Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for integral $c_j$'s. This basis has also the nice property that each element in it is point-wise bounded by $1$ in absolute value. In general, an $L^2$ function takes the form $g = \sum_n q_n f_n$ where $f_n$ are the basis functions described above. $g$ can have arbitrary bad $L^\infty$ bound, since the only requirement is that $\sum q_n^2 = 1$, but $\sum_n q_n$ can be $\infty$.

Now my question is, for the unitary or orthogonal group, what is the smallest uniform $L^\infty$ bound one can achieve on an orthogonal basis on the space of class functions? For the unitary groups, any orthogonal basis of class functions forms an orthogonal basis in the space of symmetric functions of the eigenvalues under the inner product such that the Schur polynomials are orthonormal. So the question can be phrased in terms of symmetric functions. For the orthogonal groups, one orthogonal (not normalized) basis of class functions comes from $\prod_{i=1}^n (Tr X^i)^{\alpha_i}$ where for some odd $i$, $\alpha_i$ is odd (I am not entirely sure about this, but see this paper theorem 4).

Edit: what about all $L^2$ functions in general? Can one get a uniform $L^\infty$ bound on the basis of size $e^{\mathcal{O}(n)}$ for $SO(n)$ or $U(n)$?