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The Tits building and an application to abstract central extensions of $ p$-adic algebraic groups by finite $ p$-groups

Author: B. Sury
Journal: Proc. Amer. Math. Soc. 139 (2011), 2033-2044
MSC (2010): Primary 20G25, 20G10
Published electronically: November 22, 2010
MathSciNet review: 2775381
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Abstract: For a connected, semisimple, simply connected algebraic group $ G$ defined and isotropic over a field $ k$, the corresponding Tits building is used to study central extensions of the abstract group $ G(k)$. When $ k$ is a non-Archimedean local field and $ A$ is a finite, abelian $ p$-group where $ p$ is the characteristic of the residue field of $ k$, then with $ G$ of $ k$-rank at least $ 2$, we show that the group $ H^2(G(k),A)$ of abstract central extensions injects into a finite direct sum of $ H^2(H(k),A)$ for certain semisimple $ k$-subgroups $ H$ of smaller $ k$-ranks. On the way, we prove some results which are valid over a general field $ k$; for instance, we prove that the analogue of the Steinberg module for $ G(k)$ has no nonzero $ G(k)$-invariants.

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B. Sury
Affiliation: Stat-Math Unit, Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, India

Keywords: Tits building, $p$-adic algebraic groups
Received by editor(s): November 19, 2009
Received by editor(s) in revised form: June 11, 2010
Published electronically: November 22, 2010
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.