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This paper uses the notion of the quantum dimension to obtain new results on the cohomology and representation theory of quantum groups at a root of unity. In particular, we consider the elementary theory of support varieties for quantum groups.

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Let G be a simply connected semisimple linear algebraic group over an algebraically closed field K of characteristic p > 0, δa Borel subgroup of G, and T a maximal torus of B. Let G n be the w-th Frobenius kernel of G, and G n B the closed subgroup scheme of G generated by G n and B. Then we have induction functors Z n = Ind^n S , Ind^s and Indf. The first one of these functors is exact, but the others are only left exact, so we can further construct their right derived functors Hi = R i Indg nS and H* = R ι Inds for all i ^ 0. Thanks to the transitivity of inductions we obtain that H° = H°noZ n , or more generally, H ι = Hi<>2 n .Let X(T) be the character group of T. Let χeX(T), canonically regarded as a 1-dimensional B-module. In this paper we shall reveal a connection between the G n -socle series of Z n (χ) and the G-socle series of H\x). Our main result (cf. (2.1)) is a generalization of a similar result of Andersen (cf. [3, (4.4)]). As an application of the main result, we shall also discuss the G-socle series of H°(χ) for non-generic λ in the B 2 case.This research was carried out while the first author was a visiting scholar at East China Normal University. He would like to express his appreciation to his advisor Professor Cao Xi-hua. Moreover, we are greatly indebted to the referee for giving us some useful suggestions and telling us the fact that the same result with different proof was obtained independently by John B. Sullivan in his preprint "the Euler character and cancellation theorem for Weyl modules", Feb. 1987.

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