(I did not find a solution of my problem in any forum so far. Sorry if it exists...)
I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only to apply the Weyl-Charakter-Formula on $so(5)$ ($B_2$-Type, $\text{rank }(so(5))=2$). In Fulton and Harris 'Representation Theory' they give on p.400 what I think is the same as https://en.wikipedia.org/wiki/Weyl_character_formula: \begin{align} \text{ch}_{\lambda} = \frac{\sum_{w\in\mathscr{W}} (-1)^{\ell(w)}e^{w(\lambda+\rho)}}{e^\rho \prod_{\alpha\in \Phi^+}(1-e^{-\alpha})} \end{align} So far I know/think the following:
$\Phi^+$ is the set of positive roots; I choose the simple roots to be $(-1,1)$ and $(1,0)$, so that the Weyl-Vector $\rho$ is $(0,1/2)$. The Weyl-Group $\mathscr{W}$ is isomorphic to the dihedral group of order $8$. $\ell(w)$ is the lenght of $w$ defined as the minimal number of secessary reflexions of simples roots.
Applying this on $su(2)$ was no problem, but I don't understand the meaning of $e^{\rho}$ and $e^{w(\lambda+\rho)}$. I was told once, that this is defined by $e^\rho:=e^{\rho_1}\cdot e^{\rho_2}$. Is this correct? Also this should still make sence if $e^{\beta\cdot\rho}$, where $\beta$ is a non-integral number.
Also I am confused by the fact, that the irreducible representation does only appear in $\lambda$, the highest weight in the fundamental Weyl-Chamber (Humphreys' Definition) which in turn is part of the Root-Lattice. Is that correct?
As I said, I just started learning about this, so please keep it as simple as possible. Thanks for any help! I somehoy cannot give usefull tags, so I will saty very general here...