Constructions for infinitesimal group schemes

Details Constructions for infinitesimal group schemes Constructions for infinitesimal group schemes

2008-02-20

arxiv.org/abs/0802.2918v2

Mathematics

Representation Theory

55 pages. This is a final version to appear in Transactions of the AMS. Significantly revised from the original preprint. In p

Scientific paper

Let G be an infinitesimal group scheme over a field k of positive characteristic p. We introduce the global p-nilpotent operator $\Theta_G: k[G] \to k[V(G)]$, where V(G) is the scheme which represents 1-parameter subgroups of G. This operator applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain G-modules (including those of constant Jordan type), we employ the global p-nilpotent operator to associate various algebraic vector bundles on the projective scheme $\bP(G)$, the projectivization of the scheme of one-parameter subgroups of G. These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on $\bP(G)$.

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