Tempered modules in exotic Deligne–Langlands correspondence

We present a geometric description for tempered modules of the affine Hecke algebra of type Cn with arbitrary (non-root of unity) unequal parameters, using the exotic Deligne–Langlands correspondence (Kato (2009) [18], ). Our description has several applications to the structure of the tempered modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sgn representation of the Weyl group, equivalently, via the Iwahori–Matsumoto involution, of spherical cuspidal modules. This latter combinatorial classification was expected from Heckman and Opdam (1997) [15], and determines the L2-solutions for the Lieb–McGuire system.