# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 3 Invariant Differential Operators on \(GL(n,\mathbb{R})/O(n,\mathbb{R})\)  
For any positive integer \(n\geq 1\), we let  
\[\mathcal{P}_{n}:=\{Y\in\mathbb{R}^{(n,n)}\mid Y=\,^{t}Y>0\,\}\]  
be the open convex cone in the Euclidean space \(\mathbb{R}^{N}\) with \(N=\frac{n(n+1)}{2}\). Then, \(GL(n,\mathbb{R})\) acts \(\mathcal{P}_{n}\) transitively by  
\[g\cdot Y=gY^{t}\!g,\qquad\text{where $g\in GL(n,\mathbb{R})$ and $Y\in\mathcal{P}_{n}$}. \tag{18}\]  
Since \(O(n)\) is the isotopic subgroup of \(GL(n,\mathbb{R})\) at \(I_{n}\), the symmetric space \(GL(n,\mathbb{R})/O(n)\) is diffeomorphotoc to \(\mathcal{P}_{n}\). For \(Y=(y_{ij})\in\mathcal{P}_{n}\), we put  
\[dY=(dy_{ij})\qquad\text{and}\qquad\frac{\partial}{\partial Y}\,=\,\left(\, \frac{1+\delta_{ij}}{2}\,\frac{\partial}{\partial y_{ij}}\,\right).\]  
For a fixed element \(A\in GL(n,\mathbb{R})\), we put  
\[Y_{*}=A\cdot Y=AY^{t}\!A,\quad Y\in\mathcal{P}_{n}.\]  
Then  
\[dY_{*}=A\,dY^{t}\!A\quad\text{and}\quad\frac{\partial}{\partial Y_{*}}=\,^{t} \!A^{-1}\frac{\partial}{\partial Y}\,A^{-1}. \tag{19}\]  
We can see easily that for any positive real number \(C>0\),  
\[ds^{2}_{n;C}=\,C\cdot\text{Tr}\,((Y^{-1}\!dY)^{2})\]  
is a Riemannian metric on \(\mathcal{P}_{n}\) invariant under Action (18) and its Laplace operator is given by  
\[\Delta_{n;C}=\frac{1}{C}\cdot\text{Tr}\Bigg{(}\bigg{(}Y\frac{\partial}{ \partial Y}\bigg{)}^{2}\Bigg{)},\]  
where \(\text{Tr}\,(M)\) denotes the trace of a square matrix \(M\). We also can see that  
\[d\mu_{n}(Y)=(\det Y)^{-\frac{n+1}{2}}\prod_{t\leq j}dy_{ij} \tag{20}\]  
is a \(GL(n,\mathbb{R})\)-invariant volume element on \(\mathcal{P}_{n}\). **Theorem 6**. : _A geodesic \(\alpha(t)\) joining \(I_{n}\) and \(Y\in\mathcal{P}_{n}\) has the form_  
\[\alpha(t)=\exp(tA[V]),\qquad t\in[0,1],\]_where_  
\[Y=(\exp A)[V]=\exp(A[V])=\exp(\,^{t}VAV)\]  
_is the spectral decomposition of \(Y\), where \(V\in O(n,\mathbb{R}),\ A=\operatorname{diag}(a_{1},\cdots,a_{n})\) with all \(a_{j}\in\mathbb{R}\). The distance of \(\alpha(t)\ (0\leq t\leq 1)\) between \(I_{n}\) and \(Y\) is_  
\[g\cdot Y=gY^{t}\!g,\qquad\text{where $g\in GL(n,\mathbb{R})$ and $Y\in\mathcal{P}_{n}$}. \tag{18}\]0  
Proof. : The proof can be found in [6] (pp. 16-17). We consider the following Maass-Selberg (differential) operators \(\delta_{1},\delta_{2},\cdots,\delta_{n}\) on \(\mathcal{P}_{n}\) defined by  
\[g\cdot Y=gY^{t}\!g,\qquad\text{where $g\in GL(n,\mathbb{R})$ and $Y\in\mathcal{P}_{n}$}. \tag{18}\]1  
By Formula (19), we obtain  
\[g\cdot Y=gY^{t}\!g,\qquad\text{where $g\in GL(n,\mathbb{R})$ and $Y\in\mathcal{P}_{n}$}. \tag{18}\]2  
for any \(A\in GL(n,\mathbb{R})\), so each \(\delta_{i}\ (1\leq i\leq n)\) is invariant under Action (18) of \(GL(n,\mathbb{R})\). [PERSON] [1,2] and [PERSON] [3] proved the following. **Theorem 7**. : _The algebra \(\mathbb{D}(\mathcal{P}_{n})\) of all \(GL(n,\mathbb{R})\)-invariant differential operators on \(\mathcal{P}_{n}\) is generated by \(\delta_{1},\delta_{2},\cdots,\delta_{n}\). Furthermore, \(\delta_{1},\delta_{2},\cdots,\delta_{n}\) are algebraically independent and \(\mathbb{D}(\mathcal{P}_{n})\) is isomorphic to the commutative ring \(\mathbb{C}[x_{1},x_{2},\cdots,x_{n}]\) with \(n\) indeterminates \(x_{1},x_{2},\cdots,x_{n}\)._  
Proof. : The proof can be found in [2] (pp. 64-66) and [6] (pp. 29-30). The last statement follows immediately from the work of [PERSON] [10,11] or [4] (p. 294). **Remark 2**. : _A different description of \(\mathbb{D}(\mathcal{P}_{n})\) was given by [PERSON] [4] (Chapter II, Exercise C.1, p. 337; Solution pp. 571-572). See also [4] (Chapter II, Exercise C.8, pp. 339-340) for a related topic._  
We let \(\mathfrak{g}=\mathbb{R}^{(n,n)}\) be the Lie algebra of \(GL(n,\mathbb{R})\).