The unique irreducible quotient of an affine Hecke standard module

By [F1], [F2] and [F3], all weights of $S$ have $r$-value $R$ and projection $\nu$, whereas every weight contributed by a nonidentity coset has strictly smaller $r$-value. The $A$-stable quotient $M\mathrm{/}S$ has exactly those smaller weights by the induction filtration and weight-space exactness. It follows that the sum $M^{\mathrm{t}\mathrm{o}\mathrm{p}}$ of all generalized weight spaces in $M$ with $r$-value $R$ is exactly $S$: the inclusion $S\subset M^{\mathrm{t}\mathrm{o}\mathrm{p}}$ holds by its weights, and the quotient has none of these weights. In particular $M^{\mathrm{t}\mathrm{o}\mathrm{p}}$ is an irreducible $H_P$-submodule.

If an $H$-submodule $U\subset M$ contains a nonzero vector of a generalized weight space with $r$-value $R$, then $U\cap S\mathrm{\ne }0$. This intersection is $H_P$-stable, so irreducibility gives $S\subset U$. The induced module is generated by $S$ under $H$, whence $U=M$. Consequently every proper $H$-submodule has all its weights of $r$-value strictly below $R$, and is contained in the sum of the lower generalized weight spaces as a vector subspace. The sum $J$ of all proper submodules is therefore contained in that same lower-weight subspace, is proper, and contains every proper submodule. Finite dimensionality ensures this sum is an ordinary finite sum of subspaces after a basis is selected. Thus $J$ is the unique maximal proper submodule and $M\mathrm{/}J$ is irreducible.

Since $J$ has no top weights, exactness of generalized weight spaces identifies the top subspace of $M\mathrm{/}J$ with $S$, including its $H_P$-action. It is nonzero, its projection vector is $\nu$, and every remaining weight has smaller $r$-value. Any irreducible quotient has a maximal kernel, necessarily $J$, proving uniqueness and all the claimed intrinsic top-space properties.