Kottwitz's trace formula conjecture for regular discrete series representations

From papers

Let GG be the reductive group under consideration, let π\pi be a regular discrete series representation of G(R)G(\mathbb{R}), and let eπe_{\pi} denote the stable cuspidal test function associated with π\pi as in Section. Choose a compactly supported test function fCc(G(Af))f^{\infty}\in C_c(G(\mathbb{A}_f)), with measures satisfying dgfdg=dgdg_f\,dg_{\infty}=dg, the Tamagawa measure on G(A)G(\mathbb{A}), and set f=fff=f^{\infty}f_{\infty}, where fdg=eπf_{\infty}dg_{\infty}=e_{\pi}. Kottwitz's trace formula conjecture. One has

Kot(fdg)=trRdisc(π,fdgf).\operatorname{Kot}(f\,dg)=\operatorname{tr} R_{\operatorname{disc}}(\pi,f^{\infty}dg_f).

This predicts that the stable trace formula with a regular discrete-series test function isolates the corresponding discrete automorphic representation's trace. The surrounding discussion establishes the analogous identity after summing over the relevant discrete-series packet, while the assertion for an individual regular discrete-series representation is presented as plausible and remains unresolved in the supplied text.

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Sources & referencesView supporting material

Primary source

Steven Spallone, “Stable Trace Formulas and Discrete Series Representations”, arXiv:1010.4897 (2010).

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