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I have found mention of the fact that the spherical dual of a (real reductive quasisplit) group can be canonically parametrized by a suitable finite-dimensional complex vector space. More precisely, if $G=KAN$ is an Iwasawa decomposition of the group, $W$ the Weyl group relative of $A$ and $\mathfrak{a}^\mathbf{C}$ is the complexification of the Lie algebra of $A$, then: $$\widehat{G}^{\mathrm{adm}} \simeq \mathfrak{a}^\mathbf{C}/W$$

This makes me think of the Satake parametrization in the case of the spherical representations, and I would like to unveil what is behind. Do we know the explicit isomorphism? (what is the $\lambda$ on the RHS corresponding to an admissible representation $\pi$?)Has it something to do with the Langlands classification?

(question originally on MSE where it found no attention)

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    $\begingroup$ Well, an (irreducible admissible) spherical repn imbeds (by the Casselman subrepn thm) in a (spherical) principal series that is unique mod $W$, and no non-isomorphic spherical repns imbed into the same principal series. I don't know what you mean by "explicit"... For the full dual, it can happen that two distinct irreducible imbed into the same principal series, messing up the bijection. Not so much "contained in $A$" as perhaps "dual", maybe? And we really do need the complexification, because (for example) spherical repns need not be unitary, etc. $\endgroup$
    – paul garrett
    Sep 26, 2017 at 19:52
  • $\begingroup$ @paulgarrett I have added some precisions in my question, that I believe I do not understand enough myself to ask it in a correct way. Im a interested in archimedean local duals and I would like to understand the isomorphism. Has it anything to do with the "infinitesimal equivalence" notion? $\endgroup$
    – Desiderius Severus
    Oct 12, 2017 at 5:52
  • $\begingroup$ Your revised question seems to ask how we know the subrepresentation theorem for spherical representations. To my perception, this is still a very big question, and I do not know how to answer, outside of examples in the rank-one case, where asymptotics and other features of ordinary differential equations play a key role. In higher rank, the analogue is the appendix in Casselman-Milicic, expanding a paper of Deligne on the PDE analogue. $\endgroup$
    – paul garrett
    Oct 12, 2017 at 16:29

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