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.
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Received: 20 February 1997; Revised version: 18 May 1998
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Glöckner, H. Scale functions on $p$-adic Lie groups. manuscripta math. 97, 205–215 (1998). https://doi.org/10.1007/s002290050097
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DOI: https://doi.org/10.1007/s002290050097