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Let $\mathfrak{g}$ be a complex finite dimensional semisimple Lie algbera, $W$ be the Weyl group, $\rho$ be the half sum of positive roots, $M(\eta)$ be the Verma module of weight $\eta$ and $L(\eta)$ its unique simple quotient. Then it is well-known that $[M(w\cdot(-2\rho)):L(x\cdot(-2\rho))]=P_{w_0w,w_0x}(1)$, where $P_{u,v}$ is the Kazhdan Lusztig polynomial of $W$ and $w_0$ is the longest element in $W$. The above is an example that illustrating the fact that one can study Verma modules in terms of Kazhdan Lusztig Theory.

For more evidences, we also have Kazhdan-Lusztig Conjectures for affine Lie algebra (See Masaki Kashiwara and Toshiyuki Tanisaki --- Characters of irreducible modules with non-critical highest weights over affine Lie algebras) and the paper: RONALD S. IRVING--- The socle filtration of a Verma module.

I would like to have the brief/full history of the study of Verma modules in terms of Kazhdan Lusztig Theory. If possible, please provides reference(s) for each aspect of the study of Verma modules. It would be great to have history about Verma modules over different types of Lie algberas (semisimple, affine or even Kac Moody Lie algebra).

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    $\begingroup$ Isn't the general philosophy the other way around? That is, Verma modules have a (no pun intended) "simple" structure (i.e., it is easy to write down a basis or the character), whereas the simple modules can be very complicated, so the point of KL theory is to express the simples in terms of the Vermas. $\endgroup$ – Sam Hopkins Oct 4 '19 at 14:39
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It's probably too soon to expect a good historical overview, but for example Steve Kleiman has already written a scholarly article (The development of intersection homology theory) emphasizing the original KL (Kazhdan-Lusztig) conjecture. This is available in arXiv versions or in the published version cited there.

One contributor easily overlooked is Vinay Deodhar, who probed further in what he called "KL theory". See for example his influential joint paper Deodhar–Gabber–Kac - Structure of some categories of representations of infinite-dimensional Lie algebras (MSN), or the 1991 survey Deodhar - A brief survey of Kazhdan-Lusztig theory and related topics (MSN). He soon dropped out of the research world because of illness in his immediate family, but was starting to get involved again when he himself developed serious health problems and later died. But in the late 1970s, when I met him at IAS Princeton where Jantzen was then visiting, both he and Jantzen were getting close to the Kazhdan-Lusztig viewpoint.

There is also a characteristically readable textbook by Roger Carter, Lie algebras of finite and affine type. See especially Chapters 19-20. This is only the first step toward Verma modules and simple modules for general Kac-Moody algebras, but it conveys much of the flavor of the subject.

In the direction of quantum enveloping algebras, there are of course many papers and books (including the one by Lusztig, Introduction to quantum groups (MSN)) which suggest the open-ended nature of the representation theory there. But in this and other directions, there is apparently no comprehensive history yet available. Probably it's too soon for that.

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