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We have the following set up: $K/F$ is Galois, $D$ is an algebraic group of mult. type and $E$ is an extension of groups: $$1\to D(K)\to E\to Gal(K/F)\to 1$$ Now take a linear algebraic group G over F. An algebraic 1-cocycle is a map $w \mapsto x_w$ from $E$ to $G(K)$ s.t. $x_{w_1 w_2}= x_{w_1} w_1(x_{w_2})$, where $E$ acts on $G(K)$ via $E\to Gal(K/F)$.

Furthermore we have a Galois action on morphisms $v:D(K)\to G(K)$ via $\sigma(v)(d)=\sigma(v(\sigma^{-1}(d))$ (cf here section 2.2/2.3 for more details).

In the above paper it is claimed then that for $w\in E$ which maps to $\sigma$ $$x_w\cdot\sigma(v)\cdot (x_w)^{-1}=v$$ as a cosequence of the cocycle condition. Here $v$ is the restriction of the cocycle to $D$ (i.e. $v(d)=x_d$).

But my calculation only gives me: $$(x_w\cdot\sigma(v)\cdot (x_w)^{-1})(d)=x_w\cdot\sigma(x_{\sigma^{-1}(d)})\cdot (x_w)^{-1}=x_{w\sigma^{-1}(d)}\cdot \sigma( x_{w^{-1}}) =x_{w\sigma^{-1}(d)w^{-1}}$$ and I am not able to conclude the desired result.

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It seems to me that Kottwitz's review of the notion of Galois gerb used by Langlands and Rapoport was not meant to be competely self-contained, and that it was meant only to highlight differences of a technical nature. In particular---even though he does not review this point---for any Galois gerb $E$, the induced conjugation action of $G(K/F)$ on $D(K)$ is required to coincide with the standard Galois action. In other words, given an element $w$ in $E$ that maps to $\sigma$ in $G(K/F)$, we require that$$wdw^{-1} = \sigma(d)$$for all $d$ in $D(K)$. Here the left-hand side is conjugation in the group $E$, while the right-hand side is the standard Galois action of $\sigma$ on $D(K)$. The requirement that this equality hold is part of the definition of Galois gerb.

For this reason the subscript$$w\sigma^{-1}(d)w^{-1}$$occurring at the far right end of your calculation simplifies to $d$.

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  • $\begingroup$ Yes, I already assumed these two actions to coincide, but I was not sure if this was just per definition. $\endgroup$ – Notone Oct 31 '18 at 19:49

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