The conjectural description of real discrete series for affine Hecke algebras of type B

Let $\mathcal{H}$ be the affine Hecke algebra with labels $q_2=q_1^m$ obtained from the parameters $q_{\check{\alpha}}$ as in the source, with $q_1>1$ and $q_2\geq 1$. Let $W_0r$ be the $W_0$-orbit of the center of a real residual coset of $\mathcal{H}$. Write $\Delta_{W_0r}$ for the corresponding set of modules, $\mathcal{H}_0$ for the finite Hecke algebra, $R(M)$ for the $W_0$-module obtained from an $\mathcal{H}_0$-module $M$ in the limit $q_{\check{\alpha}}^{1/2}\to1$, and $a_m$ for the stated $a$-function. Let $\hat{W}_0$ denote the irreducible $W_0$-representations, let $\epsilon$ be the sign representation, and let $\succ_m$ be an ordering on $\hat{W}_0$ refining the preorder defined by $\psi\succ\chi\Rightarrow a_m(\psi)\leq a_m(\chi)$, with similarity classes as intervals. Conjectural description. (i) $\Delta_{W_0r}$ is indexed by $\Sigma_m(W_0r)$. (ii) The modules in $\Delta_{W_0r}$ are naturally graded for the action of $\mathcal{H}_0$; their top nonzero degree is $a_m(\Sigma_m(W_0r))$, and the $\mathcal{H}_0$-representations in that degree are irreducible, with corresponding $W_0$-representations $\Sigma_m(W_0r)\otimes\epsilon$. (iii) If $M_\chi$ is the $\mathcal{H}$-module labelled by $\chi\in\hat{W}_0$ using (ii) and the stated partition of $\hat{W}_0$, so that $R(M_\chi^{\mathrm{top}})\cong\chi\otimes\epsilon$, then the matrix $(R(M_\chi),\psi)_{\psi,\chi\in\hat{W}_0}$ is block-lower triangular with blocks given by the similarity classes. (iv) The $\mathcal{H}_0$-structure of $M_\chi$ can be computed using Shoji's generalized Green functions. The statement proposes a detailed parametrization and character-theoretic description of the real discrete-series modules for these affine Hecke algebras; the supplied text does not establish it or give resolution evidence, so its status remains open.