Invariant Differential Operators and Weyl Group Invariants
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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(Z 1) and µ (Z 2) (with Z 1,Z 2 ∈ Z(G)). See Theorem 4.1.
KeywordsSymmetric Space Spherical Function Quotient Field Noncompact Type Invariant Differential Operator
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