Abstract
This paper studies the “reduction mod p” method, which constructs large classes of representations for a semisimple algebraic group G from representations for the corresponding Lusztig quantum group Uζ at a pr-th root of unity. The G-modules arising in this way include the Weyl modules, the induced modules, and various reduced versions of these modules. We present a relation between ExtGn(V,W) and ExtUζn(V′,W′), when V,W are obtained from V′,W′ by reduction mod p. Since the dimensions of Extn-spaces for Uζ-modules are known in many cases, our result guarantees the existence of many new extension classes and homomorphisms between rational G-modules. One application is a new proof of James Franklin's result on certain homomorphisms between two Weyl modules. We also provide some examples which show that the p-th root of unity case and a general pr-th root of unity case are essentially different.
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