Abstract
In this chapter we introduce two important tools of our theory, the invariant bilinear form and the generalized Casimir operator Ω. The operator Ω is a “second order” operator which, in contrast to finite-dimensional theory, does not lie in the universal enveloping algebra of g(A) and is not defined for all representations. However, Ω is defined on the so-called restricted representations, and commutes with the action of g(A) in these representations. Remarkably, one can manage to prove a number of results (including classical ones) using only Ω.
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