This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras).

Let's say $G$ is a **simple** algebraic group of rank $r$ over $\mathbb{C}$ and $\lambda$ a dominant integral weight. Consider the irreducible finite dimensional representation $V_\lambda$ with highest weight $\lambda$ and let $\mu$ be a weight occuring in $V_\lambda$. Let $W$ be the Wely group of $G$.

Suppose that **$\lambda$ is regular** (lies in interior of Weyl chamber), **$\mu$ is dominant**, and **$\lambda-\mu$ lies in the interior of the positive root cone** (the "wide cone"), then it seems to be true that

$m_{\lambda\mu}\ge 2^{r-1}$

Here $m_{\lambda\mu}$ is the dimension of $\mu$ weight space in $V_\lambda$.

**Question**: Does anyone know an elementary or straightforward proof (or maybe counterexample) of this lower bound?
For example using some multiplicity formula or combinatorial models.

I came to this possible lower bound in a very indirect way when studying geometry of certain affine Springer fibers. But I guess there should be a straightforward way to see this, for example by some weight multiplicity formula, which I'm not quite familiar with. It would also be great if this could be seen by using some combinatorial models of algebraic representations (crystals, path models, MV polytopes etc). But since I'm not familiar with all these, any comments or suggestions are welcome.

**Remarks on the number $2^{r-1}$**:

Recall $G$ is simple with rank $r$. Fix a set of simple reflections $s_1,...,s_r$ in the Weyl group $W$, then $2^{r-1}$ is the number of elements in $W$ that can be written as products of these $r$ simple reflections, each occurring precisely once. Previously I have abused terminology and called this the "number of Coxeter elements", which lead to some confusion. As far as I know, this abuse of terminology is also present in literature, so it's better to pay attention to the context when seeing these words. Thanks to Jim Humphreys for clarifying this in his answer below. Simply speaking, $2^{r-1}$ only counts those Coxeter elements that can be expressed as products of a *given* set of simple reflections, which is in general smaller than the number of all Coxeter elements. For $W=S_n$, we have $2^{r-1}=2^{n-2}$ while the Coxeter elements in $S_n$ are the $n$-cycles, so there are $(n-1)!$ of them in total.

**Update** (Nov 1st, 2017): The very indirect way I found this inequality is in here. See Corollary 4.5.2.

isthe number of Coxeter elements. A choice of Coxeter element requires precisely that you specify, for each pair of non-commuting simple reflections, which one comes first. (The order of commuting reflections doesn't matter since they commute.) There are $r-1$ such pairs, and since the edges of the Dynkin diagram form a tree, any choice of orders is obviously realizable. Eg for A_3, the four choices are $s_1s_2s_3$, $s_3s_2s_1$, $s_2s_1s_3=s_2s_3s_1$, $s_1s_3s_2=s_3s_1s_2$. In the last of these, for example, we chose $s_2$ to come after $s_1$ and $s_3$. $\endgroup$ – Hugh Thomas Jul 19 '17 at 2:54