# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 6 Invariant Differential Operators on \(\boldsymbol{GL_{n,m}/O(n,\mathbb{R})}\)  
This section is based on papers [15; 35]. We recall that the group  
\[GL_{n,m}(\mathbb{R}):=GL(n,\mathbb{R})\ltimes\mathbb{R}^{(m,n)}\]  
is the semidirect product of \(GL(n,\mathbb{R})\) and the additive group \(\mathbb{R}^{(m,n)}\) endowed with multiplication law  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]  
for all \(g,h\in GL(n,\mathbb{R})\) and \(\alpha,\beta\in\mathbb{R}^{(m,n)}\). We also recall the Minkowski-Euclid space  
\[\mathcal{P}_{n,m}:=\mathcal{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\big{\}}\times\mathbb{R}^{(m,n)}.\]  
Then, \(GL_{n,m}(\mathbb{R})\) acts on \(\mathcal{P}_{n,m}\) naturally and transitively by  
\[(g,\alpha)\cdot(Y,V):=(gY^{t}g,(V+\alpha)\,^{t}g) \tag{61}\]  
for all \((g,\alpha)\in GL_{n,m}(\mathbb{R})\) and \((Y,V)\in\mathcal{P}_{n,m}\). Since \(O(n,\mathbb{R})\) is the stabilizer of the action (61) at \((I_{n},0)\), the non-symmetric homogeneous space \(GL_{n,m}(\mathbb{R})/O(n,\mathbb{R})\) is diffeomorphic to the Minkowski-Euclid space \(\mathcal{P}_{n,m}\). We denote by \(\mathbb{D}(\mathcal{P}_{n,m})\) the algebra of all differential operators on \(\mathcal{P}_{n,m}\) invariant under Action (6.2) of \(GL_{n,m}(\mathbb{R})\). We let  
\[GL_{n,m}(\mathbb{Z}):=GL(n,\mathbb{Z})\ltimes\mathbb{Z}^{(m,n)}\]  
denote the discrete subgroup of \(GL_{n,m}(\mathbb{R})\). For a variable \((Y,V)\in\mathcal{P}_{n,m}\) with \(Y\in\mathcal{P}_{n}\) and \(V\in\mathbb{R}^{(m,n)}\), we put  
\[Y=(y_{ij})\text{ with }y_{ij}=y_{ji},\,\,\,V=(v_{kl}),\]  
\[dY=(dy_{ij}),\,\,dV=(dv_{kl}),\]  
\[[dY]=\bigwedge_{i\leq j}dy_{ij},\,\,\,\,\,\,\,[dV]=\bigwedge_{k,l}dv_{kl},\]and  
\[\frac{\partial}{\partial Y}=\left(\frac{1+\delta_{ij}}{2}\frac{\partial}{\partial y _{ij}}\right),\ \ \ \frac{\partial}{\partial V}=\left(\frac{\partial}{\partial v_{kl}}\right),\]  
where \(1\leq i,j,l\leq n\) and \(1\leq k\leq m\). For a fixed element \((g,\alpha)\in GL_{n,m}(\mathbb{R})\), we write  
\[(Y_{\star},V_{\star})=(g,\alpha)\cdot(Y,V)=\big{(}g\,Y\,^{t}g,\,(V+\alpha)\,^{ t}g\big{)},\]  
where \((Y,V)\in\mathscr{P}_{n,m}\). Then, we obtain  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]0  
and  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]1  
**Lemma 1**. : _For all two positive real numbers \(a\) and \(b\), the following metric \(ds^{2}_{n,m;a,b}\) on \(\mathscr{P}_{n,m}\) defined by_  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]2  
_is a Riemannian metric on \(\mathscr{P}_{n,m}\) which is invariant under Action (61) of \(GL_{n,m}(\mathbb{R})\). The Laplacian \(\Delta_{n,m;a,b}\) of \((\mathscr{P}_{n,m},\ ds^{2}_{n,m;a,b})\) is given by_  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]3  
_Moreover, \(\triangle_{n,m;a,b}\) is a differential operator of order 2 which is invariant under Action (61) of \(GL_{n,m}(\mathbb{R})\)._  
Proof. : The proof can be found in [15] (Lemma 8.1, p. 312). **Lemma 2**. : _The following volume element \(d\mu_{n,m}(Y,V)\) on \(\mathscr{P}_{n,m}\) defined by_  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]4  
_is invariant under Action (61) of \(GL_{n,m}(\mathbb{R})\)._  
Proof. : The proof can be found in [15] (Lemma 8.2, pp. 312-313). The Lie algebra \(\mathfrak{g}_{\star}\) of \(GL_{n,m}(\mathbb{R})\) is given by  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]5  
equipped with the following Lie bracket:  
\[(g,\alpha)\circ(h,\beta):=(gh,\alpha\,^{t}h^{-1}+\beta) \tag{60}\]6  
where \([X_{1},X_{2}]_{0}=X_{1}X_{2}-X_{2}X_{1}\) denotes the usual matrix bracket and \((X_{1},Z_{1}),\ (X_{2},Z_{2})\in\mathfrak{g}_{\star}\).