Comparison of the Sally-Shalika character formulas with the endoscopic character identities for $\mathrm{SL}_2$

Anne-Marie Aubert; Roger Plymen

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arXiv · 2410.20183

Comparison of the Sally-Shalika character formulas with the endoscopic character identities for $\mathrm{SL}_2$

Anne-Marie Aubert·Roger Plymen

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Abstract

We consider the depth-zero supercuspidal $L$-packets of $\mathrm{SL}_2(F)$ where $F$ is a non-archimedean local field of characteristic zero. We compare the explicit endoscopic character identities for $\mathrm{SL}_2(F)$ with the classical character formulas of Sally-Shalika. Our main result concerns the supercuspidal $L$-packet of size $4$. For this $L$-packet, we show how the norm $1$ groups $H_1, H_2, H_3$ in the three quadratic extensions of $F$ play a crucial role in the endoscopic character identities for $\mathrm{SL}_2(F)$.

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Anne-Marie Aubert, Roger Plymen. 2024-10-26. Comparison of the Sally-Shalika character formulas with the endoscopic character identities for $\mathrm{SL}_2$. https://arxiv.org/abs/2410.20183

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