Representation growth of compact linear groups

Abstract:

We study the representation growth of simple compact Lie groups and of $\operatorname {SL}_n(\mathcal {O})$, where $\mathcal {O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\operatorname {GL}_n(\mathcal {O})$. This amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions.

We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is $r/\kappa$, where $r$ is the rank and $\kappa$ the number of positive roots.

We then show that the twist zeta function of $\operatorname {GL}_n(\mathcal {O})$ exists and has the same abscissa of convergence as the zeta function of $\operatorname {SL}_n(\mathcal {O})$, provided $n$ does not divide $\operatorname {char}{\mathcal {O}}$. We compute the twist zeta function of $\operatorname {GL}_2(\mathcal {O})$ when the residue characteristic $p$ of $\mathcal {O}$ is odd and approximate the zeta function when $p=2$ to deduce that the abscissa is $1$. Finally, we construct a large part of the representations of $\operatorname {SL}_2(\mathbb {F}_q[[t]])$, $q$ even, and deduce that its abscissa lies in the interval $[1, 5/2]$.