We study singularity properties of word maps on semisimple algebraic groups and Lie algebras, generalizing the work of Aizenbud-Avni in the case of the commutator map. Given a word w in a free Lie algebra Lr, it induces a word map ϕw: gr → g for every semisimple Lie algebra g. Given two words w1 ∈ Lr1 and w2 ∈ Lr2, we define and study the convolution of the corresponding word maps ϕw1 ∗ ϕw2 := ϕw1 + ϕw2 : gr1+r2 → g. We show that for any word w ∈ Lr of degree d, and any simple Lie algebra g with ϕw(gr) 6= 0, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O(d6) self-convolutions of ϕw. We deduce that a group word map of length ℓ becomes (FRS) at (e, . . ., e) ∈ Gr after O(ℓ6) self-convolutions, for any semisimple algebraic group G. We furthermore bound the dimensions of the jet schemes of the fibers of Lie algebra word maps, and the fibers of group word maps in the case where G = SLn. For the commutator word w0 = [X, Y ], we show that ϕ∗w40 is (FRS) for any semisimple Lie algebra, obtaining applications in representation growth of compact p-adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form Z/pkZ. This allows us to relate them to properties of some natural families of random walks on finite and compact p-adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p-adic probabilistic Waring type problems.
MSC Codes 11P05, 14B05, 14B07, 14G05, 17B45, 20F69, 20G25, 20P05, 60B15 (Primary) 03C98, 11G25, 17B01, 20E18, 20G40, 22E50 (Secondary)
|Original language||English (US)|
|State||Published - Dec 28 2019|
ASJC Scopus subject areas