# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 4 Invariant Differential Operators on \(SL(n,\mathbb{R})/SO(n,\mathbb{R})\)  
First of all, we provide some geometric properties on \(SL(n,\mathbb{R})/SO(n,\mathbb{R})\). We let  
\[\mathfrak{P}_{n}:=\left\{Y\in\mathbb{R}^{(n,n)}\,|\,\,Y=\,^{t}Y>0,\,\det(Y)=1\,\right\}\]  
be a symmetric space associated to \(SL(n,\mathbb{R})\). Indeed, \(SL(n,\mathbb{R})\) acts on \(\mathfrak{P}_{n}\) transitively by  
\[g\circ Y=gY^{t}g,\qquad g\in SL(n,\mathbb{R}),\,\,Y\in\mathfrak{P}_{n}. \tag{29}\]  
Thus, \(\mathfrak{P}_{n}\) is a smooth manifold diffeomorphic to the symmetric space \(SL(n,\mathbb{R})/SO(n,\mathbb{R})\) through the bijective map  
\[SL(n,\mathbb{R})/SO(n,\mathbb{R})\longrightarrow\mathfrak{P}_{n},\qquad g\cdot SO (n,\mathbb{R})\mapsto g\circ I_{n}=\,g^{t}g,\quad g\in SL(n,\mathbb{R}).\]  
For \(Y\in\mathfrak{P}_{n}\), we have a partial Iwasawa decomposition  
\[Y= \begin{pmatrix}v^{-1}&0\\ 0&v^{1/(n-1)}W\end{pmatrix}\begin{bmatrix}1&{}^{t}x\\ 0&I_{n-1}\end{bmatrix} = \begin{pmatrix}v^{-1}&v^{-1}\,x\\ v^{-1}x&v^{-1}x^{t}x+v^{1/(n-1)}W\end{pmatrix}  \tag{30}\]  
where \(v>0,\,\,x\in\mathbb{R}^{(n-1,1)}\) and \(W\in\mathfrak{P}_{n-1}\). From now on, for brevity, we write \(Y=[v,x,W]\) instead of Decomposition (30). In these coordinates, \(Y=[v,x,W]\),  
\[ds_{Y}^{2}=\frac{n}{n-1}\,v^{-2}dv^{2}+2\,v^{-n/(n-1)}W^{-1}[dx]+ds_{W}^{2}\]is a \(SL(n,\mathbb{R})\)-invariant metric on \(\mathfrak{P}_{n}\), where \(dx=\,^{t}(dx_{1},\cdots,dx_{n-1})\) and \(ds_{W}^{2}\) is a \(SL(n-1,\mathbb{R})\)-invariant metric on \(\mathfrak{P}_{n-1}\). The Laplace operator \(\Delta_{n}\) of \((\mathfrak{P}_{n},ds_{Y}^{2})\) is given by  
\[\Delta_{n}=\frac{n-1}{n}\,v^{2}\frac{\partial^{2}}{\partial v^{2}}-\frac{1}{n} \frac{\partial}{\partial v}+v^{n/(n-1)}\,W\Big{[}\frac{\partial}{\partial x} \Big{]}+\Delta_{n-1}\]  
inductively, where if \(x=\,^{t}(x_{1},\cdots,x_{n-1})\in\mathbb{R}^{(n-1,1)}\),  
\[\frac{\partial}{\partial x}=\,^{t}\!\!\left(\frac{\partial}{\partial x_{1}}, \cdots,\frac{\partial}{\partial x_{n-1}}\right)\]  
and \(\Delta_{n-1}\) is the Laplace operator of \((\mathfrak{P}_{n-1},ds_{W}^{2})\),  
\[dv_{n}=v^{-(n+2)/2}\,dv\,dx\,dv_{n-1}\]  
is a \(SL(n,\mathbb{R})\)-invariant volume element on \(\mathfrak{P}_{n}\) where \(dx=dx_{1}\cdots dx_{n-1}\) and \(dv_{n-1}\) is a \(SL(n-1,\mathbb{R})\)-invariant volume element on \(\mathfrak{P}_{n-1}\). Following earlier work of [PERSON] [14], [PERSON] [23] showed that the volume of the arithmetic quotient \(SL(n,\mathbb{Z})\backslash\mathfrak{P}_{n}\) is given as follows:  
\[\text{Vol}(SL(n,\mathbb{Z})\backslash\mathfrak{P}_{n})=\int_{SL(n,\mathbb{Z} )\backslash\mathfrak{P}_{n}}dv_{n}=n\,2^{n-1}\prod_{k=2}^{n}\frac{\zeta(k)}{ \text{Vol}(S^{k-1})}, \tag{31}\]  
where  
\[\text{Vol}(S^{k-1})=\frac{2\,(\sqrt{\pi})^{k}}{\Gamma(k/2)}\]  
denotes the volume of the \((k-1)\)-dimensional sphere \(S^{k-1}\), \(\Gamma(x)\) denotes the usual Gamma function, and \(\zeta(k)=\sum_{m=1}^{\infty}m^{-k}\) denotes the Riemann zeta function. The proof of (31) can be found in [24] or [7] (pp. 27-37). We let \(\mathbb{D}(\mathfrak{P}_{n})\) be the algebra of all differential operators on \(\mathfrak{P}_{n}\) invariant under the action (29) of \(SL(n,\mathbb{R})\). It is well known (cf. [4; 10; 11]) that \(\mathbb{D}(\mathfrak{P}_{n})\) is commutative and is isomorphic to the polynomial algebra \(\mathbb{C}[x_{1},x_{2},\cdots,x_{n-1}]\) with \(n\) indeterminates \(x_{1},x_{2},\cdots,x_{n-1}\). We observe that \(n-1\) is the rank of \(SL(n,\mathbb{R})\), i.e., the rank of the symmetric space \(SL(n,\mathbb{R})/SO(n,\mathbb{R})\). In [5], using the Maass-Selberg operators \(\delta_{1},\delta_{2},\cdots,\delta_{n}\) (see Formula (21)), [PERSON], [PERSON], and [PERSON] found explicit generators \(E_{1}\), \(E_{2},\cdots,E_{n-1}\) of \(\mathbb{D}(\mathfrak{P}_{n})\). Obviously, \(E_{1},E_{2}\), \(\cdots,E_{n-1}\) are algebraically independent. We briefly sketch their method of finding generators \(E_{1},E_{2},\cdots,E_{n-1}\) of \(\mathbb{D}(\mathfrak{P}_{n})\).