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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?

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  • $\begingroup$ 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). $\endgroup$ Commented Sep 12, 2017 at 5:46

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If you only want irreducible representations (denoted by $\phi: \mathfrak{g}\to \mathrm{End}(V)$) of a reductive Lie algebra $\mathfrak{g}$, then by Schur's lemma, $\phi(a)=\lambda I$ for $a\in \mathfrak{a}$, where $I$ is the identity map in $\mathrm{End}(V)$. [Note that Schur's lemma works for irreducible representations of arbitrary finite dimensional Lie algebras, provided that the field is algebraically closed.] i.e. the central elements can only act by constant multiplication in irreps.

The real subtlety when it comes to classifying all representations of reductive Lie algebras is that there exists indecomposible representations that are not irreducible [while for semisimple Lie algebras all representations are completely reducible, so indecomposible and irreducible mean the same thing]. Take $\mathfrak{gl}(2)=\mathfrak{sl}(2)\oplus\mathbb{C}$ for example, spanned by $\{x,y,h,c\}$ where $c$ is the central element. Consider the four dimensional representation $$x=\begin{pmatrix} 0 & 1 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 1\\ 0 & 0 & 0& 0 \end{pmatrix}, y=\begin{pmatrix} 0 & 0 & 0& 0\\ 1 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 1& 0 \end{pmatrix},\\ h=\begin{pmatrix} 1 & 0 & 0& 0\\ 0 & -1 & 0& 0\\ 0 & 0 & 1& 0\\ 0 & 0 & 0& -1 \end{pmatrix}, c=\begin{pmatrix} 0 & 0 & 1& 0\\ 0 & 0 & 0& 1\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0 \end{pmatrix}. $$ You can easily show that it is reducible, but indecomposible.

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