Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A₂

Abstract

Abstract. We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various p-adic analytic and adèlic profinite groups of type A2. This has consequences for the representation zeta functions of arithmetic groups Γ ⊂ H(k), where k is a number field and H a k-form of SL3: assuming that Γ possesses the strong Congruence Subgroup Property, we obtain precise, uniform estimates for the representation growth of Γ. Our results are based on explicit, uniform formulae for the representation zeta functions of the p-adic analytic groups SL3(o) and SU3(o), where o is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral p-adic 3 × 3 matrices in gl 3 (o) and gu 3 (o), where o is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions.Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form SL3(o), SU3(o), GL3(o), and GU3(o), arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of o is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe p-adic analogues of 'Ennola duality'.

The average size of the kernel of a matrix and orbits of linear groups

“…We note that the fact that the preceding examples as well as the formula in [, (1.12)] all satisfy functional equations under the operation ‘’ does not seem to be explained by any of the results in the present article (for example, Theorem ).…”

“…In the latter case, this reflects the mysterious nature of the model theory of local fields of (small) positive characteristic. (ii)Avni et al . [, Theorems E and A.5] gave formulae for the ‘similarity class zeta functions’ associated with the groups and for . In addition to being consistent with Theorem , their formulae are valid for all local fields subject to the sole assumption that be odd in case of .…”

“…In particular, Avni et al . [, Theorems E and A.5] determined orbit‐counting zeta functions associated with the coadjoint representation of and group schemes of the form for .…”

Regular characters of classical groups over complete discrete valuation rings

“…Theorem 3.1.2. (2) implies that x 1 is regular and hence C γ1 (x 1 )(k alg ) = C g(k alg ) (x 1 ) is a k alg -vector space of dimension n = dim C Γ1 (x 1 ). By Lemma 3.1.7, the k alg -points of the image of C γr (x r ) under η * r,1 comprise a subspace of C g(k alg ) (x 1 ) of the same dimension.…”

Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: The example of GL(2, 𝔽q) and PGL(2, 𝔽q)

“…In some very few cases one can even compute exactly ζ G , this is the case for SL(2, O) [14] where O is the ring of integers of a p-adic field and more recently for SL (3, O) [3].…”