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On unitary representations of algebraic groups over local fields
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Let
G
\mathbf {G}
be an algebraic group over a local field
k
\mathbf {k}
of characteristic zero. We show that the locally compact group
G
(
k
)
\mathbf {G}(\mathbf {k})
consisting of the
k
\mathbf {k}
-rational points of
G
\mathbf {G}
is of type I. Moreover, we complete Lipsman’s characterization of the groups
G
\mathbf {G}
for which every irreducible unitary representation of
G
(
k
)
\mathbf {G}(\mathbf {k})
is a CCR representation and show at the same time that such groups belong to the family of trace class groups, recently studied by Deitmar and van Dijk.
Title: On unitary representations of algebraic groups over local fields
Description:
Let
G
\mathbf {G}
be an algebraic group over a local field
k
\mathbf {k}
of characteristic zero.
We show that the locally compact group
G
(
k
)
\mathbf {G}(\mathbf {k})
consisting of the
k
\mathbf {k}
-rational points of
G
\mathbf {G}
is of type I.
Moreover, we complete Lipsman’s characterization of the groups
G
\mathbf {G}
for which every irreducible unitary representation of
G
(
k
)
\mathbf {G}(\mathbf {k})
is a CCR representation and show at the same time that such groups belong to the family of trace class groups, recently studied by Deitmar and van Dijk.
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