Let us work over $\Bbb C$.

If I know the classification of irreducible representations of semisimple Lie algebras, how do I classify the irreducible representations of reductive Lie algebras?

Let $\mathfrak{g}$ be reductive, then we can always write $\mathfrak{g}=\mathfrak{s}\oplus \mathfrak{a}$ for $\mathfrak{s}$ a semisimple Lie algebra and $\mathfrak{a}$ an abelian Lie algebra. From this I then just need to choose how $\mathfrak{a}$ act right? So $\mathfrak{s}$ has finite dimensional highest weight irreducible representations classified by the quotients $M(\lambda)$ from the verma modules $V(\lambda)$ where $\lambda \in P^+$ (dominant weights) how does $\mathfrak{a}$ act?

Should I be considering the correspondence between $U(\mathfrak{g})$ representations and $\mathfrak{g}$ representations, and then: $$U(\mathfrak{s}\oplus \mathfrak{a})=U(\mathfrak{s})\otimes U(\mathfrak{a})\cong U(\mathfrak{s})\otimes S(\mathfrak{a})$$ and then how do I find what to quotient by to get all of the finite dimensional irreducible reps?

  • $\begingroup$ It can act in pretty much any way it wants, as long as those actions commute with each other and those from the semisimple part. This will generally leave a lot of possibilities. (Btw, your statement about the simples is not quite right). $\endgroup$ – Tobias Kildetoft Sep 12 '17 at 5:46

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