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Euler characteristics for \(p\)-adic Lie groups. (English) Zbl 0971.22011
Let \(G\) be a compact \(p\)-adic Lie group with no \(p\)-torsion, and let \(M\) be a finitely generated \(\mathbb Z_p\)-module on which \(G\) acts. Assuming that the homology group \(H_i (G,M)\) is finite for each \(i\), the associated Euler characteristic is given by \[ \chi (G,M) = \sum_i (-1)^i \text{ord}_p (H_i (G,M)) . \] In this paper the author proves that the Euler characteristics \(\chi (G,M)\) are the same for all sufficiently small open subgroups \(G_0\) of \(G\) and that the common value of these Euler characteristics is zero if every element of the Lie algebra \(\mathfrak g_{\mathbb Q_p}\) of \(G\) has centralizer of dimension at least two. When the centralizer of \(\mathfrak g_{\mathbb Q_p}\) has dimension one, he gives an explicit formula for the common value, which shows in particular that this common value is not zero for some choice of the module \(M\).

22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
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