Rad HAZU, Matematičke znanosti, Vol. 28 (2024), 245-258.

WEIRD K-ACTIONS ON U(g) FOR so(n, 1) AND su(n, 1)

Hrvoje Kraljević

Abstract.   Let g₀ be either so(n, 1) or su(n, 1), g its complexification, K a maximal compact subgroup of the adjoint group of g₀, U(g) the universal enveloping algebra of g and U(g)^K its subalgebra of K-invariants. A consequence of our results in [2] is that besides the usual adjoint action of K on U(g) there is another action of K commuting with the adjoint action and leaving U(g)^K pointwise invariant. The case g₀ = so(2, 1) ≃ su(1, 1) is trivial since K is commutative and the weird action of K coincides with the inverse of adjoint action. We investigate closely the weird action of K in the simplest nontrivial case g₀ = so(3, 1).

2020 Mathematics Subject Classification.   16S30, 20G05.

Key words and phrases.   Universal enveloping algebra, K-types, adjoint action, K-invariants, K-harmonic polynomials.

DOI: https://doi.org/10.21857/y26kecg7e9

References:

1. F. Knop, Der Zentralisator einer Liealgebra in einer einhülende Algebra, J. Reine Angew. Math. 406 (1990), 5-9.
   MathSciNet     CrossRef

2. H. Kraljević, The structure of the algebra (U(g) ⊕ C(p))^K for the groups SU(n, 1) and SO_e(n, 1), Math. Commun. 27 (2022), 11-18.
   MathSciNet