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Let $G$ be a discrete group and let $V$ be a vector space over a field of characteristic $0$ upon which $G$ acts linearly. I'm looking for the right terminology for the following situation: for all $g \in G$, there exists a linear map $n_g\colon V \rightarrow V$ that is nilpotent in the sense that some finite power of $n_g$ is $0$ such that $g$ acts on $V$ as ${\operatorname{id}} + n_g$. It is clear that this should be called a unipotent representation, and if $V$ is finite-dimensional then all unipotent representations are of this form. But is it right to use this as the definition of a unipotent representation if $V$ is infinite-dimensional? There are other things that should also probably be called unipotent (e.g. if $n_g$ is only required to be "locally nilpotent" in the sense that for all $v \in V$, there is some power of $n_g$ that takes $v$ to $0$). What term should I use?

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    $\begingroup$ Since the terminology is not clearly wrong but is confusing, I think that the simplest answer is: call it unipotent as you suggest, but emphasise explicitly that you mean $n_g$ to be "uniformly, not just locally, nilpotent". (Although that might suggest uniformity in $g$, so I would say to separate it: first define what it means for a single operator to be "uniformly unipotent" or whatever, then require that your $G$ act by such operators.) $\endgroup$
    – LSpice
    Commented Aug 15, 2021 at 19:07
  • $\begingroup$ I think that the term in the title, namely ``unipotent-like representation'', is better. $\endgroup$
    – user148212
    Commented Aug 16, 2021 at 0:43
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    $\begingroup$ @user148212: The problem with the phrase "unipotent-like" is that it suggests that these are not unipotent, but only share some features with unipotent representations. $\endgroup$
    – Linda
    Commented Aug 16, 2021 at 2:52
  • $\begingroup$ @LSpice: That will be what I do if I don't find a more standard term. $\endgroup$
    – Linda
    Commented Aug 16, 2021 at 2:52
  • $\begingroup$ If you can believe Wikipedia in en.wikipedia.org/wiki/Unipotent they call an element r of a ring unipotent if r-1 is nilpotent. So by their convention you would be fine. But I don't know how standard that is. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 16, 2021 at 20:24

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