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I would like to understand what the Plancherel measure is on the dual of a compact group, in more explicit terms than with an implicit definition. Indeed, to the extent of my knowledge it is defined as the only positive Radon measure such that a certain inversion formula holds, namely $$\int_{\hat{G}} f(\pi)d\mu(\pi) = f(1).$$

What can be said more precisely when a representation is given explicitly? For instance, consider G the group of projective units of a quaternion algebra. For a ramified place v the group is compact so that the representation is finite dimensional. Do we have that $$\mu(\pi_v) = \dim(\pi_v)^{-1},$$

as in the finite case, or something similar? Are there references constructing from scratch the Plancherel measure and manipulating it?

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    $\begingroup$ I don't think there are finite-dimensional unitary representations for a connected reductive group other than the trivial one. $\endgroup$
    – user130903
    Commented Dec 18, 2018 at 3:31
  • $\begingroup$ @Zero I modified the question to talk about compact groups, I am interested in this case and want to know if the same relation than for finite groups hold or not. $\endgroup$
    – TheStudent
    Commented Dec 18, 2018 at 5:48
  • $\begingroup$ @Zero, it depends on the ground field! For the reals, there are probably lots. For the $p$-adics, a torus can have some, but a semisimple group has none. $\endgroup$
    – LSpice
    Commented Dec 18, 2018 at 6:13
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    $\begingroup$ @LSpice What about representations of quaternion algebras at ramified places? $\endgroup$
    – TheStudent
    Commented Dec 18, 2018 at 7:27
  • $\begingroup$ @TheStudent, quaternion algebras (assuming you mean their multiplicative groups) aren't semisimple; but you are right, that even their unit groups (or adjoint quotients) have non-trivial, finite-dimensional representations. I should have specified 'non-compact' for my $p$-adic claim. $\endgroup$
    – LSpice
    Commented Dec 18, 2018 at 7:48

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