# Linear representation of general affine group over a field of characteristic zero must send translations to unipotent matrices

## Statement

Suppose $K$ is a field of characteristic zero and $L$ is any field. Suppose that $m,n$ are natural numbers, and $\rho:GA(m,K) \to GL(n,L)$ is a linear representation of the general affine group (of degree $m$ over $K$) as a $n$-dimensional representation over $L$. Then, the image under $\rho$ of any translation (i.e., any element in the base normal subgroup $K^m$) is a unipotent matrix, i.e., all its eigenvalues are equal to 1.

## Proof

### Proof idea

The proof idea is to show that the image is conjugate to every nonzero power of itself, and hence, that its multiset of eigenvalues (with multiplicities) is invariant under the operation of raising each to the same integer power. We next show that the only multiset with that property is the multiset with all entries equal to 1.