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What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?

Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\pi}$, where $A_{\pi}$ is the space of matrix coefficients associated with the representation $\pi$, and where $\pi$ runs over the irreducible unitary representations.

I've seen many sources that stop the discussion of Peter-Weyl here. My question is about how to compute this in practice. For a given compact real linear algebraic group, its irreducible (unitary) representations are classified by the theory of the highest weights. My question, therefore, boils down to whether there exists an easy way to compute the matrix coefficients for the irreducible representation associated with some weight.

As an example, how does it manifest in the case of the orthogonal group? What is the associated orthonormal basis for $L^2(G)$?

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    $\begingroup$ A generalised version of this is called a zonal spherical function. $\endgroup$
    – LSpice
    Commented Dec 7, 2020 at 0:01
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    $\begingroup$ I'm confused about how that fits in -- it appears to me that zonal spherical functions are about L^2(G/K) where K is maximal compact. I am interested in the G is compact case. How should I be thinking about this? $\endgroup$
    – Andrew NC
    Commented Dec 7, 2020 at 3:41
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    $\begingroup$ When you say "compute the matrix coefficients" of a given irrep $\pi$, what do you mean? Do you mean some explicit parametrization with respect to a given o.n. basis of $H_\pi$? Or some kind of extrinsinc condition of the form "$f\in A_\pi$ if and only if some operator annihilates $f$"? $\endgroup$
    – Yemon Choi
    Commented Dec 7, 2020 at 4:05
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    $\begingroup$ I agree that it is not enough just to know the highest weight parametrization in order to obtain a basis, but as @LSpice and I have tried to point out, as soon as you have an explicit basis $v_1, \dots, v_n$ for $H_\pi$ then you can write down the corresponding matrix coefficient functions on $G$, namely $\langle \pi( \cdot) v_j, v_i \rangle$. Do you agree that this is enough? If so, then we can reduce your original question to the problem of computing an explicit ONB for each irrep, which I acknowledge is not trivial $\endgroup$
    – Yemon Choi
    Commented Dec 7, 2020 at 4:47
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    $\begingroup$ The reason presentations stop after the PW Thm is these things are complicated. For $SU(2)$ or $SO(3)$ see, e.g., en.wikipedia.org/wiki/Wigner_D-matrix For higher dimension $SO(n)$, you might find formulas in the series of books springer.com/gp/book/9780792314660 $\endgroup$
    – Abdelmalek Abdesselam
    Commented Dec 7, 2020 at 4:47

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