Talk:Zonal spherical function
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Two mistakes in the section on the Harish-Chandra formula
[edit]K does not act on P by conjugation, and not every element of P is conjugate to A (e.g. a unipotent element cannot be conjugate to a semisimple element). ~2026-41074-10 (talk) 19:13, 23 July 2026 (UTC)
SL2R example.
[edit]I think there's a problem there. The article states that all zonal spherical functions are principal ones, but there are also the complementary ones (as is stated in the introduction). ~2026-41074-10 (talk) 20:38, 23 July 2026 (UTC)
Disputed status
[edit]Having noticed several inaccuracies (which I do not know personally how to best resolve) I flagged this for expert attention. Not being an expert myself, I suspect edits have been made be persons who are not necessarily experts in harmonic analysis.
A side note: should the page be called "zonal spherical function" or simply "spherical function"? Is the term "zonal" sufficiently well-established? I do not see it in any of the major textbooks I have consulted. ~2026-41074-10 (talk) 20:43, 23 July 2026 (UTC)
Another problem
[edit]I believe there is another problem: zonal spherical functions are described as the space of characters on the Hecke algebra. On the other hand, positive definiteness is taken to be a part of the definition of the zonal function. However, the space of such characters requires boundedness, but not positive-definiteness. ~2026-41074-10 (talk) 17:45, 24 July 2026 (UTC)