_{\flat}=\{\,X\in\mathbb{R}^{(n,n)}\mid X={}^{t}X,\,\,\operatorname {Tr}(X)=0\,\}.\]  
We propose the following natural problems. **Problem 10**. : _Find a complete list of explicit generators of \(\operatorname{Pol}(\mathfrak{p}_{\circ})^{K_{\
atural}}\)._  
**Problem 11**. : _Find all the relations among a set of generators of \(\operatorname{Pol}(\mathfrak{p}_{\circ})^{K_{\
atural}}\)._  
**Problem 12**. : _Find an easy or effective way to express the images of the above invariant polynomials under the Helgason map \(\Psi_{n,m}\) explicitly._  
**Problem 13**.