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The Arthur trace formula and its variations provide general results for reductive groups, however to the extent of my knowledge only few specific instances of the formula have been really worked out in details, i.e. with precise decomposition of the spectrum, as for $\mathrm{GL}(n)$.

Is there such a formula for $\mathrm{GSp(4)}$ or instances of use of the general trace formula in that setting? Or should I rather come back to the classification of representations of $\mathrm{GSp(4)}$ (perhaps as in Roberts and Schmidt)?

Thanks for any idea!

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  • $\begingroup$ Do you know Arthur's paper "Automorphic representations of GSp(4)"? I'm not certain what you are asking but this might answer your question. $\endgroup$
    – Dan Petersen
    Nov 9, 2017 at 13:38
  • $\begingroup$ @DanPetersen I know it, and should have mentioned it when I quoted Robert and Schmidt: the classification of the automorphic spectrum of $\mathrm{GSp(4)}$ is indeed known, opening the path to working out the trace formula for this group. Hence my question: is there any place where such a project has been carried on? (for any kind of results) $\endgroup$
    – Desiderius Severus
    Nov 9, 2017 at 13:46
  • $\begingroup$ I guess you mean Robert_s_ and Schmidt, as in Brook_s_ Robert_s_. (Hmm... that didn't give me italicized s's.) $\endgroup$
    – Kimball
    Nov 9, 2017 at 18:00

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There are many kinds of trace formulas on a given group $G$, and different things you could mean by decomposition of the spectrum. As mentioned in the comments, there's Arthur's article in the Shalika volume, though I am not sure whether those results are unconditional as of yet. On the other hand, Arthur's work on the trace formula for classical groups including SO(5) $\simeq$ PGSp(4).

Precisely what Arthur is interested is functoriality (wrt the standard represenentation) to GL($N$). As a result you get a description of the discrete part of the $L^2$ spectrum for $G$. In the case of SO(5) (i.e., GSp(4) with trivial central character), Ralf Schmidt wrote down what this explicitly tells you about the automorphic representations appearing discretely in his Packet structure preprint.

If you are interested in the non-discrete spectrum, there are some things you should be able to say, but I don't know of an explicit description of this part of the spectrum.

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  • $\begingroup$ I believe those are enough clues to try to address a problem involving a trace formula for GSp(4), thanks for the references! $\endgroup$
    – Desiderius Severus
    Dec 5, 2017 at 8:48

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