The full Ext version requires the Koszul grading of category O

Scope warning. The grading used on this page is the auxiliary grading of the polynomial Soergel lifts: it is what makes the character theorem read off graded standard-support ranks, and it is forgotten by the Soergel functor, which is an ungraded equivalence on projectives (Indecomposable projectives correspond to indecomposable Soergel modules). Consequently the results proved here are the ordinary (ungraded) Verma-flag, composition and character statements, together with the exact shift-to-$q=1$ conversion (Projective Verma multiplicities are Kazhdan–Lusztig values at one, Simple Verma composition multiplicities are Kazhdan–Lusztig values, The Kazhdan–Lusztig character formula in regular integral category O); they do NOT construct a grading on category $\mathcal{O}$, and in particular they assert nothing about graded multiplicities, about the $\mathrm{E}\mathrm{x}\mathrm{t}$-algebra of $\mathcal{O}$, or about Koszul duality. Individual coefficients of the Kazhdan–Lusztig polynomials need not be ordinary category-$\mathcal{O}$ multiplicities: their value at $q=1$ is the ordinary multiplicity. This does not rule out a graded refinement whose graded multiplicities sum to that value. The graded version (a Koszul grading, graded BGG reciprocity and the $\mathrm{E}\mathrm{x}\mathrm{t}$ form of the multiplicity theorem, in the sense of the Beilinson–Ginzburg–Soergel programme) remains reserved for the separate Koszul-duality frontier of this library and is deliberately not smuggled into this page.