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Representation zeta functions of compact \(p\)-adic analytic groups and arithmetic groups. (English) Zbl 1281.22005

For a group \(G\), let \(r_n(G)\) be the number of isomorphism classes of \(n\)-dimensional irreducible complex representations of \(G\). In this paper, the authors study the representation zeta function \(\zeta_{G}(s):=\sum^{\infty}_{n=1}r_n(G)n^{-s}\) for a rigid group \(G\). One of the reasons to investigate such a zeta function is that the abscissa of convergence \(\alpha(G)\) of \(\zeta_G(s)\) describes the representation growth of \(G\).
The authors first study \(\zeta_{G}(s)\) when \(G\) is a \(p\)-adic analytic pro-\(p\) group which arises from the completions of a global Lie lattice over the ring of integers of a number field; Establishing a method of \(\mathfrak{p}\)-adic integrals generalizing Igusa local zeta functions, they provide a universal formula, obtain a local functional equation and find the finite set of the real parts of possible poles of the zeta function. Furthermore, they obtain explicit formulas when \(G\) is a principal congruence subgroup of \(\mathrm{SL}_3(\mathfrak{o})\) and of \(\mathrm{SU}_3(\mathfrak{O},\mathfrak{o})\). Here \(\mathfrak{o}\) is a compact discrete valuation ring of characteristic zero and \(\mathfrak{O}\) an unramified quadratic extension of \(\mathfrak{o}\).
Next, the authors study \(\zeta_{\Gamma}(s)\) when \(\Gamma\) is an arithmetic subgroup of a connected, simply connected simple algebraic group of type \(A_2\) defined over a number field. For such a group \(\Gamma\), it is known that if \(\Gamma\) has the congruence subgroup property CSP, then \(\zeta_{\Gamma}(s)\) admits an Euler product decomposition. From this fact, combining the above local results together with approximative Clifford theory, they prove that \(\alpha(\Gamma)=1\) when \(\Gamma\) has the CSP. Moreover, as a corollary, they show that Serre’s conjecture, which asserts that an arithmetic subgroup of a connected, simply connected simple algebraic group has the CSP if and only if the latter is of higher-rank, implies Larsen and Lubotzky’s conjecture. Namely, if \(\Gamma_1\) and \(\Gamma_2\) are any two irreducible lattices in a higher-rank semisimple group being products of groups of type \(A_2\), then \(\alpha(\Gamma_1)=\alpha(\Gamma_2)\).

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
20F69 Asymptotic properties of groups
22E40 Discrete subgroups of Lie groups
11M41 Other Dirichlet series and zeta functions
20C15 Ordinary representations and characters
20G25 Linear algebraic groups over local fields and their integers
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References:

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