Base change maps for unipotent algebra groups

TL;DR

This paper constructs canonical base change maps for irreducible representations of unipotent algebra groups over finite fields, advancing geometric character theory and representation classification.

Contribution

It introduces a new method to define base change maps for algebra groups' representations, compatible with Galois actions, enhancing understanding of their structure over field extensions.

Findings

Constructed canonical injective base change maps between representation sets.

Maps commute with Galois group actions, ensuring compatibility with field extensions.

Provides a framework for studying representations of algebra groups via geometric methods.

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.