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Proof of the Aubert-Baum-Plymen-Solleveld conjecture for split classical groups. (English) Zbl 1388.22016

Brumley, Farrell (ed.) et al., Around Langlands correspondences. International conference, Université Paris Sud, Orsay, France, June 17–20, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 691, 257-281 (2017).
In the paper under the review, the author studies the Aubert-Baum-Plymen-Solleveld (ABPS) conjecture for split classical groups. Roughly, this conjecture provides a connection between the Bernstein components appearing in the decomposition of the set of irreducible representations and the geometric structure of the spectral extended quotient of the torus by the Weyl group. More details on the ABPS conjecture can be found in [A.-M. Aubert et al., Jpn. J. Math. (3) 9, No. 2, 99–136 (2014; Zbl 1371.11097)].
The author proves the ABPS conjecture by establishing its Galois version, obtained by replacing the representations by their Langlands parameters, thus presenting a relation between the ABPS conjecture and the Langlands correspondence.
We note that the ABPS conjecture has also been proved in [M. Solleveld, Represent. Theory 16, 1–87 (2012; Zbl 1272.20003)], using different methods.
For the entire collection see [Zbl 1369.11002].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11R39 Langlands-Weil conjectures, nonabelian class field theory
20C33 Representations of finite groups of Lie type
11F85 \(p\)-adic theory, local fields
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References:

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