What projective groups whose characters form the basis of their corresponding $L^2$ space?

Question

For instance, the cyclic group $\Bbb Z/n\Bbb Z$ has n characters in the form $x\mapsto \exp(2\pi i ax/n)$ for some $a\in \Bbb Z/n\Bbb Z$. The characters form a basis for $L^2(\Bbb Z/n\Bbb Z)$, which has dimension n (the Peter–Weyl theorem).

Are there some projective groups having the same property of $\Bbb Z/n\Bbb Z$? My ultimate goal is to see the link between projective geometry and harmonic analysis.

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Answer 1

There is no relation. Don't read too much into these informal analogies. There is no natural group structure on the set of all bases of a vector space, and so, it makes no sense to speak of "isomorphism" with a projective group. (The same with the set of all orthonormal bases of a Hilbert space).