# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 6 Invariant Differential Operators on \(\boldsymbol{GL_{n,m}/O(n,\mathbb{R})}\)
If \(P\in\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\), then  
\[(g,\alpha)\cdot(Y,V):=(gY^{t}g,(V+\alpha)\,^{t}g) \tag{61}\]0  
where \(f\in C^{\infty}(\mathscr{P}_{n,m})\). We refer to [4] (pp. 280-289). In general, it is very hard to express \(\Phi_{n,m}(P)\) explicitly for a polynomial \(P\in\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\). We take a coordinate \((X,Z)\) in \(\mathfrak{p}_{\star}\) such that  
\[X=\left( \begin{array}{cccc}x_{11}&\frac{1}{2}x_{12}&\ldots&\frac{1}{2}x_{1n }\\ \frac{1}{2}x_{12}&x_{22}&\ldots&\frac{1}{2}x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{1}{2}x_{1n}&\frac{1}{2}x_{2n}&\ldots&x_{mn}\end{array} \right)\in \mathfrak{p}\quad\text{and}\quad Z=(z_{kl})\in\mathbb{R}^{(m,n)}.\]  
Here,  
\[\mathfrak{p}:=\{\,X\in\mathbb{R}^{(n,n)}\mid X=\,^{t}X\,\}.\]  
We define the polynomials \(\alpha_{j},\ \beta_{pq}^{(k)},\ R_{jp}\), and \(S_{jp}\) on \(\mathfrak{p}_{\star}\) by  
\[\alpha_{j}(X,Z) = \mathrm{Tr}\left(X^{j}\right),\quad 1\leq j\leq n, \tag{71}\] \[\beta_{pq}^{(k)}(X,Z) = \left(Z\,X^{k}\,{}^{t}Z\right)_{pq^{\prime}},\quad 0\leq k\leq n-1, \ 1\leq p\leq q\leq m,\] (72) \[R_{jp}(X,Z) = \mathrm{Tr}\left(X^{j}(\,{}^{t}Z\,{}^{p})\,\right),\quad 0\leq j \leq n-1,\ 1\leq p\leq m,\] (73) \[S_{jp}(X,Z) = \mathrm{det}\left(X^{j}(\,{}^{t}Z\,{}^{p})\,\right),\quad 0\leq j \leq n-1,\ 1\leq p\leq m, \tag{74}\]  
where \(\left(ZX^{t}Z\right)_{pq}\) denotes the \((p,q)\)-entry of \(ZX^{t}Z\). We propose the following natural problems. **Problem 2**. : _Find a complete list of explicit generators of \(\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\)._  
**Problem 3**. : _Find all the relations among a set of generators of \(\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\)._  
**Problem 4**. : _Find an easy or effective way to express the images of the above invariant polynomials under the Helgason map \(\Phi_{n,m}\) explicitly._  
**Problem 5**. : _Decompose \(\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\) into \(O(n,\mathbb{R})\)-irreducibles._  
**Problem 6**. : _Find a complete list of explicit generators of the algebra \(\mathbb{D}(\mathscr{P}_{n,m})\). Or construct explicit \(GL_{n,m}(\mathbb{R})\)-invariant differential operators on \(\mathscr{P}_{n,m}\)._  
**Problem 7**. : _Find all the relations among a set of generators of \(\mathbb{D}(\mathscr{P}_{n,m})\)._  
**Problem 8**. : _Is \(\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\) finitely generated? Is \(\mathbb{D}(\mathscr{P}_{n,m})\) finitely generated?_  
**Problem 9**. : _Find the center \(\mathscr{X}_{n,m}\) of \(\mathbb{D}(\mathscr{P}_{n,m})\)._  
[PERSON] [36] proved the following theorem. **Theorem 12**. : \(\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\) _is generated by \(\alpha_{j}\ (1\leq j\leq n)\) and \(\beta_{pq}^{(k)}\ (0\leq k\leq n-1,\ 1\leq p\leq q\leq m)\)._  
Proof. : We refer to [36] (Theorem 3.1). According to the above theorem, he solved Problem 2 and Problem 8. He also solved Problem 3 in [36] (Theorem 3.2). We present some invariant differential operators on \(\mathscr{P}_{n,m}\). We define the differential operators \(D_{j},\ \Omega_{pq}\), and \(L_{p}\) on \(\mathscr{P}_{n,m}\) by  
\[D_{j}=\,\mathrm{Tr}\Bigg{(}\bigg{(}2Y\frac{\partial}{\partial Y}\bigg{)}^{j} \Bigg{)},\quad 1\leq j\leq n, \tag{75}\]  
\[\Omega_{pq}^{(k)}=\,\Bigg{\{}\,\frac{\partial}{\partial V}\bigg{(}2Y\frac{ \partial}{\partial Y}\bigg{)}^{k}Y^{l}\bigg{(}\frac{\partial}{\partial V} \bigg{)}\,\Bigg{\}}\,,\quad 0\leq k\leq n-1,\ 1\leq p\leq q\leq m \tag{76}\]  
and  
\[L_{p}=\,\mathrm{Tr}\Bigg{(}\bigg{\{}Y^{l}\bigg{(}\frac{\partial}{\partial V} \bigg{)}\frac{\partial}{\partial V}\bigg{\}}^{p}\Bigg{)},\ \ 1\leq p\leq m. \tag{77}\]  
Here, for matrix \(A\) we denote by \(A_{pq}\) the \((p,q)\)-entry of \(A\). Also, we define the invariant differential operators \(\mathfrak{S}_{jp}\) by  
\[\mathfrak{S}_{jp}=\,\mathrm{Tr}\Bigg{(}\bigg{(}\,2Y\frac{\partial}{\partial Y }\bigg{)}^{j}\bigg{\{}Y^{l}\bigg{(}\frac{\partial}{\partial V}\bigg{)}\frac{ \partial}{\partial V}\bigg{\}}^{p}\Bigg{)}, \tag{78}\]  
where \(1\leq j\leq n\) and \(1\leq p\leq m\). **Remark 14**.