Degenerate Affine Hecke Algebra as Quotient of U(sl_n) via BGG Category O — E8 Intelligence Research

FINDING: Degenerate affine Hecke algebra H(GL_n) of type A is realized as a quotient of the universal enveloping algebra U(sl_n) via exact functors on BGG category O, with Verma modules mapping to standard modules. | MATH: Let g = sl_n, h ⊂ b ⊂ g Cartan/Borel. BGG category O decomposes into blocks O_χ indexed by central characters χ. The degenerate affine Hecke algebra H_n (Drinfeld) is generated by S_n (symmetric group) and polynomial generators x_1,...,x_n with relations: s_i x_j = x_j s_i (|i-j|>1), s_i x_i = xᵢ₊₁ s_i − 1, s_i xᵢ₊₁ = x_i s_i + 1, and x_i x_j = x_j x_i. The functor Φ: O → H_n-mod sends Verma M(λ) ↦ standard module, simple L(λ) ↦ irreducible or 0. Key parameter: the "level shift" is the central element z = Σ x_i acting on standard modules by the dot-action λ + ρ evaluated at the trivial character; the shift by ρ = half-sum of positive roots is the quantum/affine parameter. | CONNECTION: Root system Aₙ₋₁ with Weyl group S_n; the shift by ρ = (1/2)Σα>0 α is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com