We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes– Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with t...

Consider the random walk Gn:=gn…g1, n?1, where (gn)n?1 is a sequence of independent and identically distributed elements with law ? on general linear group GL(V) V=Rd. Under suitable conditions ?, we establish Cramér type moderate deviation expansions local limit theorems deviations for coefficients ?f,Gnv?, v?V f?V?. Our approach based Hölder regularity invariant measure Markov chain Gn?x=RGnv...