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.
 A: The following comments are not related to physics, but might give some motivation for studying them.

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*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.


*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.
