The representation zeta function of a FAb compact \(p\)-adic Lie group vanishes at \(-2\). (English) Zbl 1292.22006
Let \(G\) be a finitely generated profinite group with the FAb property (that is \(H/[H,H]\) is finite for every open subgroup \(H\) of \(G\)). Denote by \(\text{Irr}(G)\) the set of characters \(\chi\) of its complex irreducible representations. The representation zeta function is defined as
\[
\zeta_G(s)=\sum\limits_{\chi \in \text{Irr}(G)} \chi (1)^{-s}.
\]
The authors prove for a certain class of groups including all infinite FAb compact \(p\)-adic Lie groups (\(p\geq 3\)) that \(\zeta_G(-2)=0\). This property contrasts the one known for a finite group \(G\) where \(\zeta_G(-2)=|G|\).
Reviewer: Anatoly N. Kochubei (Kyïv)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11M41 | Other Dirichlet series and zeta functions |
| 20E18 | Limits, profinite groups |
| 20C15 | Ordinary representations and characters |
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