Finite dimensional, irreducible representations of simply connected, complex semisimple algebraic groups can be classified by their highest weight. I was wondering if there is an analogous classification of representations of reductive groups.
This can be made precise as follows: let $\mathfrak g = \operatorname{Lie}(G)$ with $G$ a complex simply connected algebraic group. Let $\mathfrak t$ be a Cartan subalgebra of $\mathfrak g$ with set of simple roots $\Delta$ and simple coroots $\Delta^{\vee} \subset \mathfrak t^{\ast}$. We define a partial order on $\mathfrak t^{\ast}$ by saying that $\lambda \geq \mu$ if $\lambda - \mu$ is a nonnegative linear combination of simple coroots.
We say that $\mu \in \mathfrak t^{\ast}$ is dominant if $\langle \mu,\alpha \rangle \geq 0$ for all $\alpha \in \Delta$, and we say that $\mu$ is integral if $\langle \mu, \alpha \rangle$ is an integer for all $\alpha \in \Delta$.
Then there is a bijection between isomorphism classes of irreducible representations of $G$, ones of $\mathfrak g$, and dominant integral weights. The explicit correspondence is we begin with a representation $(\pi, V)$ of $G$, and let $(d\pi, V)$ be the associated tangent space map on $\mathfrak g$. Now $V$ has an associated weight space decomposition $V = \bigoplus\limits_{\mu \in \mathfrak t^{\ast}} V_{\mu}$, where $$V_{\mu} = \{ v \in V : d\pi(X)v = \mu(X)v \textrm{ for all } X \in \mathfrak t\}$$ There is a unique maximal dominant, integral element $\mu \in \mathfrak t^{\ast}$ for which $V_{\mu}$ is nonzero, and this $\mu$ determines $\pi$ and $d\pi$ up to isomorphism.
If we assume that $G$ is connected and semisimple, but not necessarily simply connected, then the classification is the same, but $\pi \mapsto d \pi$ is no longer a bijection.
Question: Is there a classification along similar lines to irreducible, finite dimensional representations of connected, complex reductive groups?
When $G$ is reductive, but not semisimple, the notion of highest weight still makes sense, since we can write the Cartan subalgebra as the direct sum of that of the derived group of $G$ and the Lie algebra of the center.
