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Kazhdan-Lusztig parameters and extended quotients

The Kazhdan-Lusztig parameters are important parameters in the representation theory of $p$-adic groups and affine Hecke algebras. We show that the Kazhdan-Lusztig parameters have a definite geometric structure, namely that of the extended quotient $T//W$ of a complex torus $T$ by a finite Weyl group $W$. More generally, we show that the corresponding parameters, in the principal series of a reductive $p$-adic group with connected centre, admit such a geometric structure. This confirms, in a special case, our recently formulated geometric conjecture. In the course of this study, we provide a unified framework for Kazhdan-Lusztig parameters on the one hand, and Springer parameters on the other hand. Our framework contains a complex parameter $s$, and allows us to interpolate between $s = 1$ and $s = \sqrt q$. When $s = 1$, we recover the parameters which occur in the Springer correspondence; when $s = \sqrt q$, we recover the Kazhdan-Lusztig parameters.