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Depth and the local Langlands correspondence. (English) Zbl 1388.22010
Ballmann, Werner (ed.) et al., Arbeitstagung Bonn 2013. In memory of Friedrich Hirzebruch. Proceedings of the meeting, Bonn, Germany, May, 22–28, 2013. Basel: Birkhäuser/Springer (ISBN 978-3-319-43646-3/hbk; 978-3-319-43648-7/ebook). Progress in Mathematics 319, 17-41 (2016).
Let $$F$$ be a non-archimedean local field, and let $$G$$ be the group of $$F$$-points of a connected reductive algebraic group over $$F$$. The local Langlands correspondence (LLC) is a conjectural correspondence between the set $$\text{Irr}(G)$$ of (equivalence classes of) smooth irreducible complex representations of $$G$$ and the set $$\Phi(G)$$ of (equivalence classes of) Langlands parameters for $$G$$. This correspondence $$\text{Irr}(G)\rightarrow\Phi(G)$$ is not in general one-to-one; the fibers are referred to as $$L$$-packets.
For $$\pi\in\text{Irr}(G)$$, A. Moy and G. Prasad [Invent. Math. 116, No. 1–3, 393–408 (1994; Zbl 0804.22008)] associated to any irreducible smooth representation $$\pi$$ of $$G$$ a nonnegative real number $$d(\pi)$$ known as the depth of $$\pi$$. This number is defined in terms of certain filtrations of the parahoric subgroups of $$G$$. One also has a notion of the depth $$d(\phi)$$ of a Langlands parameter $$\phi$$, namely the smallest real number $$r$$ such that $$\phi$$ restricts trivially to the $$r$$th ramification subgroup of $$\text{Gal}(F^{\text{sep}}/F)$$. There is a growing body of evidence that in a very large collection of cases, the LLC preserves depth, i.e., $d(\pi) = d(\phi)$ for $$\pi\in\text{Irr}(G)$$ and $$\phi\in\Phi(G)$$ that correspond under the LLC. In particular, for $$G = \text{GL}_n(F)$$, this equality was stated in [J.-K. Yu, Fields Institue Monographs 26, 53–77 (2009; Zbl 1179.22020)] and proved in [the authors, Res. Math. Sci. 3, Paper No. 32, 34 p. (2016; Zbl 1394.22015)]. The article under review extends this result to inner forms $$G$$ of $$\text{GL}_n(F)$$. The authors also prove depth preservation for a large class of Langlands parameters of inner forms $$G$$ of $$\text{SL}_n(F)$$, namely the essentially tame parameters; when $$\text{char}(F)\nmid n$$, all parameters of $$G$$ are essentially tame.
The proof for inner forms $$G$$ of $$\text{GL}_n(F)$$ proceeds by reducing to the case of essentially square-integrable representations using the Langlands classification. The authors then prove a uniform relation between $$d(\pi)$$ and another invariant of $$\pi$$, its conductor, using arguments along the lines of those in [J. Lansky and A. Raghuram, Proc. Am. Math. Soc. 131, No. 5, 1641–1648 (2003; Zbl 1027.22017)]. Since the LLC for $$G$$ is the composition of the Jacquet-Langlands correspondence $$\text{Irr(G)}\rightarrow \text{Irr}(\text{GL}_n(F))$$ and the LLC for $$\text{GL}_n(F)$$ (for which depth preservation has already been established), it suffices to prove that the Jacquet-Langlands correspondence preserves conductors. The authors do this by proving results on the Jacquet-Langlands correspondence first stated by Deligne-Kazhdan-Vigneras.
For the entire collection see [Zbl 1364.00032].

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 20G25 Linear algebraic groups over local fields and their integers
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