2025/05/12 by Pal, Souvik, Teja, G. Krishna
#17B22 #17B67 #17B70 #52B20 #52B99 #FOS: Mathematics #Primary: 17B10 #Representation Theory (math.RT) #Secondary: 17B20
paper · doi:10.48550/arxiv.2505.08102
Fix any Borcherds-Kac-Moody ℂ-Lie algebra (BKM LA) \mathfrakg=\mathfrakg(A) of BKM-Cartan matrix A, and Cartan subalgebra \mathfrakh⊂ \mathfrakg. In this paper, we obtain explicit weight formulas of any highest weight \mathfrakg-module V with top weight λ∈ \mathfrakh^* : 1) Generalizing and extending those in one stroke from Kac-Moody (KM) case, of simples V=L(λ) by Khare [Trans. Amer. Math. Soc. 2017] and Dhillon-Khare [Adv. Math. 2017 & J. Algebra. 2022] and recently of all V by Khare-Teja; via parabolic and higher order Verma V. 2) Uniform for all (\mathfrakg, λ, V); seemingly novel even for integrable (L(λ) and all intermediate) V for dominant integral λ∈ P+. 3) As Weyl-orbit formulas (of finite-dim. L(λ)s) for several V; and for our candidates of parabolic Vermas over BKM LAs. 4) Using our concepts of holes (1-dim. weight-spaces lost) in V, and P±-dominant weights to cover Chevalley-Serre relations in generic simple Vs. We define P± to be the set of μ∈ \mathfrakh^* paired with simple co-roots for Aii≥ 0 as usual, but notably by negative multiples of \frac|Aii|2 if Aii<0. By-products of working with P±: study of simples L(λ) ∀ λ∈ P±, notably of L(ρ) for Weyl vectors ρ∈ P± ∖ P+, and their Verma covers; all novel to our best knowledge. For Weyl-Kac-Borcherds character type formulas of these L(λ)s over negative rank-2 \mathfrakg and of L(ρ) over negative An type \mathfrakg ∀ n∈ ℕ, we explore : i) their presentations; ii) their Verma modules' structures; iii) problems on maximal vectors or Verma embeddings, from Kac-Kazhdan [Adv. Math. 1979], for our BKM P± setting.