# Ext and representations with fixed central characters

In this paper (http://arxiv.org/pdf/1108.3668v2.pdf) Adler and Prasad compute certain Ext groups. On page 2 they write,

"Since extensions of representations of abelian groups are well understood through the cohomology $H^i(Z^n,C)$ of $Z^n$, it is no loss of generality when considering extensions Ext$^i(π_1, π_2)$ to restrict oneself to the subcategory $R_χ(G)$ of the category $R(G)$ of all smooth representations of G, consisting of those representations on which the center of $G$ acts via a given character $χ$, which we can also assume to be unitary."

Could someone please explain this sentence? For example, in another paper Vigneras worked with the subcategory of smooth representations with fixed central character. Could that be extended easily to the category of smooth representations?