# Motivation for studying infinite-dimensional representations of Lie algebras

I come from a group-theoretic background, where the primary motivation for studying representations are the numerous reconstruction theorems, such the Pontryagin duality theorem or the Tanaka-Krein duality. This theorems are exclusively concerned with finite-dimensional representations. On the other hand, infinite-dimensional representations are quite a popular topic this days, specially in Lie theory. Is there any concrete reason for this?

Of course, many people are interested in infinite-dimensional representations simply because they are poorly understood in comparison with their finite-dimensional counterparts, but I'm looking for applications of infinite-dimensional representations of Lie algebras in other areas of mathematics. I've heard over the years that "infinite-dimensional representations are important in mathematical physics", but I was never offered a concrete application.

• See here. The last section is on applications to mathematical physics. Commented Aug 4, 2022 at 15:24

1. The reason for studying only finite-dimensional representations in the examples you described is because they usually assume some kind of finiteness condition. If you're interested in Lie groups or $$p$$-adic groups such as $$\operatorname{GL}_n(\mathbb{R})$$ or $$\operatorname{GL}_n(\mathbb{Q}_p)$$, then you end up studying infinite-dimensional representations. Of course, you're usually going to assume at least some finiteness hypothesis, but it won't finite dimensionality. In the first case one usually translates to the language of $$(\mathfrak{g},K)$$-modules and in the second case one usually does not use Lie algebras.
2. Even for studying finite-dimensional representations, one can use some infinite-dimensional ones. For a Lie algebra $$\mathfrak{g}$$, one can define the category $$\mathcal{O}$$. It's a $$\mathbb C$$-linear category of finite length which is generated by the Verma modules and the finite-dimensional irreducible ones. Using this category, one can study the Kazhdan-Lusztig conjectures and more generally reduce questions about the finite-dimensional irreducible representations to questions about Verma modules, which are usually simpler.