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In this article we describe a long-term departmental change effort in one mathematics department. The change began with one instructor adopting the Extreme Apprenticeship instructional model. This modest shift served as the catalyst for a series of subsequent, systemic improvements. We believe that this innovation and the resultant change demonstrate how instructional change can serve as a catalyst for broader change, rather than a change that focuses solely on instruction. We use four frames from the literature on organizational development to characterize the changes that have occurred in this department. This indepth case study describes the department's current culture and how it developed, and we suggest that this explanation could serve as a guide for other departments seeking change.

We study the representation growth of simple compact Lie groups and of SLn(O), where O is a compact discrete valuation ring, as well as the twist representation growth of GLn(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/κ, where r is the rank and κ the number of positive roots.We then show that the twist zeta function of GLn(O) exists and has the same abscissa of convergence as the zeta function of SLn(O), provided n does not divide char O. We compute the twist zeta function of GL 2 (O) when the residue characteristic p of 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 SL 2 (Fq[[t]]), q even, and deduce that its abscissa lies in the interval [1, 5/2]. that the abscissa of ζ SL2(O) (s) is 1 whenever char O = 0. Our computations of the abscissa ofζ GL2(O) (s) in this case, together with Proposition 3.4, give a new proof of this fact. We also show that the abscissa ofζwith q even. This does not follow from any previously known results and our computation is substantially harder than in the cases where char O = 2.The zeta function of SL 2 (F q [[t]]), q even. In Section 5, we assume that char O = 2, that is, O = F q [[t]] with q even. We give a Clifford theory construction of the representations of SL 2 (F q [[t]]/(t r )) for r even, which is completely explicit apart from the order of certain finite groups V (β, θ) (see Definition 5.7) and certain integers c ∈ {1, 2, 3} (see Lemma 4.21). We use this construction to approximate the zeta function ζ SL 2 (O) (s), and show that its abscissa lies between 1 and 5/2 (Theorem 5.9). The lower bound 1 follows from a general result of Larsen and Lubotzky [21, Proposition 6.6], but we give an independent proof of this.Section 6 is devoted to a proof of Lemma 4.21, which is crucial for our results aboutζ GL 2 (Fq[[t]]) (s) and ζ SL 2 (Fq[[t]]) (s) when q is even. The lemma gives the number of solutions, up to a factor c ∈ {1, 2, 3}, in F q [[t]]/(t i ), i ≥ 1, to the equation∆ τ ] mod (t) is a scalar plus a regular nilpotent matrix. The number of solutions depends in a delicate way on a new invariant, which we call the odd depth, of the twist orbit (i.e., orbit modulo scalars) of the matrix [ 0 1∆ τ ] (see Definition 4.19). Remark. After the present paper had been accepted for publication, Hassain M and Pooja Singla [24] announced results about the representations of SL 2 (O), p = 2, which in particular imply that the abscissa of ζ SL2(Fq[[t]]) (s) is 1. Notation.We let N stand for the set of natural numbers, not including 0.In Sections 4 and 5, we will use the Vinogradov notation f (r) ≪ g(r) for two functions f (r), g(r) of r (or of l = ⌈r/2⌉). Note that f (r) ≪ g(r) is equiv...