\subsection{Avoidance of Non-Extremal Subspaces}
\label{section: avoidance}

	The purpose of this section is to prove a useful property of the orbit of a highest weight vector under a semisimple group.
    This property will allow us to obtain precise expansion rates in the situations we are interested in.

	Suppose $G$ is a semisimple Lie group with Lie algebra $\mf{g}$, and $\mbf{S}$ is a maximal split torus in $G$ which we also identify with its Lie algebra.
    Denote by $\Delta \subset \mbf{S}^\ast$ the set of roots on which we fix an order and denote by $\Delta^+$ the subset of positive roots.
    Define the following subalgebras of $\mf{g}$
    \[ \mf{n}^+ = \bigoplus_{\a \in \Delta^+} \mf{g}_\a ,\qquad \mf{b} = \mf{g}_0 \oplus \mf{n}^+ \]
    where $\mf{g}_\a$ denotes the root space corresponding to $\a$.
    Denote by $N^+$ and $B$ the subgroups of $G$ whose Lie algebras are $\mf{n}^+$ and $\mf{b}$ respectively.
    
    
    We let $\mc{W}$ denote the Weyl group of $(G,\mbf{S},\Delta)$ and recall that $\mc{W}$ acts naturally on $\mbf{S}^\ast$.
    The Bruhat decomposition of $G$~\cite[Section 3, Theorem 1]{Bourbaki-4-6} implies
    \begin{equation} \label{eqn: general Bruhat}
    G = \bigcup_{\mrm{w} \in \mc{W}} B \mrm{w} B
    \end{equation}
    
    
    Given a representation $V$ of $G$ and a linear functional $\mu\in \mbf{S}^\ast$, we denote by $V^\mu$ the weight subspace of $V$ with weight $\mu$.
    
    \begin{proposition} \label{propn: avoidance}
    Suppose $V$ is an irreducible representation of $G$ with highest weight $\l$. 
    Then, for all $0\neq v \in  V^{ \l} $,
    \begin{equation*}
    G\cdot v \quad \bigcap \bigoplus_{ \mu \in \mbf{S}^\ast\backslash \mc{W}\cdot\l  } V^\mu = \emptyset
    \end{equation*}
    \end{proposition}
    
	\begin{proof}
	Let $0\neq v \in  V^{ \l} $ and $g\in G$.
    Denote by $\pi :V \r \bigoplus_{\mrm{w} \in \mc{W} } V^{\mrm{w} \cdot \l}$ the projection parallel to the weight spaces of $\mbf{S}$.
    It suffices to show that $\pi(gv) \neq 0$.
    
    
    Using the Bruhat decomposition~\eqref{eqn: general Bruhat}, we can write
    \[ g = b_1 \mrm{w} b_2  \]
    for some $b_1,b_2 \in B$ and $\mrm{w}\in \mc{W}$.
    The group $B$ stabilizes the line $\R\cdot v$.
    In particular, we have that $gv \in b_1 \mrm{w} V^{\l} \subseteq b_1 V^{\mrm{w}\cdot \l} $.
    
    We can further decompose $b_1$ as follows.
    \[ b_1 = n^+m \]
    where $n^+\in N^+$ and $m \in C_G(\mbf{S})$ commutes with $\mbf{S}$.
    In particular, $m$ preserves the eigenspaces of $\mbf{S}$ and thus we have 
    \begin{equation}\label{eqn: orbit hits extremal weight}
     gv \in b_1 V^{\mrm{w}\cdot \l} = n^+ V^{\mrm{w}\cdot \l} 
    \end{equation} 

    
    Let $Y \in \mf{n}^+$ be such that $n^+ = \exp(Y)$.
    Denote by $\rho : G \r \mrm{GL}(V)$ the representation of $G$ on $V$ and let $d\rho: \mf{g} \r \mf{gl}(V)$ denote its derivative.
    Then, since $Y$ is nilpotent, so is $d\rho(Y)$.
    In particular, $\rho(n^+) = \exp(d\rho(Y))$ is a polynomial in $d\rho(Y)$ of the form
        \begin{equation} \label{eqn: n+ is nilpotent}
        	\rho(n^+) = \mrm{I} + d\rho(Y) + \cdots + \frac{d\rho(Y)^k}{k!}
        \end{equation}
        for some $k\in \N$, where $\mrm{I}$ is the identity map. From the standard representation theory of semisimple Lie groups, we have
        \begin{equation*}
        	d\rho(\mf{g}_\a) V^\mu \subseteq V^{\a + \mu}
        \end{equation*}
        for any root $\a \in \Delta$ and any weight $\mu \in \mbf{S}^\ast$.
        Thus, for each $1\leq j \leq k$, we see that
        \[ d\rho(Y)^j V^{\mrm{w}\cdot \l} \subseteq V^{\k}, \qquad \k = \mrm{w}\cdot \l + \sum_{\a \in \Delta^+} k_\a \a \]
        for some non-negative integers $k_\a$, at least one of which is non-zero, and, in particular, $V^\k \cap V^{\mrm{w}\cdot \l} = \set{0}$.
        Hence, in view of~\eqref{eqn: n+ is nilpotent}, for all $w\in V^{\mrm{w}\cdot \l}$,  we have
        \[\pi^{\mrm{w}\cdot \l}( \rho(n^+)w) = w,\]
        where $\pi^{\mrm{w}\cdot \l}:V \r V^{\mrm{w}\cdot \l} $ denotes the projection parallel to the eigenspaces of $\mbf{S}$.
        Combined with~\eqref{eqn: orbit hits extremal weight}, this shows that $\pi(gv) \neq 0$ as desired.
	\end{proof}
{"config": "arxiv", "file": "1806.06832/LinearExpansion/Avoidance.tex"}