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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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