A new proof of the Kac-Kazhdan conjecture
Tomoyuki Arakawa2006, International Mathematics Research Notices
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Abstract
Letḡ be a complex simple Lie algebra of rank , g the affine Lie algebra associated withḡ: g =ḡ⊗C[t, t −1 ] ⊕ CK ⊕ CD, where K is the central element and D is the degree operator. A weight λ of g is called a critical weight if λ + ρ, K = 0. A critical weight λ is called generic if λ + ρ, α ∨ ∈ Z >0 for all α ∈ ∆ re + , where ∆ re + is the set of positive real roots of g. Let L(λ) be the irreducible g-module of highest weight λ. Theorem 1. If λ is a generic critical weight, then ch L(λ) = e λ α∈∆ re + (1 − e −α) −1. Theorem 1 was conjectured by V. Kac and D. Kazhdan [KK]. It has been proved by different methods: for sl 2 by M. Wakimoto [Wk], N. Wallach [Wl]; for the classical type affine Lie algebras by T. Hayashi [H] and R. Goodman and N. Wallach [GW]; for general affine Lie algebras by B. Feigin and E. Frenkel [FF1] 1 , J. M. Ku [Ku] and G. Kuroki [Kr]. See also the results in finite characteristic by O. Mathieu [M]; for affine Lie superalgebras by M. Gorelik [G] The purpose of this note is to give a yet another proof of Theorem 1. We show that Theorem 1 easily follows as a corollary of a general result [A2] on the quantized Drinfeld-Sokolov reduction [FF2, FKW].
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