By M. Aizenman (Chief Editor)

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**Extra resources for Communications in Mathematical Physics - Volume 250**

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Tn±1 ] ⊕ D ⊕ z be a toroidal Lie algebra corresponding to ◦ a semisimple Lie algebra g. We describe all Borel subalgebras of g which contain the ◦ ◦ ◦ Cartan subalgebra h =h ⊕D ⊕z, where h is a fixed Cartan subalgebra of g. We show that each such Borel subalgebra b determines a parabolic decomposition g = n− ⊕ k ⊕ n+ , where k is a proper toroidal subalgebra of g and b ⊂ k ⊕ n+ . Our first main result is that, for any weight λ which does not vanish on z ∩ [k, k], an arbitrary subquotient of the Verma module Mg (λ) is induced from its k ⊕ n+ –submodule of n+ –invariant vectors.

Mimachi, M. Yoshida and then, by the change of b → a −1 q −n+1 , we have 0≤i≤n (q −n ; q)i (a −1 q −n+1 ; q)i −( i ) q 2 −aq 2n−1 (q; q)i i = a n q n(n−1) . On the other hand, we have k k Ak−p Xp = (g − 1)k p=1 = p=1 (g − 1)k (a; g)k (g; g)k k p=1 p (−1)p g ( 2 ) (g; g)p (a; g)k−p (g; g)k−p (a −1 g 1−k ; g)p (g −k ; g)p −(p) g 2 (−ag 2k−1 )p , (g; g)p since p (ag k−p ; g)p (a −1 g 1−k ; g)p g ( 2 ) (−ag k−p )p 1 = = . (a; g)k−p (a; g)k (a; g)k Therefore, we have k Ak−p Xp = p=1 (g − 1)k (a k g k(k−1) − 1).

Denote by a · · · b the sum of all the possible (DC)-monomials in {a, . . , b} for 1 ≤ a < b ≤ n, and put Xn = 1···n , n ≥ 2. 36 K. Mimachi, M. Yoshida For instance, X2 = 1 + 12 , X3 = (1 + 123 )(1 + 12 + 23 ), X4 = (1 + 1234 )(1 + 12 + 23 + 34 + 12 34 + (1 + 12 + 23 ) 123 + (1 + 23 + 34 ) 234 ). Similarly, put [[01 · · · k]] :=[01 · · · k] × {the sum of all the possible (DC)-monomials in 0, 1, . . , k of length at most k}, [[n − k + 1 · · · n ]] :=[n − k + 1 · · · n ] × {the sum of all the possible (DC)-monomials in n − k + 1, .

### Communications in Mathematical Physics - Volume 250 by M. Aizenman (Chief Editor)

by Charles

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