# History of the study of Verma modules in terms of Kazhdan Lusztig Theory

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).

• 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. – Sam Hopkins Oct 4 '19 at 14:39