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Carter: Lie algebras of finite and affine type. ): Kac-Moody and Virasoro algebras. • Kumar: Kac-Moody groups, their flag carieties and representation theory. • E. Frenkel: Langlands correspondence for loop groups. • Pressley, Segal: Loop Groups. • Humphreys: Lie algebras. • Segal, Wilson: ? (probably IHES) • Deodhar, Gabher, Kac: Adv. Math. 45 (1982). • Kumar: J. Algebra 108 (1987) B Notational reference Witt the Witt algebra over C Vir the Virasoro algebra over C, a central extension of Witt Lie(G) the Lie algebra of the Lie group G, g = Te G g either a general Lie algebra or a finite-dimensional, simple one g a central extension of a Lie algebra g; 0 → z → g → g → 0 Lg the loop algebra g ⊗ C[t , t −1 ] Lg =Lg d Cd , where d = t dt ∈ Der C[t , t −1 ] Note: H 2 L g; C ∼ =C∼ = H 2 L g; C Lg = L g ⊕ Cd ⊕ Cc, the central extension of L g, affine Kac-Moody Lie algebra L (A) the Kac-Moody Lie algebra corresponding to the generalised Cartan matrix A U (g) the universal enveloping algebra of the Lie algebra g; U (g) = T (g) (x ⊗ y − y ⊗ x − [x, y]) 〈−, −〉 a positive-definite sesquilinear form on a representation V , non-degenerate if V is unitary A the affine, untwisted gen.

First calculate the “leading term” and get ∞ mult(α)P (η−nα) α∈∆re + n=1 hα . e. λ + ρ, β = 21 〈β, β〉. So if this cannot be satisfied, then M (λ) is irreducible and so F (λ) is non-degenerate; hence det F η is a product of linear factors h β + 〈ρ − 21 β, β〉 , and the formula for leading terms shows that β = nα for some α ∈ ∆+ , which we call a quasi-root. So we get products of h α + 〈ρ − 21 α, α〉 . 50 Infinite-dimensional Lie algebras 51 The trick. 2), and F η is a product of finitely many linear terms of the form h α + ρ − nα 2 ,α .

U 0 ∧ u −1 ∧ u −2 ∧ · · · : u −m = v −m for m 46 0 , Infinite-dimensional Lie algebras 47 where v 0,−1,−2,... is the vacuum vector in degree 0. ) ∼ = PΩ −−→ Gr , u 0 ∧ u −1 ∧ u −2 ∧ · · · → Cu i . i ≤0 There are two operators, F (0) → F (1) and F (0) → F (−1) , respectively via V andV ∗ (the restricted dual, cf. 3), defined as follows. v i 0 ∧ v i 1 ∧ · · · = v ∧ v i 0 ∧ v i 1 , and (−1) j v i 0 ∧ v i 1 ∧ · · · ∧ v i j −1 f (v i j ) ∧ v i j +1 ∧ · · · . v i 0 ∧ v i 1 ∧ · · · = j ≥0 The element E i j acts on F (0) via v i v ∗j .