Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says that $\{t_i\}_{i=1}^{n}$ are numbers such that the given a $ g\in SO(2n)$ it is conjugate to $e^{\sum_{i=1}^n t_i H_i}$
Then apparently the character of this $g$ in this representation is given as,
$$\chi (h_i,t_i) = \frac{\det (\sinh [ t_i(h_j + n-j)]) + \det (\cosh [ t_i(h_j + n-j)]) }{\det (\sinh [t_i(n-j)]) } $$
But when actually trying to put this into use I am faced with two confusions,
Given a $g$ how do I determine "a" set of $t_i$? Is there a standard choice of $H_i$ if I am given $g$ in its "defining" representation of a rotation matrix in $\mathbb{R}^{2n}$?
I would like to know of a general method of getting "a" required set of $t_i$ from a given (defining) matrix representation of $g$.
For concreteness (and my immediate necessity!) you can consider for $g$ the rotation matrix in $\mathbb{R}^{2n}$ which rotates by an angle $\alpha$ in the $1$-$2$ plane and keeps everything else fixed. Then the $g$ given is a $2n \times 2n$ matrix such that the first $2\times 2$ block on the diagonal is $\{\{\cos \alpha, \sin \alpha\},\{-\sin \alpha, \cos \alpha \}\}$ and all other entries are $0$.
Is there some symmetry here which ensures that the WCF will give the same answer for whatever is the choice of basis in the Cartan of the Lie algebra? (..these different choice of basis will clearly change the values of $t_i$..)
If someone can give a reference to this particular formula for the character then that would be great.
In this MO discussion about this ARupanski had suggested the following way to get a valid set of $t_i$,
Take a $SO(2n)$ matrix say $g$ and then all $2n$ eigenvalues come in pairs $\{a_i, 1/a_i\}$. Now consider the n-vector $v = \{a_i\}$
- Is there a way in which one can think of this $v$ to be sitting in the span of the fundamental weights of $SO(2n)$?
I think the claim is that $t_i$ is determined by writing $v = \sum_{i=1} ^n t_i w_i$ where $w_i$ are the fundamental weights.
Can someone kindly help understand as to why this is supposed to give a required set of $t_i$?
Does this work beyond $SO(2n)$ groups?