Arithmetic Groups, Base Change, and Representation Growth

作者：Avni Nir^*; Klopsch Benjamin^*; Onn Uri^*; Voll Christopher^*

来源：Geometric and Functional Analysis, 2016, 26(1): 67-135.

DOI：10.1007/s00039-016-0359-6

Consider an arithmetic group , where is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of S-integers O (S) of a number field K with respect to a finite set of places S. For each , let denote the number of irreducible complex representations of of dimension at most n. The degree of representation growth is finite if and only if has the weak Congruence Subgroup Property. We establish that for every with the weak Congruence Subgroup Property the invariant is already determined by the absolute root system of . To show this we demonstrate that the abscissae of convergence of the representation zeta functions of such groups are invariant under base extensions . We deduce from our result a variant of a conjecture of Larsen and Lubotzky regarding the representation growth of irreducible lattices in higher rank semi-simple groups. In particular, this reduces Larsen and Lubotzky's conjecture to Serre's conjecture on the weak Congruence Subgroup Property, which it refines.

• 出版日期2016-2
• 单位西北大学