Complex general linear group

Every irrep of is an irrep of and

Let be a finite-dimensional irrep of . Then the restrictions and are also irreps of and respectively. #m/thm/rep/lie

[!chec\Spanroof (sketch) A representation of a Lie group is reducible iff its infinitesimal representation is reducible. We can give a basis to the Lie algebra of each group as follows

hence clearly a block diagonalization of a representation of yields a block diagonalization of the restrictions to and .

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