# Lang Steinberg over separably closed field

Let $$K=K^{sep}$$ be a separably closed field with $$K|\mathbb{F}_q$$, where $$\mathbb{F}_q$$ is the field with $$q$$ elements. Let $$\mathbb{G}$$ be a connected linear algebraic group over $$\mathbb{F}_q$$. Lastly, let $$B\in\mathbb{G}(K)$$ be a $$K$$-valued point of $$\mathbb{G}$$. Define the Lang map relative to B $$f_B:\mathbb{G}(K)\rightarrow\mathbb{G}(K), A\mapsto A^{-1}\cdot B\cdot\mathbb{G}(Frob_q)(A),$$ where $$Frob_q:K\rightarrow K, x\mapsto x^q$$. My question is: Is $$f_1$$ surjective?

I guess, I figured out a way to deduce this from the theorem of Lang-Steinberg: Let $$K^{alg}|K$$ be an algebraic closure. The map $$f_{1,alg}:\mathbb{G}(K^{alg})\rightarrow\mathbb{G}(K^{alg}), A\mapsto A^{-1}\cdot\mathbb{G}(Frob_q)(A)$$ is surjective by [Steinberg, Endomorphisms of Linear algebraic groups Theorem 10.1]. On the other hand, by [https://ivv5hpp.uni-muenster.de/u/pschnei/publ/lectnotes/Theorie-des-Anstiegs.pdf, Satz 2.1 (german lecture notes)] it follows that $$\tilde{f}_1:GL_n(K)\rightarrow GL_n(K), A\mapsto A^{-1}\cdot GL_n(Frob_q)(A)$$ is surjective. Since $$\mathbb{G}$$ is a linear algebraic group, we can fix an embedding $$\mathbb{G}\subset GL_n$$. Now let $$A_0\in\mathbb{G}(K)$$ be arbitrary. Since $$f_{1,alg}$$ is surjective, we find $$B\in\mathbb{G}(K^{alg})$$ with $$f_1(B)=A_0$$. Since $$\tilde{f}_1$$ is surjective, there also exists $$\tilde{B}\in GL_n(K)$$, such that $$\tilde{f}_1(\tilde{B})=A_0$$. Let $$\tilde{f}_{1,alg}:GL_n(K^{alg})\rightarrow GL_n(K^{alg}), A\mapsto A^{-1}\cdot GL_n(Frob_q)(A).$$ One calculates that $$B\in f_{1,alg}^{-1}(\{A_0\})\subset\tilde{f}_{1,alg}^{-1}(\{A_0\})=GL_n(\mathbb{F}_q)\cdot \tilde{B}\subset GL_n(K).$$ So $$B\in\mathbb{G}(K^{alg})\cap GL_n(K)=\mathbb{G}(K)$$, hence $$f_1$$ is surjective.
• I guess with this method you can prove the following for every integral domain $A$, which is a $\mathbb{F}_q$-algebra: the Lang map is surjective on the $A$-valued points for any connected linear algebra group over $\mathbb{F}_q$, if it is surjective on the $A$-valued points for any $GL_n$. – Estus Feb 18 at 18:15