The endomorphism algebra of a general finite principal series

Let $H_q(W_{\eta }):={⨂}_r{H}_q(S_{n_r})$, presented by the disjoint union of the type-A generators of the factors together with the type-A relations inside each factor and commutation between different factors. By [F3] the generators $T_s$, $s$ simple in $W_{\eta }$, satisfy the quadratic relations with parameter $q$; by [F2] they satisfy the braid relations inside each block and the commutation relations between blocks. Hence the assignment sending the generators of $H_q(W_{\eta })$ to the corresponding $T_s\in {\mathrm{End}}_G(I(\eta ))$ extends to a unital $\mathbb{C}$-algebra homomorphism $\beta :H_q(W_{\eta })\to {\mathrm{End}}_G(I(\eta ))$.

The map $\beta$ is an isomorphism. It is surjective: by [F1] the $B_w$, $w\in W_{\eta }$, form a basis of the target, and $B_w={\lambda }_w^{-1}{T}_w$ with ${\lambda }_w\mathrm{\ne }0$ by [F3]; by [F2] each $T_w$ is a product of the generators $T_s$ along a reduced expression of $w$ (products in each block are length-additive and factors from different blocks commute), so every $B_w$ lies in the image. Both algebras have the same finite dimension: the target has dimension $\mathrm{\mid }{W}_{\eta }\mathrm{\mid }={\prod }_r{n}_r!$ by [F5], and the source has basis the tensor products of the standard bases of the factors, of cardinality ${\prod }_r{n}_r!$ by [F4]. A surjection of finite-dimensional vector spaces of equal dimension is an isomorphism. Under $\beta$ the tensor basis element ${⨂}_r{T}_{w_r}$ maps to ${\prod }_r{T}_{w_r}=T_w={\lambda }_w{B}_w$ by the length-additive rule of [F2], so the Hecke basis of ${\mathrm{End}}_G(I(\eta ))$ is exactly $\{T_w={\lambda }_w{B}_w:w\in W_{\eta }\}$ with ${\lambda }_w={\prod }_r{a}_r(-1{)}^{-\mathrm{\ell }(w_r)}$ by [F3]. The construction of $\beta$ used only [F1]-[F5], none of which uses AC.

By [F7] the algebra ${\mathrm{End}}_G(I(\eta ))$ is semisimple and a product of matrix algebras, with simple-module dimensions given by the matrix sizes. By [F4] and [F6], and using $\beta$, $H_q(W_{\eta })\cong {⨂}_r{H}_q(S_{n_r})\cong {⨂}_r\mathbb{C}[S_{n_r}]\cong \mathbb{C}[W_{\eta }]\cong \mathbb{C}[W_{\chi }]$; an algebra isomorphism preserves the number and dimensions of simple modules.

For arbitrary $\chi$, choose the sorting $\sigma \in S_n$ with $\eta =\sigma \cdot \chi$ as in the Given data and a module isomorphism $J:I(\chi )\to I(\eta )$, which exists by [F5]; conjugation by $J$ is an algebra isomorphism ${\mathrm{End}}_G(I(\eta ))\to {\mathrm{End}}_G(I(\chi ))$, so transporting the Hecke basis of step 2.1 gives a Hecke basis of ${\mathrm{End}}_G(I(\chi ))$ indexed by $W_{\chi }={\sigma }^{-1}{W}_{\eta }\sigma$ with the transported lengths, and ${\mathrm{End}}_G(I(\chi ))\cong H_q(W_{\eta })\cong H_q(W_{\chi })$. By [F5] a character $w\cdot \chi$ in the same $S_n$-orbit gives an isomorphic principal series module $I(w\cdot \chi )\cong I(\chi )$, hence an isomorphic endomorphism algebra. The transported basis is not asserted to coincide with the raw ambient-length basis $\{B_v:v\in W_{\chi }\}$: it carries transported lengths and the $\rho$-normalization, as recorded in Length-additive products of the standard intertwiners.

Steps 1.1 and 2.1 identify ${\mathrm{End}}_G(I(\eta ))$ with $H_q(W_{\eta })$ by an explicit AC-free argument and give the Hecke basis $T_w={\lambda }_w{B}_w$; step 3.1 gives semisimplicity and the abstract isomorphism ${\mathrm{End}}_G(I(\chi ))\cong \mathbb{C}[W_{\chi }]$ preserving simple-module dimensions, and step 3.2 transports all of this to arbitrary $\chi$ and records orbit invariance. AC is needed only in the Tits-deformation conclusion of step 3.1, as declared; the identification with $H_q(W_{\chi })$ in steps 1.1 and 2.1 is choice-free.