Base change maps for unipotent algebra groups

Abstract

If A is a finite dimensional nilpotent associative algebra over a finite field k, the set G=1+A of all formal expressions of the form 1+a, where a is an element of A, has a natural group structure, given by (1+a)(1+b)=1+(a+b+ab). A finite group arising in this way is called an algebra group. One can also consider G as a unipotent algebraic group over k. We study representations of G from the point of view of ``geometric character theory'' for algebraic groups over finite fields (cf. G. Lusztig, ``Character sheaves and generalizations'', math.RT/0309134). The main result of this paper is a construction of canonical injective ``base change maps'' between - the set of isomorphism classes of complex irreducible representations of G', and - the set of isomorphism classes of complex irreducible representations of G'', which commute with the natural action of the Galois group Gal(k''/k), where k' is a finite extension of k and k'' is a finite extension of k', and G', G'' are the finite algebra groups obtained from G by extension of scalars.