Let $Q_1 = \{(x_1, \cdots, x_n) \in \mathbb{R}^n ~|~ \sum_{i} x_i^2 = 1, x_1 \geq 0, \cdots, x_n \geq 0\}$ (i.e. the set of unit vectors in the first $2^n$-tant of $\mathbb{R}^n$). The $n \times n$ DFT matrix $W_n$ is the matrix whose $(j, k)$th entry is $\omega^{jk} / \sqrt{n}$, where indexing begins at zero. I was wondering if any nice characterizations of $$U_n = W_n Q_1 = \{W_n q ~|~ q \in Q_1\}$$ were known. I know that it is a sector of the $n$-dimensional complex unit sphere, but not much more.
My question is motivated by a problem where $q$ is chosen uniformly at random from $Q_1$, and I want to calculate expectations of various quantities that are much more simply expressed in Fourier space. I know that the distribution of $q$ will be uniform over $U_n$. So if I wanted to calculate the expectation of a quantity $\tilde A(q)$ (expressed in Fourier space), I would just have to calculate $$\frac{1}{|U_n|} \int_{U_n} \tilde A(q) dq = 2^n \int_{U_n} \tilde A(q) dq$$ The only problem is that I don't know what $U_n$ is, or if it's feasible to integrate over $U_n$ at all.
Thanks!