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On bounded generalized Harish-Chandra modules
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Let ???? be a complex reductive Lie algebra and ????⊂???? be any reductive in ???? subalgebra. We call a (????,????)-module M bounded if the ????-multiplicities of M are uniformly bounded. In this paper we initiate a general study of simple bounded (????,????)-modules. We prove a strong necessary condition for a subalgebra ???? to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded (????,????)-module, and then establish a sufficient condition for a subalgebra ???? to be bounded (Theorem 5.1). As a result we are able to classify the maximal bounded reductive subalgebras of ????=sl(n).
Title: On bounded generalized Harish-Chandra modules
Description:
Let ???? be a complex reductive Lie algebra and ????⊂???? be any reductive in ???? subalgebra.
We call a (????,????)-module M bounded if the ????-multiplicities of M are uniformly bounded.
In this paper we initiate a general study of simple bounded (????,????)-modules.
We prove a strong necessary condition for a subalgebra ???? to be bounded (Corollary 4.
6), i.
e.
to admit an infinite-dimensional simple bounded (????,????)-module, and then establish a sufficient condition for a subalgebra ???? to be bounded (Theorem 5.
1).
As a result we are able to classify the maximal bounded reductive subalgebras of ????=sl(n).
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