Relative asymptotic weight multiplicity conjecture

Establish that for every finite-dimensional simple Lie algebra g and every nonzero dominant integral weight λ in the dominant weight lattice P+, the following relative asymptotic multiplicity condition holds: for any weights μ1 and μ2 in the coset λ + Q (where Q is the root lattice of g), the limit of the ratio of weight multiplicities m_{nλ}(μ1)/m_{nλ}(μ2) equals 1 as n → ∞ along integers n satisfying nλ ∈ λ + Q, where m_{nλ}(μ) denotes the multiplicity of weight μ in the irreducible g-module L(nλ).

Background

The paper relates large-color limits of colored invariants of torus knots T(p,p′) to characters of principal affine W-algebras. After renormalizing by the trailing monomial, the Rosso–Jones formula expresses these invariants in terms of weight multiplicities of irreducible modules L(λ). Matching the limits with W-algebra characters requires that certain ratios of weight multiplicities stabilize to 1 as the color n grows.

This requirement is formalized as the 'relative asymptotic multiplicity condition' (equation (31)), asserting that for any μ1, μ2 in λ + Q, the ratio m_{nλ}(μ1)/m_{nλ}(μ2) tends to 1 along n with nλ ∈ λ + Q. Conjecture 22 posits that this condition holds for all nonzero λ ∈ P+ and all finite-dimensional simple Lie algebras g. The paper verifies the conjecture in several families and ranks (e.g., types A, B, C, D, and G2), and Theorem 18 derives the main limit formulas assuming this conjecture.