Invariant Differential Operators and Weyl Group Invariants

In this research announcement we describe the relationship between the algebra Z(G) of bi-invariant differential operators on a simple noncompact Lie group G and the algebra D(G/K )of invariant differential operators on the symmetric space G/K associated with G (cf. §4). The natural map µ: Z(G) → D(G/K) turns out to be surjective except in exactly four cases. These cases involve the exceptional groups and for them the relationship between Z(G) and D(G/K) is quite complicated. However for all cases of G/K, each D ∈ D(G/K) is the “ratio” of two µ(Z₁) and µ (Z₂) (with Z₁,Z₂ ∈ Z(G)). See Theorem 4.1.