On bounded generalized Harish-Chandra modules

Let 𝔤 be a complex reductive Lie algebra and 𝔨 ⊂ 𝔤 be any reductive in 𝔤 subalgebra. We call a ( 𝔤 , 𝔨 ) -module M bounded if the 𝔨 -multiplicities of M are uniformly bounded. In this paper we initiate a general study of simple bounded ( 𝔤 , 𝔨 ) -modules. We prove a strong necessary condition for a subalgebra 𝔨 to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded ( 𝔤 , 𝔨 ) -module, and then establish a sufficient condition for a subalgebra 𝔨 to be bounded (Theorem 5.1). As a result we are able to classify the maximal bounded reductive subalgebras of 𝔤 = sl ( n ) .