# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 4 Invariant Differential Operators on \(SL(n,\mathbb{R})/SO(n,\mathbb{R})\)
Then, we obtain  
\[SL(n,\mathbb{Z})\backslash SL(n,\mathbb{R})/SO(n)\cong SL(n,\mathbb{Z}) \backslash GL(n,\mathbb{R})/(O(n)\cdot\mathbb{R}^{\times}),\]  
where \(SO(n):=SO(n,\mathbb{R})=SL(n,\mathbb{R})\cap O(n)\). We let  
\[\mathfrak{S}_{n}:=SL(n,\mathbb{Z})\backslash SL(n,\mathbb{R})/SO(n)=SL(n, \mathbb{Z})\backslash\mathfrak{P}_{n},\quad\mathfrak{P}_{n}:=SL(n,\mathbb{R}) /SO(n)\]  
be the locally symmetric space. Therefore, we obtain the following isomorphism:  
\[\mathfrak{S}_{n}\cong SL(n,\mathbb{Z})\backslash\mathfrak{H}_{n}.\]  
\(\mathfrak{P}_{n}\) parameterizes special principally polarized real tori of dimension \(n\) (cf. [15]). The arithmetic quotient \(\mathfrak{S}_{n}\) is the moduli space of isomorphism classes of special principally polarized real tori of dimension \(n\). Unfortunately, \(\mathfrak{S}_{n}\) does not admit the structure of a real algebraic variety and does not admit a compactification which is defined over the rational number field \(\mathbb{Q}\) (cf. [16] or [17]). **Remark 7**. : _In [25; 26], [PERSON] and [PERSON] constructed the geodesic compactification, the standard compactification, and a maximal Satake compactification of the locally symmetric space \(\mathfrak{S}_{n}\)._  
**Remark 8**. : _[PERSON] [27] studied [PERSON]'s law for the cuspidal spectrum of \(SL(n,\mathbb{R})\). In [28], [PERSON] and [PERSON] studied the cuspidal spectrum of \(\mathfrak{S}_{n}\). In [29], [PERSON] and [PERSON] introduced the analytic torsion for \(\mathfrak{S}_{n}\)._  
**Proposition 1**. : _We let \(n\geq 2\). Following the coordinates of Definition 4, we put_  
\[d^{*}x=\prod_{1\leq i<j\leq n}dx_{ij}\quad\text{and}\quad dy^{*}=\prod_{k=1}^{ n-1}y_{k}^{-n(n-k)-1}dy_{k}.\]  
_Then,_  
\[d^{*}z=d^{*}x\cdot d^{*}y\]  
_is the left \(SL(n,\mathbb{R})\)-invariant volume element on \(\mathfrak{H}_{n}\)._  
Proof. : The proof can be found in [7] (Proposition 1.5.3, pp. 25-26). **Theorem 10**. : _We let \(n\geq 2\). Then, volume \(\mathbf{Vol}(\Gamma^{n}\backslash\mathfrak{H}_{n})\) of \(\Gamma^{n}\backslash\mathfrak{H}_{n}\) is given by_  
\[\mathbf{Vol}(\Gamma^{n}\backslash\mathfrak{H}_{n})=\int_{\Gamma^{n} \backslash\mathfrak{H}_{n}}d^{*}z=n\cdot 2^{n-1}\cdot\prod_{k=2}^{n}\frac{ \zeta(k)}{\mathbf{Vol}(S^{k-1})},\]  
_where \(\Gamma^{n}=SL(n,\mathbb{Z})\) and_  
\[GL(n,\mathbb{R})=\mathfrak{H}_{n}\cdot O(n)\cdot Z_{n},\]0  
_denotes the volume of the \((k-1)\)-dimensional sphere \(S^{k-1}\), \(\zeta(k)=\sum_{n=1}^{\infty}n^{-k}\) is the Riemann zeta function, and \(\Gamma(p)\) denotes the usual Gamma function._  
Proof. : The proof can be found in [7] (Theorem 1.6.1, pp. 27-37). **Remark 9**. : _Since \(\mathfrak{H}_{n}\) is diffeomorphic to \(\mathfrak{P}_{n}\),_  
\[GL(n,\mathbb{R})=\mathfrak{H}_{n}\cdot O(n)\cdot Z_{n},\]1  
_The calculation of [PERSON] [7] (Theorem 1.6.1, pp. 27-37) is different from that of [PERSON] [24]._  
For any \(\
u=(\
u_{1},\
u_{2},\cdots,\
u_{n-1})\), we define the function \(I_{\
u}:\mathfrak{H}_{n}\longrightarrow\mathbb{C}\) by  
\[GL(n,\mathbb{R})=\mathfrak{H}_{n}\cdot O(n)\cdot Z_{n},\]2  
where  
\[b_{ij}:= \begin{cases}\ ij,&\text{if }\ i+j\leq n\\ (n-i)(n-j),&\text{if }\ i+j\geq n.\end{cases} \]  
Then, we see that \(I_{\
u}(z)\) is an eigenfunction of \(\mathbb{D}(\mathfrak{H}_{n})\). Let us write  
\[DI_{\
u}(z)=\lambda_{D}\cdot I_{\
u}(z)\quad\text{for every }D\in\mathbb{D}( \mathfrak{H}_{n}) \tag{42}\]  
since  
\[\lambda_{D_{1}D_{2}}=\lambda_{D_{1}}\lambda_{D_{2}}\qquad\text{for all }D_{1},D_{2}\in\mathbb{D}(\mathfrak{H}_{n}).\]Function \(\lambda_{D}\) (viewed as a function of \(D\)) is a character of \(\mathbb{D}(\mathfrak{H}_{n})\) which is called the Harish-Chandra character. Following [PERSON] (cf. [7], (Definition 5.1.3, pp. 115-116), the notion of a Maass form is defined in the following way. Definition 5: We let \(n\geq 2\). We put \(\Gamma^{n}=SL(n,\mathbb{Z})\).