# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 7 Invariant Differential Operators on \(SL_{n,m}(\mathbb{R})/SO(n,\mathbb{R})\)  
We recall that the group  
\[SL_{n,m}(\mathbb{R}):=SL(n,\mathbb{R})\ltimes\mathbb{R}^{(m,n)}\]  
is the semidirect product of \(SL(n,\mathbb{R})\) and the additive group \(\mathbb{R}^{(m,n)}\) endowed with multiplication law  
\[(g,\alpha)\circ(h,\beta):=(gh,a^{\,t}h^{-1}+\beta) \tag{80}\]  
for all \(g,h\in SL(n,\mathbb{R})\), and \(\alpha,\beta\in\mathbb{R}^{(m,n)}\). We also recall the homogeneous space  
\[\mathfrak{P}_{n,m}:=\mathfrak{P}_{n}\times\mathbb{R}^{(m,n)}=\big{\{}\,Y\in \mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0,\ \det Y=1\,\big{\}}\times\mathbb{R}^{(m,n)}.\]Then, \(SL_{n,m}(\mathbb{R})\) acts on \(\mathfrak{P}_{n,m}\) naturally and transitively by  
\[(g,\alpha)\cdot(Y,V):=(gY^{t}\!g,(V+\alpha)^{t}\!g) \tag{81}\]  
for all \((g,\alpha)\in SL_{n,m}(\mathbb{R})\) and \((Y,V)\in\mathfrak{P}_{n,m}\). Since \(SO(n,\mathbb{R})\) is the stabilizer of Action (81) at \((I_{n},0)\), the non-symmetric homogeneous space \(SL_{n,m}(\mathbb{R})/SO(n,\mathbb{R})\) is diffeomorphic to the non-symmetric space \(\mathfrak{P}_{n,m}\). We denote by \(\mathbb{D}(\mathfrak{P}_{n,m})\) the algebra of all differential operators on \(\mathfrak{P}_{n,m}\) invariant under Action (81) of \(SL_{n,m}(\mathbb{R})\). We let  
\[SL_{n,m}(\mathbb{Z}):=SL(n,\mathbb{Z})\ltimes\mathbb{Z}^{(m,n)}\]  
denote the discrete subgroup of \(SL_{n,m}(\mathbb{R})\). From now on, we write \(G_{\circ}=SL_{n,m}(\mathbb{R})\) for brevity. We let  
\[\mathfrak{sl}(n,\mathbb{R})=\ \Big{\{}\ X\in\mathbb{R}^{(n,n)}\,|\,\operatorname{ Tr}(X)\ =\ 0\,\Big{\}}\]  
be the Lie algebra of \(SL(n,\mathbb{R})\). Then, it is easy to see that the Lie algebra \(\mathfrak{g}_{\circ}\) of \(G_{\circ}\) is given by  
\[\mathfrak{g}_{\circ}=\Big{\{}\ (X,Z)\,|\ X\in\mathfrak{sl}(n,\mathbb{R}),\ Z \in\mathbb{R}^{(m,n)}\ \Big{\}} \tag{82}\]  
equipped with the following Lie bracket:  
\[[(X_{1},Z_{1}),\ (X_{2},Z_{2})]_{\circ}=([X_{1},X_{2}]_{0},\ Z_{2}^{t}X_{1}-Z_{ 1}^{t}X_{2}), \tag{83}\]  
where \([X_{1},X_{2}]_{0}:=X_{1}X_{2}-X_{2}X_{1}\) denotes the usual matrix bracket and \((X_{1},Z  
In addition, we have  
\[\mathfrak{g}_{\circ}=\mathfrak{k}_{\circ}\oplus\mathfrak{p}_{\circ}\quad\text{ ( direct sum}\,). \tag{88}\]  
We note that the restriction of the Killing form \(B_{\circ}\) to \(\mathfrak{k}_{\circ}\) is negative definite and the restriction of \(B_{\circ}\) to the abelian subalgebra \(\mathfrak{r}=\{\,(0,Z)\in\mathfrak{g}_{\circ}\,\}\) is identically zero. Since \(\mathfrak{r}\) is the radical of \(B_{\circ}\), \(B_{\circ}\) is degenerate (see Formula (86)). An Iwasawa decomposition of the group \(SL_{n,m}(\mathbb{R})\) is given by  
\[G_{\circ}=N_{\circ}A_{\circ}K_{\circ}, \tag{89}\]  
where  
\[N_{\circ}=\left\{\,\left( \begin{pmatrix}1&*&\cdots&\cdots&*\\ 0&1&\cdots&\cdots&*\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&1&*\\ 0&0&0&0&1\end{pmatrix} ,\,a\,\right)\in G_{\circ}\,\bigg{|}\,\,\,a\in\mathbb{ R}^{(m,n)}\,\right\}\]  
and  
\[A_{\circ}=\{\,(\text{diag}(a_{1},\cdots,a_{n}),0)\in G_{\circ}\,|\,\,a_{i}\in \mathbb{R},\,\prod_{k=1}^{n}a_{k}=1,\,1\leq i\leq n\,\}.\]  
An Iwasawa decomposition of the Lie algebra \(\mathfrak{g}_{\circ}\) of \(G_{\circ}\) is given by  
\[\mathfrak{g}_{\circ}=\mathfrak{n}_{\circ}+\mathfrak{a}_{\circ}+\mathfrak{k}_ {\circ}, \tag{90}\]  
where  
\[\mathfrak{n}_{\circ}=\left\{\,\left( \begin{pmatrix}0&*&\cdots&\cdots&*\\ 0&0&\cdots&\cdots&*\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&0&*\\ 0&0&0&0&0\end{pmatrix} ,\,a\,\right)\in\mathfrak{g}_{\circ}\,\bigg{|}\,\,\,a \in\mathbb{R}^{(m,n)}\,\right\}\]  
and  
\[\mathfrak{a}_{\circ}=\{\,(\text{diag}(c_{1},\cdots,c_{n}),0)\in\mathfrak{g}_{ \circ}\,|\,\,c_{i}\in\mathbb{R},\,\,\sum_{k=1}^{n}c_{k}=0,\,\,\,1\leq i\leq n \,\}.\]  
In fact, \(\mathfrak{a}_{\circ}\) is the Lie algebra of \(A_{\circ}\) and \(\mathfrak{n}_{\circ}\) is the Lie algebra of \(N_{\circ}\).