Generalized exponents via Hall-Littlewood symmetric functions

The generalized exponents of finite-dimensional irreducible representations of a compact Lie group are important invariants first constructed and studied by Kostant in the early 1960s. Their actual computation has remained quite enigmatic. What was known ([K] and [Hs, Theorem 1]) suggested to us that their computation lies at the heart of a rich combinatorially flavored theory. This note announces several results all tied together by Theorem 2.3 below, which selects the natural generalizations of the Hall-Littlewood symmetric functions, rather than the irreducible characters, as the best basis of the character ring. Full details will appear elsewhere.