I am not sure if this is the right site for this question, but it seems "too simple" a question to post on MO, so I chose to put it here.

In 1.1.13 of Milne's translation of Deligne's paper titled Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, which can be found here, he states (here $G$ is an algebraic group, say over $\mathbb{R}$):

...for some faithful family $V_i$ of representations of $G$...

My question is: what is a faithful family of representations? Does it mean that each $V_i$ is a faithful representation of $G$, or does is mean that $G\rightarrow\prod\operatorname{GL}_{V_i}$ is injective.

I believe it is the latter because the former would be a family of faithful representations and not a faithful family of representations. Or am I just being too pedantic?

Thanks for any help.


1 Answer 1


The definition is as follows:

Definition: Let $\mathcal{F}$ be a set of representations of $G$. We say that the family $\mathcal{F}$ is faithful if the direct sum representation $ \rho:= \bigoplus_{\phi\in \mathcal{F}}\phi $ is injective.

Reference: Here, section $2$.

  • 2
    $\begingroup$ Yes, indeed: if we were optimistic, the modifier "faithful" seems to modify "family". So it's not "family of faithful repns"... $\endgroup$ Apr 2 at 19:23
  • $\begingroup$ Great! Thanks a lot! $\endgroup$ Apr 3 at 6:16

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