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A nonabelian Fourier transform for tempered unipotent representations
- Part of
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- Journal:
- Compositio Mathematica / Volume 161 / Issue 1 / January 2025
- Published online by Cambridge University Press:
- 12 March 2025, pp. 13-73
- Print publication:
- January 2025
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- Article
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We define an involution on the elliptic space of tempered unipotent representations of inner twists of a split simple
$p$-adic group 
$G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence for the conjecture, including proofs for 
${\mathsf {SL}}_n$ and 
${\mathsf {PGL}}_n$.
