# Reference for parabolic root systems

Let $$G$$ be a connected reductive group with maximal split torus $$A_0$$, and $$P = MN$$ a parabolic subgroup with Levi $$M$$ containing $$A_0$$. Let $$A_M$$ be the split component of $$\mathfrak a_M^{\ast} = X(A_M) \otimes \mathbb R$$. Then the linear span of $$\Phi(A_M,G) \subseteq \mathfrak a_M^{\ast}$$ is usually not a genuine root system. But we still have a notion of positive roots, namely $$\Phi(A_M,N)$$, and a set of simple roots $$\Delta_P \subset \Phi(A_M,N)$$.

There is some nice affine geometry one can relate to these sorts of systems. For example, one can consider the hyperplanes $$H_{\alpha} = \{ h \in \mathfrak a_M : \langle h,\alpha \rangle = 0\}$$ for $$\alpha \in \Phi(A_M,G)$$, and the chambers of $$\mathfrak a_M$$ cut out by these hyperplanes are in bijection with the parabolic subgroups of $$G$$ which having $$M$$ as a Levi subgroup.

Is there any general reference for these "parabolic" root systems? It seems one must deal with these systems when learning about the trace formula or Harish Chandra's results about harmonic analysis on reductive groups.

Try section "Relative roots" here: http://math.stanford.edu/~conrad/249BW16Page/handouts/249B_2016.pdf

They originate from the paper of Borel and Tits, I guess: http://www.numdam.org/item/PMIHES_1965__27__55_0/

• Well, usually one considers only the case when P is a minimal parabolic subgroup, in this situation relative roots form indeed a root system, possibly non-reduced. The general case is considered in Stavrova's paper: ams.org/journals/spmj/2009-20-04/S1061-0022-09-01064-4/… section "Relative roots" Another useful reference is Azad, Barry, Seitz "On the structure of parabolic subgroups" (they use term "a shape" insted of "a relative root"). Nov 1, 2018 at 9:50
• Jan 22, 2020 at 17:37