# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 3 Invariant Differential Operators on \(GL(n,\mathbb{R})/O(n,\mathbb{R})\)
.\]  
For any \(k\in K\),  
\[(k\cdot q_{i})(X)=q_{i}(k^{-1}Xk)=\operatorname{Tr}(k^{-1}X^{i}k)=q_{i}(X), \quad i=1,2,\cdots,n.\]  
Thus, \(q_{i}\in\operatorname{Pol}(\mathfrak{p})^{K}\) for \(i=1,2,\cdots,n\). By a classical invariant theory (cf. [12; 13]), we can prove that the algebra \(\operatorname{Pol}(\mathfrak{p})^{K}\) is generated by the polynomials \(q_{1},q_{2},\cdots,q_{n}\) and that \(q_{1},q_{2},\cdots,q_{n}\) are algebraically independent. Using Formula (26), we can show without difficulty that  
\[\Phi_{n}(q_{1})=\operatorname{Tr}\biggl{(}2Y\frac{\partial}{\partial Y} \biggr{)}.\]However, \(\Phi_{n}(q_{i})\)\((i=2,3,\cdots,n)\) are still not known explicitly. We propose the following conjecture. **Conjecture 8**. : _For any \(n\),_  
\[\Phi_{n}(q_{i})=\mathrm{Tr}\Bigg{(}\bigg{(}2Y\frac{\partial}{\partial Y}\bigg{)} ^{i}\Bigg{)},\quad i=1,2,\cdots,n.\]  
**Remark 3**. : _The above conjecture is true for \(n=1,2\)._  
The fundamental domain \(\mathfrak{M}_{n}\) for \(GL(n,\mathbb{Z})\) in \(\mathscr{P}_{n}\) which was found by [PERSON] [14] is defined as a subset of \(\mathscr{P}_{n}\) consisting of \(Y=(y_{ij})\in\mathscr{P}_{n}\) satisfying the following conditions (M.1)-(M.2) (cf. [2], (p. 123)):  
1. \(aY^{\,t}a\geq y_{kk}\) for every \(a=(a_{i})\in\mathbb{Z}^{n}\) in which \(a_{k},\cdots,a_{n}\) are relatively prime for \(k=1,2,\cdots,n\). 2. \(y_{k,k+1}\geq 0\) for \(k=1,\cdots,n-1\). We say that a point of \(\mathfrak{M}_{n}\) is _Minkowski reduced_ or simply _M-reduced_. \(\mathfrak{M}_{n}\) is a convex cone through the origin bounded by a finite number of hyperplanes and is closed in \(\mathscr{P}_{n}\) (cf. [2], pp. 123-124). Thus, we see that \(\mathfrak{M}_{n}\) is a semi-algebraic set with real analytic structure. We let  
\[\mathfrak{R}_{n}:=GL(n,\mathbb{Z})\backslash GL(n,\mathbb{R})/O(n,\mathbb{R})= GL(n,\mathbb{Z})\backslash\mathscr{P}_{n}\]  
be the locally symmetric space. \(\mathscr{P}_{n}\) parameterizes principally polarized real tori of dimension \(n\) (cf. [15]). The arithmetic quotient \(\mathfrak{R}_{n}\) is the moduli space of isomorphism classes of principally polarized real tori of dimension \(n\). Unfortunately \(\mathfrak{R}_{n}\) 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]). We offer the definition of automorphic forms for \(GL(n,\mathbb{Z})\) given by [PERSON] (cf. [6], (p. 182)). For \(s=(s_{1},\cdots,s_{n})\in\mathbb{C}^{n}\), [PERSON] [3] (pp. 57-58) introduced the power function \(p_{s}:\mathscr{P}_{n}\longrightarrow\mathbb{C}\) defined by  
\[p_{s}(Y):=\prod_{j=1}^{n}(\det Y_{j})^{s_{j}},\quad Y\in\mathscr{P}_{n}, \tag{27}\]  
where \(Y_{j}\in\mathscr{P}_{j}\)\((1\leq j\leq n)\) is the \(j\times j\) upper left corner of \(Y\). It is known that \(p_{s}(Y)\) is a joint eigenfunction of \(\mathbb{D}(\mathscr{P}_{n})\), i.e., \(p_{s}(Y)\) is an eigenfunction of each invariant differential operator in \(\mathbb{D}(\mathscr{P}_{n})\) (cf. [6], (pp. 39-40)). **Definition 3**. : _A real analytic function \(f:\mathscr{P}_{n}\longrightarrow\mathbb{C}\) is said to be a automorphic form for \(GL(n,\mathbb{Z})\) if it satisfies the following Conditions (A1)-(A3):_  
1. \(f(\gamma Y^{\,t}\gamma)=f(Y)\) _for all_ \(Y\in\mathscr{P}_{n}\) _and all_ \(\gamma\in GL(n,\mathbb{Z})\)_;_
2. \(f\) _is an eigenfunction of all_ \(D\in\mathbb{D}(\mathscr{P}_{n})\)_, i.e.,_ \(Df=\lambda_{D}f\) _for some eigenvalue_ \(\lambda_{D}\)_;_
3. \(f\) _has at most polynomial growth at infinity, i.e.,_ \[|f(Y)|\leq C\,|\,p_{s}(Y)|\qquad\text{for some $s\in\mathbb{C}^{n}$ and $C>0$.}\]We set \(\Gamma_{n}=GL(n,\mathbb{Z})\). We denote by \(\mathbf{A}(\Gamma_{n},\lambda)\) the space of all automorphic forms for \(\Gamma_{n}\) with a given eigenvalue system \(\lambda\). An automorphic form \(f\) in \(\mathbf{A}(\Gamma_{n},\lambda)\) is called a cusp form for \(\Gamma_{n}\) if for any \(k\) with \(1\leq k\leq n-1\). We have  
\[\int_{X\in T^{(k,n-k)}}f\left(