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Betti numbers of congruence groups. (Appendix: On representations of compact \(p\)-adic groups by Ze’ev Rudnick). (English) Zbl 0843.11027
In general very little is known on automorphic multiplicities. Up to now there are mainly asymptotic assertions on varying the representation or the group. This remarkable paper contains a new point of view. Varying the group it is given an assertion on the shape of the function (group \(\mapsto\) automorphic multiplicity).
More precisely let \(G\) be a real or \(p\)-adic semisimple Lie group and \(\Gamma \subset G\) a cocompact lattice. The right regular representation of \(G\) on \(L^2 (\Gamma \setminus G)\) decomposes into irreducibles with finite multiplicities \(m_\Gamma (\pi)\). A representation \(\pi\) is called cohomological if its isotype in \(L^2 (\Gamma \setminus G)\) contributes to the cohomology of \(\Gamma\). Let \(\Gamma (N)\) be a family of congruence subgroups and \(H_N\) the finite group \(\Gamma/ \Gamma (N)\). Then \(H_N\) acts on the isotypes \(L^2 (\Gamma (N) \setminus G) (\pi)\) and has finite multiplicities \(m_{H_N} (\rho, \pi)\), \(\rho\in \widehat {H}_N\) there. For \(d\geq 1\) define the \(d\)-dimensional part of \(m_\Gamma (\pi)\) by \[ {}^d m_{\Gamma (N)} (\pi):= \sum _{\substack{ \rho\in \widehat {H}_N\\ \dim\rho =d }} m_{H_N} (\rho, \pi). \] Now call a sequence \(a_n\) polynomial periodic sequences if there are periodic \(b_j (n)\), \(j= 1, \dots, q\) such that \(a_n= \sum^q_{j=0} b_j (n) n^j\). The main theorem of the paper asserts that if \(G\) is \(p\)-adic then for fixed \(d\) and \(\pi\) the sequence \(N\mapsto^d m_{\Gamma (N)} (\pi)\) is polynomial periodic. The same holds for real \(G\) if \(\pi\) is cohomological. The proof uses a result of A. Lubotsky and A. Magid [Varieties of representations of finitely generated groups, Mem. Am. Math. Soc. 336 (1985; Zbl 0598.14042)] stating that all irreducible representations of \(\Gamma\) of dimension \(d\) factor through a fixed quotient \(\Delta\) which is abelian by finite. The representations of \(\Delta\) can be parametrized by the tori of characters of finite index subgroups. The representations with finite image correspond to torsion points in those tori. The representations corresponding to an isotype \(\pi\) make up algebraic sets in these tori and the torsion points in them can be found in a union of subgroups. This finally gives the claim.

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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