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First assume that there are at least two components. Then there are no h-invariant components, so we must show that F = 0 in this case. 7) shows that F (b) = 0. So we only need show that F is zero on the algebra Ah of h-invariants of A. If there is 1 = k ∈ h such that k fixes each Ai then the algebra of k-invariants B is a semi-simple algebra admitting h / k . By induction we see that F (B) = 0, so we are done as Ah ⊂ B. So without loss there is no such k. So h has order |I |, the number of components.

Math. American Math. Soc. 56 II, 295–316 (1994) [EZ] Eichler, M. : On the Theory of Jacobi Forms I. Progress in Math. Vol. L. : Verma modules over the Virasoro algebra. , Vol. B. : Spinor construction of vertex operator algebras, triality (1) and E8 . Contemp. Math. , Huang, Y. : On axiomatic approaches to vertex operator algebras and modules. Memoirs Am. Math. Soc. , Lepowsky, J. : A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc. Natl. Acad.

Discrete series of the Virasoro algebra and the moonshine module. Proc. Symp. Pure. Math. American Math. Soc. 56 II, 295–316 (1994) [EZ] Eichler, M. : On the Theory of Jacobi Forms I. Progress in Math. Vol. L. : Verma modules over the Virasoro algebra. , Vol. B. : Spinor construction of vertex operator algebras, triality (1) and E8 . Contemp. Math. , Huang, Y. : On axiomatic approaches to vertex operator algebras and modules. Memoirs Am. Math. Soc. , Lepowsky, J. : A natural representation of the Fischer-Griess Monster with the modular function J as character.