# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 4 Invariant Differential Operators on \(SL(n,\mathbb{R})/SO(n,\mathbb{R})\)
We let \(\{\xi_{\alpha}\mid 1\leq\alpha\leq N_{0}\,\}\) be a basis of \(\mathfrak{p}_{0}\). If \(P\in\operatorname{Pol}(\mathfrak{p}_{0})^{K}\), then  
\[dv_{n}=v^{-(n+2)/2}\,dv\,dx\,dv_{n-1}\]5  
where \(f\in C^{\infty}(\mathfrak{P}_{n})\). We refer to [4] (Theorem 4.6, pp. 285-286) or Formula (17) for more detail. In general, it is very hard to express \(\Psi_{n}(P)\) explicitly for a polynomial \(P\in\operatorname{Pol}(\mathfrak{p}_{0})^{K}\). If we repeat a partial decomposition process for \(Y\in\mathfrak{P}_{n}\), we obtain the Iwasawa decomposition  
\[Y=y^{-1}\text{diag}\Big{(}1,y_{1}^{2},(y_{1}y_{2})^{2},\cdots,(y_{1}y_{2} \cdots y_{n-1})^{2}\Big{)}\left[ \begin{pmatrix}1&x_{12}&\cdots&x_{1n}\\ 0&1&\cdots&x_{2n}\\ 0&0&\ddots&\vdots\\ 0&0&0&1\end{pmatrix} \right], \tag{40}\]  
where \(y>0,\,y_{j}\in\mathbb{R}\,(1\leq j\leq n-1)\) and \(x_{ij}\in\mathbb{R}\,(1\leq i<j\leq n)\). Here, \(y=y_{1}^{2(n-1)}\cdots y_{n-1}^{2}\) and \(\text{diag}(a_{1},\cdots,a_{n})\) denotes the \(n\times n\) diagonal matrix with diagonal entries \(a_{1},\cdots,a_{n}\). In this case, we denote \(Y=(y_{1},\cdots,y_{n-1},x_{12},\cdots,x_{n-1,n})\). We define \(\Gamma_{n}=GL(n,\mathbb{Z})/\{\pm I_{n}\}\). We observe that \(\Gamma_{n}=SL(n,\mathbb{Z})/\{\pm I_{n}\}\) if \(n\) is even and \(\Gamma_{n}=SL(n,\mathbb{Z})\) if \(n\) is odd. An automorphic form for \(\Gamma_{n}\) is defined to be a real analytic function \(f:\mathfrak{P}_{n}\longrightarrow\mathbb{C}\) satisfying the following conditions (AF1)-(AF3):  
1. \(f\) is an eigenfunction for all \(SL(n,\mathbb{R})\)-invariant differential operators on \(\mathfrak{P}_{n}\). 2. \(f(\gamma Y^{t}\gamma)=f(Y)\quad\text{for all }\gamma\in\Gamma_{n}\) and \(Y\in\mathfrak{P}_{n}\). 3. There exist a constant \(C>0\) and \(s\in\mathbb{C}^{n-1}\) with \(s=(s_{1},\cdots,s_{n-1})\) such that \(|f(Y)|\leq C\,|p_{-s}(Y)|\) as the upper left determinants \(\det Y_{j}\longrightarrow\infty\), \(j=1,2,\cdots,n-1\), where \[p_{-s}(Y):=\prod_{j=1}^{n-1}(\det Y_{j})^{-s_{j}}\] is [PERSON]'s power function (cf. [3; 6]). We denote by \(\mathbf{A}(\Gamma_{n})\) the space of all automorphic forms for \(\Gamma_{n}\). A cusp form \(f\in\mathbf{A}(\Gamma_{n})\) is an automorphic form for \(\Gamma_{n}\) satisfying the following conditions:  
\[\int_{X\in(\mathbb{R}/\mathbb{Z})^{(j\mu-j)}}f\left(Y\left[ \begin{pmatrix}I_{j} &X\\ 0&I_{n-j}\end{pmatrix} \right]\right)dX=0,\quad 1\leq j\leq n-1.\]  
Here, \((\mathbb{R}/\mathbb{Z})^{(j\mu-j)}\) denotes the set of all \(j\times(n-j)\) matrices with entries in the one-dimensional real torus \(\mathbb{R}/\mathbb{Z}\). We denote by \(\mathbf{A}_{0}(\Gamma_{n})\) the space of all cusp forms for \(\Gamma_{n}\). Definition 4: For any positive integer \(n\geq 2\), we define \(\mathfrak{H}_{n}\) to be the set of all \(n\times n\) real matrices of the form \(z=x\cdot y\), where  
\[x= \begin{pmatrix}1&x_{12}&x_{13}&\cdots&x_{1n}\\ 0&1&x_{22}&\cdots&x_{2n}\\ 0&0&\ddots&\vdots&\vdots\\ 0&0&0&0&1\end{pmatrix} \]  
and  
\[y=\operatorname{diag}(y_{1}y_{2}\cdots y_{n-1},y_{1}y_{2}\cdots y_{n-2}, \cdots,y_{1},1)\]  
with \(x_{ij}\in\mathbb{R}\) for \(1\leq i<j\leq n\) and \(y_{i}\geq 0\) for \(1\leq i\leq n-1\). We can show that \(\mathfrak{H}_{n}\) is diffeomorphic to \(\mathfrak{P}_{n}\). In fact, we have the Iwasawa decomposition  
\[GL(n,\mathbb{R})=\mathfrak{H}_{n}\cdot O(n)\cdot Z_{n},\]  
where \(Z_{n}(\cong\mathbb{R}^{\times})\) is the center of \(GL(n,\mathbb{R})\) (cf. [7], (Proposition 1.2.6, pp. 11-12)). Here,  
\[O(n):=O(n,\mathbb{R})=\{k\in GL(n,\mathbb{R})|\ ^{t}kk=k^{t}k=I_{n}\,\}\]  
denotes the real orthogonal group of degree \(n\). We see easily that  
\[\mathfrak{H}_{n}\cong GL(n,\mathbb{R})/(O(n)\cdot\mathbb{R}^{\times}),\]  
where \(\cong\) denotes the diffeomorphism. It is seen that \(GL(n,\mathbb{R})\) acts on \(\mathfrak{H}_{n}\) by left translation (cf. [7], (Proposition 1.2.10, p. 14)).