Parabolic affine Hecke algebras and their weight characters

Let $H$ be the complex affine Hecke algebra of a reduced based root datum with positive real parameters and Bernstein PBW as in General Bernstein PBW and the Iwahori–Matsumoto comparison. To keep the page's dual convention explicit, put $L=Y$, $B={\mathrm{\Delta }}^{\vee }\subset L$, and write ${\beta }^{\vee }=\alpha \in X$ when $\beta ={\alpha }^{\vee }\in B$. Thus $A=\mathbb{C}[L]$ is the Bernstein subalgebra, $T=\mathrm{Hom}(L,{\mathbb{C}}^{\times })$, and $E=\mathrm{Hom}(L,\mathbb{R})=X\otimes \mathbb{R}$. The Weyl group $W=W_0$ acts on all these objects. Index parabolic subsets by $P\subseteq B$; they correspond bijectively to subsets of $\mathrm{\Delta }$ by $\alpha \leftrightarrow {\alpha }^{\vee }$.

Write $W_P=⟨s_{\beta }:\beta \in P⟩$, and let $H_P$ denote the subalgebra of $H$ generated by $A$ and $N_{s_{\beta }}$ for $\beta \in P$. In particular $H_{\mathrm{\varnothing }}=A$ and $H_B=H$. The notation $H_P$ here always retains the full lattice $L$. For an $H_P$-module $V$, its parabolic induction is $I_P(V)=H{\otimes }_{H_P}V$ with $H$ acting on the first factor.

All modules in the classification below are nonzero, unital, complex and finite dimensional. For an $A$-module $V$ and $t\in T$, the generalized weight space is

$$V_t=\{v\in V:({\theta }_{\mathrm{\ell }}-t(\mathrm{\ell }){)}^{\dim V}v=0\text{ for every }\mathrm{\ell }\in L\}\mathrm{.}$$

The weight set is $\mathrm{Wt}(V)=\{t:V_t\mathrm{\ne }0\}$. The logarithmic absolute value $\lambda (t)\in E$ is $\lambda (t)(\mathrm{\ell })=\log \mathrm{\mid }t(\mathrm{\ell })\mathrm{\mid }$. For an irreducible $H_P$-module, the central weight character ${\mathrm{cc}}_P(V)$ is its $W_P$-orbit of $A$-weights; the fact that there is exactly one such orbit is proved in Parabolic induction, central orbit blocks and weight filtrations ↗. This terminology specifies the character of $A^{W_P}$; no equality with a larger center is needed.