# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 2 Preliminaries  
Throughout this section, we let \(G\) be a connected real Lie group of finite dimension \(n\) and let \(K\) be a subgroup of \(G\). We let \(\mathfrak{g}(\text{resp. }\,\mathfrak{k})\) be the Lie algebra of \(G(\text{resp.}\,K)\). The symmetric algebra \(S(\mathfrak{g})\) is defined to be the algebra of complex-valued polynomial functions on the dual space \(\mathfrak{g}^{*}\). If \(X_{1},\cdots,X_{n}\) is a basis of \(\mathfrak{g}\), \(S(\mathfrak{g})\) can be identified with the algebra of all polynomials  
\[\sum_{(k)}a_{k_{1}\cdots k_{n}}X_{1}^{k_{1}}\cdots X_{n}^{k_{n}},\quad(k)=(k_ {1},\cdots,k_{n})\in(\mathbb{Z}_{+})^{n}.\]  
Here, \(\mathbb{Z}_{+}\) denotes the set of all non-negative integers. For an element \(g\in G\), \(L_{g}\) (resp. \(R_{g}\)) is the left (resp. right) translation by \(g\) defined by  
\[L_{g}(h)=gh\ (\text{resp.}\ R_{g}(h)=hg)\qquad\text{for all }h\in G.\]  
If \(X\in\mathfrak{g}\), we let \(\widetilde{X}\) denote a differential operator on \(G\) defined by  
\[(\widetilde{X}f)(g):=\frac{\partial}{\partial t}\Big{|}_{t=0}f(g\cdot\exp(tX)),\qquad f\in\mathbb{C}^{\infty}(G),\ g\in G.\]\(\mathbb{E}(G)\) denotes the algebra of all differential operators on \(G\). A differential operator \(D\in\mathbb{E}(G)\) is said to be left-invariant if  
\[D(f\circ L_{g})=D(f)\circ L_{g}\qquad\text{for all $f\in C^{\infty}(G)$ and $g\in G$.}\]  
We let \(\mathbb{D}(G)\) be the algebra of all left-invariant differential operators on \(G\) and let \(\mathbf{Z}(G)\) be the center of \(\mathbb{D}(G)\). It is easily seen that  
\[\widetilde{\operatorname{ad}(X)}Y=\widetilde{X}\widetilde{Y}-\widetilde{Y} \widetilde{X}\qquad\text{for all $X,Y\in\mathfrak{g}$.}\]  
Now, for any \(X\in\mathfrak{g}\), we define the map \(\operatorname{ad}(X):\mathbb{D}(G)\longrightarrow\mathbb{D}(G)\) by  
\[\operatorname{ad}(X)D:=\widetilde{X}D-D\widetilde{X}\qquad\text{for all $D\in \mathbb{D}(G)$.} \tag{9}\]  
Obviously \(\operatorname{ad}(X)\) is a derivation of the algebra \(\mathbb{D}(G)\). We define  
\[e^{\operatorname{ad}(X)}(D):=\sum_{k=0}^{\infty}\frac{1}{k!}\left(\operatorname {ad}(X)\right)^{k}(D),\qquad D\in\mathbb{D}(G). \tag{10}\]  
**Definition 1**. : _The coset space \(G/K\) is said to be_ reductive _if there exists a subspace \(\mathfrak{p}\) such that_  
\[\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{p}\qquad\text{and}\qquad \operatorname{Ad}_{G}(k)\mathfrak{p}\subset\mathfrak{p}\quad\text{for all $k\in K$.} \tag{11}\]  
_Here \(\operatorname{Ad}_{G}:G\longrightarrow GL(\mathfrak{g})\) denotes the adjoint representation of \(G\) on \(\mathfrak{g}\)._  
**Theorem 1**. : _We assume the coset space \(G/K\) is reductive. Then there exists a unique linear bijection (called the_ symmetrization_)_  
\[\lambda:S(\mathfrak{g})\longrightarrow\mathbb{D}(G) \tag{12}\]  
_such that \(\lambda(X^{m})=\widetilde{X}^{m}\)\((X\in\mathfrak{g},\ m\in\mathbb{Z}^{+})\). If \(X_{1},\cdots,X_{n}\) is any basis of \(\mathfrak{g}\) and \(P\in S(\mathfrak{g})\); then, for any \(f\in C^{\infty}(G)\),_  
\[(\lambda(P)f)(g)=\left[P(\partial_{1},\cdots,\partial_{n})f(g\cdot\exp(t_{1} X_{1}+\cdots+t_{n}X_{n})\right]_{t=0}, \tag{13}\]  
_where \(\partial_{i}=\partial/\partial t_{i}\)\((1\leq i\leq n)\) and \(t=(t_{1},\cdots,t_{n})\in\mathbb{R}^{n}\). Here, the suffix \([\ \cdot\ ]_{t=0}\) means the evaluation at \(t=0\) after differentiation._  
Proof. : The proof can be found in [4] (Theorem 4.3, pp. 280-281). **Definition 2**. : _We fix an element \(g\in G\).