I was beginning to read Bruhat and Tits article Groupes Reductifs sur un Corps Locale and was confused on a point in the beginning of the section on affine root systems.

$\mathbf{A}$ is a finite dimensional real affine space with corresponding vector space $^v\mathbf{A}$, and we are given a locally finite collection $\mathfrak H$ of hyperplanes in $\mathbf{A}$. The choice of a scalar product on $^v \mathbf{A}$ gives us a notion of orthogonality in $\mathbf{A}$. About each hyperplane $L$, we then have an orthogonal reflection $s_L$ in $\mathbf{A}$ about $L$. The group $W$ generated by all the $s_L$, with the discrete topology, is assumed to act properly on $\mathbf{A}$, and $\mathfrak H$ is assumed to be closed under the action of $W$.

enter image description here

In this section, they define an affine root to be a closed half space bounded by a hyperplane. If $\alpha$ is an affine root, $\alpha^{\ast}$ is apparently the other closed half space of the same wall.

Now $\alpha_+$ is defined to be the "intersection of all affine roots containing a neighborhood of $\alpha$." They then say that $\alpha_+$ is an affine root which is equipollent to $\alpha$, which means that $\alpha = \alpha_+ + v$ for some $v \in ^v \mathbf{A}$.

I don't understand what this definition of $\alpha_+$ is saying. Why should $\alpha_+$ be anything other than $\alpha$ itself? If $\alpha_+$ is different from $\alpha$, why should it also be an affine root according to the above definition, and why should it be equipollent to $\alpha$? How does this relate to the other definition of affine roots, for example in Bourbaki Chapter VI of Lie Groups and Lie Algebras?


1 Answer 1


The intersection is only taken over halfspaces whose boundary is in $\mathfrak H$. So $\alpha_+$ is the smallest affine root which strictly contains $\alpha$. For example, if $\mathbf A=\mathbb R$ and $\mathfrak H=\mathbb Z$ then $\{\pm x+n\ge 0\}^+=\{\pm x+n+1\ge 0\}$.

The difference between affine roots in this sense and roots in the ordinary sense is that the former have only a direction but no length. So, Bruhat-Tits could have defined an affine root also as a ray $\mathbb R_{\ge0}f$ where $f$ is an affine linear function whose zero set $\{f=0\}$ appears in $\mathfrak H$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.