What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?

We already know all the complex finite dimensional linear representation of SU(N).

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What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?

We already know all the complex finite dimensional linear representation of SU(N).

continuityof the repn. Continuity certainly does implysmoothness, because even in infinite-dimensional (continuous) repns, Garding's argument proves density of smooth vectors, which in a finite-dimensional repn must be everything. A refinement of such an argument proves thatrealanalytic vectors are dense in "nice" repns, and finite-dimensional are nice enough so that all vectors are real-analytic. But, as Richard S. notes, certainly not complex-analytic. Then it's about highest weights? Clarify? $\endgroup$ – paul garrett Nov 21 '12 at 3:38