zbMATH — the first resource for mathematics

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.

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
Full Text: DOI