reference containing the list of irreducible finite dimensional representation of real general linear group

Question

It seems that it is not easy to find a reference containing a classification and construction of finite dimensional irreducible representations of $GL_n(\mathbb{R})$. One way to look at it is via $(\mathfrak{g},K)$ module, and we do have a classification theorem in terms of highest weight, but it is not obvious to get the explicit constructions from this,especially in higher rank case.

So I'm wondering is there some good reference containing explicit constructions of all irreducible finite dimensional representations of $GL_n(\mathbb{R})$?

Answer 1

Let's pull back to irreps of $GL_1({\mathbb R}) \times SL_n({\mathbb R})$. By Schur's lemma, the first factor acts by scalars, so the representation is of the form $V \otimes W$ where $V$ is a character of $GL_1({\mathbb R}) \cong Z_2 \times {\mathbb R}$ and $W$ is an irrep of $SL_n({\mathbb R})$. In particular $V\otimes W$ is already an irrep of just the $SL_n$ factor. From there it's an irrep of ${\mathfrak sl}_n({\mathbb R})$, hence of its complexification, hence of $SL_n({\mathbb C})$.

So really it comes down to, once we have an old familiar representation of $SL_n({\mathbb C})$, what can the representation $V$ look like. Well, the kernel of $Z_2 \times {\mathbb R} \times SL_n({\mathbb R}) \to GL_n({\mathbb R})$ is trivial for $n$ odd, $(-1,1,-{\bf 1})$ for $n$ even. So your extra freedom is to let the ${\mathbb R}_+$ part of the scalars act by any character, plus for $n$ odd you can let $-{\bf 1}$ act as $\pm 1$ as you wish. Not a whole lot.

Answer 2

To expand a little on Allen's useful answer:

1) The question doesn't specify over which field (presumably $\mathbb{R}$ or $\mathbb{C}$) the f.d. irreps of $G=GL_n(\mathbb{R})$ should be studied, but in this particular case it doesn't matter much. From the viewpoint of Steinberg's 1967-68 Yale lectures, the special linear group is a Chevalley group; so its structure and representations in characteristic 0 can be studied first over $\mathbb{Q}$ followed by extension of scalars.

2) The f.d. representation theory of semisimple, or more generally reductive, Lie groups has been well understood for a century, going back to work of Elie Cartan and Hermann Weyl. But most of the difficulty involves simple groups (semisimple with finite center), while passage to general linear groups over $\mathbb{C}$ just involves easy modifications by powers of the determinant. So the textbook literature has mostly concentrated on simple Lie groups, though books for physicists usually focus more on $GL_n(\mathbb{C})$ and the combinatorics of partitions along with Schur-Weyl duality. Adapting this to the real group $G$ is fairly straightforward, as Allen indicates.

3) Anyway, it's not easy to specify a standard source for the specific question raised here. That's partly because modern research efforts have focused mostly on the less-understood world of infinite dimensional representations.

4) The question asks for a "list" and a "construction". Allen basically gives a list, taking for granted the classical parametrization of f.d. irreps of the simple groups over $\mathbb{C}$ (or restrictions to $\mathbb{R}$) by highest weights in general or partitions in this special case. But constructions are not so easy even for general or special linear groups, where the best concrete approach seems to be via Schur-Weyl duality.