Notes on the step algebra

These are informal notes of my talk during the Atlas of Lie groups workshop at AIM in Palo Alto, July 2003. All the ideas below are due to others, the primary references are Mickelsson [M] and Zhelebenko [Z1], [Z2]. Zhelebenko’s papers contain statements without proofs, therefore one should probably verify the results independently. Assuming the results, one gets a fairly explicit algorithm for computing signatures on the K-types of a Harish-Chandra module. (See notes on David Vogan’s Montreal talk.) Let g be a reductive Lie algebra with Cartan decomposition g = k ⊕ p. Fix a Cartan subalgebra t of k and positive root system ∆+ (t, k), and let k+ = ⊕α∈∆+kα. Let U = U(g) be the universal enveloping algebra of g, and let I = Uk+ be the left ideal generated by k+. We define