Kirillov's Orbit Method forp-Groups and Pro-pGroups

Arithmetic Groups, Base Change, and Representation Growth

“…The assumptions then imply that H ⊂ G(O L,q ) is saturable for q / ∈ S (see [17, Theorem A]), in particular good in the sense of Definition 4.5. As saturable pro-p groups of dimension at most p are potent, the assertions follow from [16,Theorem 5.2].…”

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

“…By Corollary , is a saturable subalgebra of . The Hausdorff series shows that for (see [, equation (3)]) whence for all . Thus, and Corollary shows that .…”

“…In particular, if is a finite ‐group of nilpotency class less than , then the Kirillov orbit method establishes a bijection between the ordinary irreducible characters of and the coadjoint orbits of on the dual of its associated Lie ring; cf. and see for applications of such techniques to the enumeration of characters and conjugacy classes.…”

Transporting cohomology in Lazard correspondence

“…under the equivalence class defined as in (6). Constructing this short exact sequence is equivalent to saying that there is a crossed module f : g →g, that is, f is a homomorphism of Lie algebras together with an action ofg over g denoted by η :g → Der(g) such that for g 1 , g 2 ∈ g andg ∈g…”

“…This isomorphism of categories is given by the exponential and logarithm functors. In the context of complex representations, the orbit method of A. Kirillov can be applied to nilpotent pro-p groups of nilpotency class smaller than p. This method establishes a bijection between the complex irreducible representations of the group and the orbits of the coadjoint action of the group in the dual space of its Lie algebra (see [9] and [6]). …”