Representation theory of $\mathfrak{gl}(n,\mathbb{C})$

Question

I am looking for a reference on the representation theory of the algebra $\mathfrak{gl}(n,\mathbb{C})$. Anything I find relates to the group. I know that representations of the group can be described by Young diagrams due to their relation to representations of the symmetric group. How much of this theory carries over to the algebra?

Answer 1

There are several books on representation theory of semisimple and reductive Lie algebras, e.g., Fulton and Harris, Humphreys, Knapp and others. Since $\mathfrak{gl}_n(\mathbb{C})$ is a reductive Lie algebra, we know by the MSE-question

Irreducible representation and reductive Lie algebra

that we can also use the representation theory of the simple Lie algebra $\mathfrak{sl}_n(\mathbb{C})$.

Answer 2

Algebraic representations of $\mathrm{GL}(n, \mathbb{C})$ are entirely described by Young diagrams and powers of the one-dimensional determinant or inverse determinant representation. The category is semisimple, and each simple is a Young diagram with at most $n$ rows, tensored with some power of $\det^{-1}$. Each representation of $\mathrm{GL}(n, \mathbb{C})$ descends to a representation of $\mathfrak{gl}(n, \mathbb{C})$, and so this gives a partial description of the representation theory of $\mathfrak{gl}(n, \mathbb{C})$. However, if you just start with the Lie algebra $\mathfrak{gl}(n, \mathbb{C})$ and try to find representations, you will find many many more than you would have found with the original group. For example, there are simple infinite-dimensional modules for $\mathfrak{gl}(n, \mathbb{C})$, as well as modules with highest weights which are not integral.

There is a book by Humphreys on BGG Category $\mathcal{O}$ if you are interested in learning about a wide class of representations of the Lie algebra $\mathfrak{gl}(n, \mathbb{C})$. There is also a way to isolate representations of "dominant integral weights", which are essentially those which come from a representation of the original group $\mathrm{GL}(n, \mathbb{C})$.