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EXTENSIONS OF SIMPLE MODULES FOR THE ALGEBRAIC GROUP OF TYPE G2
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Let G be a connected simply-connected simple algebraic group of type G2 over an algebraically closed field K of characteristic p>-13,and G1 the kernel of the Frobenius morphism F on G. In the present paper we show how one can obtain the extensions of any two simple modules for G by using the same method as[7] and[9]. Note that there are 6 positive roots in the root system of G, and 12 alcove types corresponding to the 12 alcoves in the restricted box, hence there are 12 "generic decomposition patterns" which indicate composition factor multiplicities in the Weyl modules of G. In particular, all patterns involve the same number of alcoves and the same distribution of multiplicities, and the total number of composition factors is 119. Moreover, composition factor multiplicities are from 1 to 4. All these facts make the calculation quite complicated. We shall omit almost all details of proofs which are not essentially defferent from those in[7]in order to make the size of this paper not too large. The paper is organized as follows. In Section 1 we introduce our basic notations and preliminaries on the extensions of simple modules for G and G1. In Section 2 and 3 we determine the extensions of any two simple Gmodules with "small" highest weights, and the extensions of any two simple G1-modules. Section 4 is devoted to determining the G-socle of the tensor product of two simple G-modules. Finally in Section 5 we obtain our main results on Ext1 G. The authors wish to thank the referee for his helpful comments, especially for his pointing out both mistakes of (8) and the definition of B(λ) in the original version. (The same mistakes also occur in[7].)
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