Representation growth of compact special linear groups of degree two

• Published in: Advances in mathematics (New York. 1965) Vol. 396; p. 108164
• Main Authors: Hassain, M., Singla, Pooja
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 12-02-2022
• Subjects: Compact special linear groups of degree two; Group algebras; Odd level representations of [formula omitted]; Representation growth; Odd level representations of SL2(o); Group algebras; 20H05; Compact special linear groups of degree two; 20G05; 20G25; 20F69; 20C15; Representation growth
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2021.108164

We study the finite-dimensional continuous complex representations of SL2 over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of convergence of the representation zeta function is 1, resolving the last remaining open case of this problem. We additionally prove that, contrary to the expectation, the group algebras of C[SL2(Z/(22rZ))] and C[SL2(F2[t]/(t2r))] are not isomorphic for any r>1. This is the first known class of reductive groups over finite rings wherein the representation theory in the equal and mixed characteristic settings is genuinely different. From our methods, we explicitly obtain the primitive representation zeta polynomials of SL2(F2[t]/(t2r)) and SL2(Z/22rZ) for 1≤r≤3.