A new approach to computing generators for U(g)K

Let ( G , K ) be the complex symmetric pair associated with a real reductive Lie group G 0 . We discuss an algorithmic approach to computing generators for the centralizer of K in the universal enveloping algebra of g . In particular, we compute explicit generators for the cases G 0 = SU ( 2 , 2 ) , SL ( 3 , R ) , SL ( 4 , R ) , Sp ( 4 , R ) , and the exceptional group G 2 ( 2 ) .