A strong generalization of Helgason’s theorem

Abstract

Let G G be a simple Lie group with K A N KAN an Iwasawa decomposition of G G , and let M M be the centralizer of A A in K K . Suppose K 1 {K_1} is a fixed, closed, normal, analytic subgroup of K K , and set P ( K 1 ) {\mathbf {P}}({K_1}) equal to the set of all parabolic subgroups P P of G G which contain M A N MAN such that K 1 P = G {K_1}P = G and K 1 ∩ P {K_1} \cap P is normal in the reductive part of P P . Suppose π : G → G L ( V ) \pi :G \to GL(V) is an irreducible representation of G G . Then, if P ( K 1 ) ≠ ∅ {\mathbf {P}}({K_1}) \ne \emptyset , we obtain necessary and sufficient conditions for V K 1 {V^{{K_1}}} , the space of K 1 {K_1} -fixed vectors, to be ≠ ( 0 ) \ne (0) . Moreover, reciprocity formulas are obtained which determine dim ⁡ V K 1 \dim {V^{{K_1}}} .