[Paper Review] Base change maps for unipotent algebra groups

Mitya Boyarchenko|ArXiv.org|Jan 7, 2006

Advanced Topics in Algebra6 references5 citations

TL;DR

This paper constructs canonical injective base change maps for irreducible representations of unipotent algebra groups over finite fields, using the theory of strongly Heisenberg representations and norm maps on K₁-groups. The key contribution is proving the existence of these injective maps, which generalize known constructions in orbit method settings and provide a new framework for geometric character theory in the absence of explicit representation classifications.

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.

Motivation & Objective

• To address the foundational problem of constructing base change maps for irreducible representations of unipotent algebra groups over finite fields.
• To establish the existence of canonical, injective base change maps for finite algebra groups, particularly in cases where explicit representation classifications are unavailable.
• To extend the theory of base change beyond orbit method settings by introducing strongly Heisenberg representations as a key technical tool.
• To investigate the surjectivity and Galois-equivariance of base change maps in the absence of a geometric object parametrizing representations.

Proposed method

• Introduces the notion of a strongly Heisenberg representation as a generalization of Heisenberg representations, defined via central actions and compatible with base change.
• Applies a reduction process that decomposes arbitrary irreducible representations into components supported on algebra subgroups, leveraging the structure of the group algebra and its nilpotent ideals.
• Constructs norm maps on K₁-groups of finite extensions of the base field to define base change maps for 1-dimensional representations.
• Uses Lang’s theorem and Galois descent to lift invariance properties of representations under Frobenius actions to higher extensions.
• Employs Mackey’s criterion and Frobenius reciprocity to relate induced representations and prove injectivity and surjectivity results.
• Establishes a direct limit system of representation sets equipped with Galois actions, aiming to realize the limit as a geometric object over the base field.

Experimental results

Research questions

• RQ1 Do canonical, Galois-equivariant base change maps exist for irreducible representations of unipotent algebra groups over finite fields?
• RQ2 Are these base change maps injective, and under what conditions are they surjective?
• RQ3 Can the direct limit of representations under base change maps be realized as the set of geometric points of a geometric object defined over the base field?
• RQ4 How does the theory of strongly Heisenberg representations facilitate the construction of base change maps in the absence of orbit method classifications?

Key findings

• The paper constructs canonical injective base change maps for irreducible representations of finite algebra groups, proving their existence and Galois-equivariance.
• The base change maps for 1-dimensional representations are shown to be surjective under certain conditions, which implies surjectivity for general representations via induction.
• The injectivity of base change maps is established via a reduction process that decomposes representations into components on algebra subgroups and uses Frobenius invariance.
• The construction relies on K₁-theory and norm maps, providing a new algebraic-geometric framework independent of the orbit method.
• The paper proves that if base change maps for 1-dimensional representations are surjective, then so are those for arbitrary irreducible representations, under the given reduction framework.
• A corollary establishes that any Galois-invariant representation on an algebra subgroup has a Galois-invariant irreducible summand, supporting the consistency of the construction.