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The notion of Verma-type modules for affine Lie algebras is related to the concept of Borel subalgebras. The literature is extensive when the affine algebra is untwisted and all constructions come from Borel subalgebras of the adjacent finite-dimensional simple Lie algebra. However, I did not find any reference in the case of twisted affine algebras.

What is a reference for Borel (parabolic) subalgebras of twisted affine algebras?

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I think the main point is that all affine Lie algebras (twisted or untwisted) have similar structure involving a triangular decomposition with associated Verma modules. What sources are you starting with? There are countless research papers as well as some books and conference reports dealing with these matters. A random example: Pavel I. Etingof and Alexander A. Kirillov, Jr. Representations of affine Lie algebras, parabolic differential equations, and Lam´e functions. Duke Math. J. 74 (1994), no. 3, 585–614. –  Jim Humphreys Oct 28 '12 at 12:36
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