Central content sectors of the cyclotomic Hecke algebra

Work in the charged finite-$e$ specialization of Affine type-A presentations and charged cyclotomic quotients at a dominant weight $\mathrm{\Lambda }$ and rank $d$, and let $H=H_d^{\mathrm{\Lambda }}$. For the canonical CRT projectors supplied by Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras, denote by $e(\nu )$ the joint residue projector attached to the residue tuple $\nu$ and by $e_{\alpha }:={\sum }_{\nu \text{ of content }\alpha }e(\nu )$ their sum at content $\alpha$ (Affine type-A presentations and charged cyclotomic quotients).

Content sector. Put $H_{\alpha }^{\mathrm{\Lambda }}:=e_{\alpha }{H}_d^{\mathrm{\Lambda }}=e_{\alpha }{H}_d^{\mathrm{\Lambda }}{e}_{\alpha },$ an ideal and unital algebra with unit $e_{\alpha }$, called the central content sector of content $\alpha$; it may be the zero algebra, and at this stage it is NOT asserted to be a primitive block. The two descriptions coincide because $e_{\alpha }$ is a central idempotent.

Named nilpotent generators. On $H_{\alpha }^{\mathrm{\Lambda }}$ put $y_r:={\sum }_{\nu \text{ of content }\alpha }(x_r-{\nu }_r)e(\nu )\text{(degenerate branch)},y_r:={\sum }_{\nu \text{ of content }\alpha }(1-q_H^{-{\nu }_r}{X}_r)e(\nu )\text{(nondegenerate branch)}\mathrm{.}$ By the spectral supplier these named elements are commuting nilpotents on the sector, and they provide the local variables in which the sector is compared with the corresponding KLR block.

Boundary conventions. For $\alpha =0$ one has $e_{\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{t}\mathrm{y}}=1$ and $H_0^{\mathrm{\Lambda }}=F$ at rank zero. Later primitivity statements may identify the nonzero sectors with the blocks of $H$, but that identification does not change the objects named here. This definition introduces no new primitivity or basis assertion of its own and uses no choice principle.