Representations of quantum groups at even roots of unity

Let g be a finite dimensional complex simple Lie algebra and U(g) its enveloping algebra. The quantum group of Drinfeld and Jimbo is a Hopf algebra denoted U_q(g) defined on Chevalley-like generators over C[q, q^-1]. Through “specialization" of q at different ε{lunate} ∈ C one obtains a parameterized family of Hopf algebras. When ε{lunate} = 1 one recovers the classical universal enveloping algebra. Moreover, when ε{lunate} is not a root of unity Lusztig (Adv. Math.70 (1988), 237-249) and Rosso, (Comm. Math. Phys. 117 (1988), 581-593) have shown independently that the representation theory of U_ε{lunate}(= U_ε{lunate}(g)) is analogous to that of U(g). When we fix ε{lunate} a primitive root of unity the situation changes considerably. At odd roots of unity the representations of U_ε{lunate} are partitioned by conjugacy classes of the algebraic group G with Lie algebra g [DC-K]. This paper relates the representations of U_ε{lunate} at even roots of unity to conjugacy classes in the group G^∨-the Langlands dual of G. The partition is somewhat finer than that at odd roots of unity and requires a more detailed analysis. This correspondence is then used to study the representations at even roots of unity. In particular, we obtain a “triangulability" result which allows us to calculate the degree of U_ε{lunate} using a deformation argument.