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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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