# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 6 Invariant Differential Operators on \(\boldsymbol{GL_{n,m}/O(n,\mathbb{R})}\)
The adjoint representation \(\operatorname{Ad}_{\star}\) of \(GL_{n,m}(\mathbb{R})\) is given by  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]7where \((g,\lambda)\in GL_{n,m}(\mathbb{R})\) and \((X,Z)\in\mathfrak{g}_{*}\) and the adjoint representation \(\mathrm{ad}_{*}\) of \(\mathfrak{g}_{*}\) on \(\mathfrak{g}_{*}\) is given by  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]8  
We see that the Killing form \(B_{\star}\) of \(\mathfrak{g}_{*}\) is given by  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]9  
We let  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]0  
Then, the Lie algebra \(\mathfrak{k}_{\star}\) of \(K_{\star}\) is  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]1  
We let \(\mathfrak{p}_{\star}\) be the subspace of \(\mathfrak{g}_{*}\) defined by  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]2  
Then, we have the following relation:  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]3  
In addition, we have  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]4  
\(K_{\star}\) acts on \(\mathfrak{p}_{\star}\) via the adjoint representation \(\mathrm{Ad}_{\star}\) of \(GL_{n,m}(\mathbb{R})\) by  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]5  
where \(k_{\star}=(k,0)\in K_{\star}\) with \(k\in O(n,\mathbb{R})\) and \((X,Z)\in\mathfrak{p}_{\star}\). For brevity, we set \(K=O(n,\mathbb{R})\). Then, Action (68) induces the action of \(K\) on the polynomial algebra \(\mathrm{Pol}(\mathfrak{p}_{\star})\) of \(\mathfrak{p}_{\star}\) and the symmetric algebra \(S(\mathfrak{p}_{\star})\). We denote by \(\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\) (resp. \(S(\mathfrak{p}_{\star})^{K}\)) the subalgebra of \(\mathrm{Pol}(\mathfrak{p}_{\star})\) (resp. \(S(\mathfrak{p}_{\star})\)) consisting of all \(K\)-invariants. The following inner product \((,)_{\star}\) on \(\mathfrak{p}_{\star}\) defined by  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]6  
gives an isomorphism as vector spaces  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]7  
where \(\mathfrak{p}_{\star}^{*}\) denotes the dual space of \(\mathfrak{p}_{\star}\) and \(f_{X,Z}\) is the linear functional on \(\mathfrak{p}_{\star}\) defined by  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]8  
We let \(\mathbb{D}(\mathscr{P}_{n,m})\) be the algebra of all differential operators on \(\mathscr{P}_{n,m}\) that are invariant under Action (61) of \(GL_{n,m}(\mathbb{R})\). It is known that there is a canonical linear bijection of \(S(\mathfrak{p}_{\star})^{K}\) onto \(\mathbb{D}(\mathscr{P}_{n,m})\). Identifying \(\mathfrak{p}_{\star}\) with \(\mathfrak{p}_{\star}^{*}\) by the above Isomorphism (68), we obtain a canonical linear bijection  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]9of \(\mathrm{Pol}(\mathfrak{p}_{\star})^{K}\) onto \(\mathbb{D}(\mathscr{P}_{n,m})\). The map \(\Phi_{n,m}\) is described explicitly as follows. We put \(N_{\star}=n(n+1)/2+mn\). We let \(\{\eta_{\star}\mid 1\leq\alpha\leq N_{\star}\,\}\) be a basis of \(\mathfrak{p}_{\star}\).