Scale functions on $p$-adic Lie groups |
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Authors: | Helge Glöckner |
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Institution: | Technische Universit?t Darmstadt, Fachbereich Mathematik, Schlo?gartenstr. 7, D-64289 Darmstadt, Germany. e-mail: gloeckner@mathematik.tu-darmstadt.de, DE
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Abstract: | Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis 14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ
p
, s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ
p
, the scale functions of G and H:=G(K) are related by s
H
∣
G
=s
G
K
:ℚ
p
]. More generally, we clarify the relations between the scale function of
a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients.
Received: 20 February 1997; Revised version: 18 May 1998 |
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Keywords: | Mathematics Subject Classification (1991):22E20 (main) 20G25 22D05 |
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