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The principal series of \(p\)-adic groups with disconnected center. (English) Zbl 1383.20033
Let \(\mathcal{G}\) be a split connected reductive group over a local non-Archimedean field \(F\), possibly with disconnected center. In this paper, the authors construct a local Langlands classification for all the principal series representations of \(\mathcal{G}\) under the assumption that the residual characteristic does not take on certain small values. They follow a similar approach to that of M. Reeder [Represent. Theory 6, 101–126 (2002; Zbl 0999.22021)], based on A. Roche’s realization of types, and his equivalence of categories with Iwahori-Hecke algebras [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 3, 361–413 (1998; Zbl 0903.22009)]. Furthermore, they show that these representations are nicely parameterized by suitable extended quotient, in line with the ABPS conjecture (the Aubert-Baum-Plymen-Solleveld cojecture).
More specifically, let \(\mathcal{T}\) be a maximal torus in \(\mathcal{G}\), and let \(G\) and \(T\) be the Langlands dual groups of \(\mathcal{G}\) and \(\mathcal{T}\), respectively. Denote by \(\mathbf{Irr}(\mathcal{G},\mathcal{T})\) the space of all \(\mathcal{G}\)-representations in the principal series. The authors consider Langlands parameters of the form \(\Phi: F^\times \times \mathrm{SL}_2(\mathbb{C}) \to G\), and for such \(\Phi\) they define a Kazhdan-Lusztig-Reeder parameter as the ordered pair \((\Phi, \rho)\), where \(\rho\) is a “geometric” representation of \(\pi_0(Z_G(\Phi))\). Let \(\mathcal{W}^G=W(G,T)\). The authors prove that there exists a commutative, bijective triangle \[ \begin{aligned} & (\mathbf{Irr}\mathcal{T}/\!/\mathcal{W}^G)_2 \\ & \swarrow \searrow \\ \mathbf{Irr}(\mathcal{G},\mathcal{T}) & \longrightarrow \{\text{KLR parameters for }G \}/G \end{aligned} \] Here, \((\mathbf{Irr}\mathcal{T}/\!/\mathcal{W}^G)_2\) is the extendend quotient of the second kind; the right slanted map is natural, and via the bottom map any \(\pi \in \mathbf{Irr}(\mathcal{G},\mathcal{T})\) canonically determines a Langlands parameter \(\Phi\) for \(\mathcal{G}\). The triangle above is obtained as the union of corresponding triangles for all Bernstein components.
The authors explicitly verify the desiderata for the local Langlands correspondence proposed by A. Borel [Proc. Symp. Pure Math. 33, 27–61 (1979; Zbl 0412.10017)]. In particular, the constructed correspondence is functorial with respect to homomorphisms of reductive groups.

MSC:
20G25 Linear algebraic groups over local fields and their integers
20G05 Representation theory for linear algebraic groups
22E50 Representations of Lie and linear algebraic groups over local fields
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