Representation Theory of Semidirect Products: The Preliminaries (Pt. III)
Point of Post: This is a continuation of this post.
The interesting thing about this ‘dualization’ of is its relation ship between the original and the aforementioned . Namely:
Theorem: For any and one has that .
Proof: Let be arbitrary, then for any one has
And, since was arbitrary it follows that is in as claimed. Since was arbitrary the conclusion follows.
From this we have the following corollary:
Corollary: Let and the associated orbit under the action of on induced by . Then, is -invariant.
Proof: We first show that the space in question is invariant under for every . Let be an element of . Then, there exists for every such that . So, by our last theorem
Since was arbitrary the conclusion follows.
We next show that the space in question is invariant under for every . Indeed, if is as above then
Since and were arbitrary this part of the proof follows.
To finish the proof we merely note that for any one has for some and and so for any in the direct sum in question one has
From this we conclude as a corollary:
Corollary: Let be an irrep. Then, there exists precisely one orbit of the action for which if and only if .
Proof: We know since
that there exists some such that . We claim that is the orbit we seek. Indeed, to see that for we note that since is non-zero and -invariant we may conclude from the irreducibility of that is the full space . Thus, one has that and so of course . Now, if then for some . Thus, by an earlier theorem we have that and so and so .
From this it’s easy to see that
Corollary: If is the non-zero orbit described in the above proof then for every . In particular, .
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.
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