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Transactions of the American Mathematical Society

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Invariant ideals of abelian group algebras and representations of groups of Lie type

Authors: D. S. Passman and A. E. Zalesskii
Journal: Trans. Amer. Math. Soc. 353 (2001), 2971-2982
MSC (2000): Primary 16S34, 20G05; Secondary 20E32, 20F50
Published electronically: March 15, 2001
MathSciNet review: 1828481
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Abstract: This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for $G$ an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group $A$ is the additive group of a finite-dimensional vector space $V$ over a locally finite field $F$ of prime characteristic $p$, and the automorphism group $G$ is a simple infinite absolutely irreducible subgroup of $GL(V)$. Thus $G$ is isomorphic to an infinite simple periodic group of Lie type, and $G$ is realized in $GL(V)$ via a twisted tensor product $\phi$ of infinitesimally irreducible representations. If $S$ is a Sylow $p$-subgroup of $G$ and if $\langle v\rangle$ is the unique line in $V$ stabilized by $S$, then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus $T_G$ on $\langle v\rangle$. At present, we are able to handle the case where $\phi$ is a rational representation with character field equal to $F$.

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Additional Information

D. S. Passman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

A. E. Zalesskii
Affiliation: School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

Received by editor(s): May 31, 2000
Published electronically: March 15, 2001
Additional Notes: Much of this work was performed during a visit by the second author to the University of Wisconsin, Madison. He is grateful to the members of the Mathematics Department for their kind hospitality. The visit was made possible thanks to the financial support of EPSRC. The first author’s research was supported in part by NSF Grant DMS-9820271
Dedicated: Dedicated to the memory of our friend, Richard E. Phillips
Article copyright: © Copyright 2001 American Mathematical Society