# Problems in Invariant Differential Operators on Homogeneous Manifolds
## 2 Preliminaries
In fact, we have_  
\[(\widetilde{X}f)(g):=\frac{\partial}{\partial t}\Big{|}_{t=0}f(g\cdot\exp(tX)),\qquad f\in\mathbb{C}^{\infty}(G),\ g\in G.\]6  
_where \(Q\in I(\mathfrak{p})\) has degree \(<\) degree(\(P_{1}\)) + degree(\(P_{2}\))._  
(2) _If_ \(P_{1},\cdots,P_{d}\) _are generators of_ \(I(\mathfrak{p}_{C})\)_, then_ \(\Theta(P_{1}),\cdots,\Theta(P_{d})\) _are generators of_ \(\mathbb{D}(G/K)\)_. Here,_ \(\mathfrak{p}_{C}\) _denotes the complexification of_ \(\mathfrak{p}\)_._  
(3) _If_ \(I(\mathfrak{p})\) _has a finite system of generators_ \(P_{1},\cdots,P_{d}\) _and we put_ \(D_{i}=\Theta(P_{i})\ (1\leq i\leq d)\)_, then each_ \(D\in\mathbb{D}(G/K)\) _can be written_  
\[(\widetilde{X}f)(g):=\frac{\partial}{\partial t}\Big{|}_{t=0}f(g\cdot\exp(tX)),\qquad f\in\mathbb{C}^{\infty}(G),\ g\in G.\]7  
_where \((n)=(n_{1},\cdots,n_{d})\in\mathbb{Z}_{+}^{d}\)._  
**Theorem 5**. : (1) _We let \((G,K)\) be a symmetric pair (i.e., \(G/K\) is a symmetric space), \(G\) semisimple, and \(K\) a maximal compact subgroup of \(G\). Then \(\mathbb{D}(G/K)\) is a commutative algebra. Here, \(\mathbb{D}(G/K)\) denotes the algebra of all invariant differential operators on \(G/K\). (2) We let \(H\) be a connected Lie group of finite dimension and let \(H^{\diamond}\) be the diagonal in \(H\times H\). Under the bijection_  
\[(\widetilde{X}f)(g):=\frac{\partial}{\partial t}\Big{|}_{t=0}f(g\cdot\exp(tX)),\qquad f\in\mathbb{C}^{\infty}(G),\ g\in G.\]8  
_of \((H\times H)/H^{\diamond}\) onto \(H\), we have the identification_  
\[(\widetilde{X}f)(g):=\frac{\partial}{\partial t}\Big{|}_{t=0}f(g\cdot\exp(tX)),\qquad f\in\mathbb{C}^{\infty}(G),\ g\in G.\]9  
_Here, \(\mathbb{D}((H\times H)/H^{\diamond})\) (resp. \(\mathbb{D}(H)\)) denotes the algebra of invariant differential operators on \((H\times H)/H^{\diamond}\) (resp. \(H\)) and \(\mathbf{Z}(H)\) denotes the center of \(\mathbb{D}(H)\)._  
Proof. : The proof can be found in [4] (Theorem 5.7, pp. 294-295). We let \(M=G/K\) be a symmetric space of the noncompact type, i.e., \(G\) is a connected semisimple Lie group with finite center and \(K\) a maximal compact subgroup of \(G\). We let \(\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{p}\) be the Cartan decomposition of the Lie algebra \(\mathfrak{g}\) of \(G\). We let \(\mathfrak{a}\subset\mathfrak{p}\) be a maximal abelian subspace of \(\mathfrak{p}\) and let \(\mathfrak{a}^{+}\subset\mathfrak{a}\) be a fixed Weyl chamber. We let \(G=KAN\) be an Iwasawa decomposition of \(G\). We denote by \(\mathbb{D}(A)\) the algebra of invariant differential operators on \(A\cdot o\). Here, \(o=e\cdot K\) is the origin of \(G/K\) (\(e\) is the identity element of \(G\)) and \(A\cdot o:=\{a\cdot o\ |\ a\in A\ \}\) denotes the \(A\)-orbit of \(o\) in \(G/K\). We let \(W\) be the Weyl group of \(G\), that is, the Weyl group of the root system of \(G\). We recall the linear bijection \(\lambda:S(\mathfrak{g})\longrightarrow\mathbb{D}(G)\) in Theorem 1. We see that \(S(\mathfrak{a})\) can be identified with \(\mathbb{D}(A)\). We let \(\mathbb{D}_{W}(A)\) be the set of all \(W\)-invariant differential operators on the orbit \(A\cdot o\) and \(I(\mathfrak{a})\) the set of all \(W\)-invariants in \(S(\mathfrak{a})\). Then, there exists a bijection of \(\mathbb{D}(G/K)\) onto \(\mathbb{D}_{W}(A)\) (cf. see [4] (Theorem 5.13, pp. 300-302)). Furthermore, there exists a surjective homomorphism of \(\mathbb{D}_{K}(G)\) onto \(\mathbb{D}_{W}(A)\) with kernel  
\[D(f\circ L_{g})=D(f)\circ L_{g}\qquad\text{for all $f\in C^{\infty}(G)$ and $g\in G$.}\]0  
We refer to [4] (Theorem 5.18, p. 306) for more detail. Combining all these results, we conclude that if \(G/K\) is a symmetric space of the noncompact type, then \(\mathbb{D}(G/K)\) is a polynomial algebra in \(r\) algebraically independent generators \(\delta_{1},\cdots,\delta_{r}\) whose degrees \(d_{1},\cdots,d_{r}\) are canonically determined by \(G\). We note that \(r=\dim A\) is the rank of \(G\) or the rank of \(G/K\).