(a) is a restatement of (2.4). (b) Since the ${\stackrel{\sim }{H}}_k$ action and the $U_h{𝔤𝔩}_n$ action commute on $L\left(\mu \right)\otimes V^{\otimes k}$ it follows that the decomposition in (3.11) is a decomposition as $(U_h{𝔤𝔩}_n,{\stackrel{\sim }{H}}_k)$ bimodules, where the ${\stackrel{\sim }{H}}_k^{\lambda /\mu }$ are some ${\stackrel{\sim }{H}}_k\text{-modules.}$ Comparing the $L\left(\lambda \right)$ components on each side of $$\begin{array}{ccc}\underset{\lambda }{⨁}L\left(\lambda \right)\otimes {\stackrel{\sim }{H}}_{\ell }^{\lambda /\mu }& \cong & L\left(\mu \right)\otimes V^{\otimes \ell }=L\left(\mu \right)\otimes V^{\otimes (\ell -1)}\otimes V\cong (\underset{\nu }{⨁}L\left(\nu \right)\otimes {\stackrel{\sim }{H}}_{\ell -1}^{\nu /\mu })\otimes V\\ & \cong & \underset{\lambda }{⨁}\underset{\lambda /\nu =▫}{⨁}L\left(\nu \right)\otimes {\stackrel{\sim }{H}}_{\ell -1}^{\nu /\mu }\cong \underset{\lambda }{⨁}(L\left(\lambda \right)\otimes \left(\underset{\lambda /\nu =▫}{⨁}{\stackrel{\sim }{H}}_{\ell -1}^{\nu /\mu }\right))\end{array}$$ gives $$\begin{array}{cc}{\stackrel{\sim }{H}}_{\ell }^{\lambda /\mu }\cong \underset{\lambda /\nu =▫}{⨁}{\stackrel{\sim }{H}}_{\ell -1}^{\nu /\mu },& \text{(4.2)}\end{array}$$ for any $\ell \in {\mathbb{Z}}_{\ge 0}$ and skew shape $\lambda /\mu$ with $\ell$ boxes. Iterate (4.2) (with $\ell =k,k-1,\dots \text{)}$ to produce a decomposition $${\stackrel{\sim }{H}}_k^{\lambda /\mu }=\underset{T\in {\hat{H}}_k^{\lambda /\mu }}{⨁}{\stackrel{\sim }{H}}_1^T,$$ where the summands ${\stackrel{\sim }{H}}_1^T$ are 1-dimensional vector spaces. This determines a basis (unique up to multiplication of the basis vectors by constants) $\left\{v_T\hspace{0.17em}\right|\hspace{0.17em}T\in {\hat{H}}_k^{\lambda /\mu }\}$ of ${\stackrel{\sim }{H}}_k^{\lambda /\mu }$ which respects the decompositions in (4.2) for $1\le \ell \le k\text{.}$

Combining (3.1), (3.2) and (3.4) gives that $X^{{\epsilon }_i}$ acts on the $L\left(\lambda \right)$ component of the decomposition (3.10) by the constant $$q^{⟨\lambda ,\lambda +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩-⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩}=q^{2c(\lambda /\nu )}$$ since if $\lambda =\nu +{\epsilon }_j,$ so that $\lambda$ is the same as $\nu$ except with an additional box in row $j,$ then $\nu \subseteq \lambda ,$ $\lambda /\nu =▫$ and $$\begin{array}{ccc}⟨\lambda ,\lambda +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩-⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩& =& ⟨\nu +{\epsilon }_j,\nu +{\epsilon }_j+2\rho ⟩-⟨\nu ,\nu +2\rho ⟩-(1+2(n-1))\\ & =& 2{\nu }_j+⟨{\epsilon }_j,{\epsilon }_j+2\rho ⟩-2n+1=2{\nu }_j+(1+2(n-j))-2n+1\\ & =& 2({\nu }_j+1)-2j=2c(\lambda /\nu )\text{.}\end{array}$$ Hence, $$X^{{\epsilon }_i}{v}_T=q^{2c\left(T\left(i\right)\right)}{v}_T,\text{for}\hspace{0.17em}1\le i\le k,$$ where $T\left(i\right)$ is the box containing $i$ in $T\text{.}$

The remainder of the proof, including the simplicity of the ${\stackrel{\sim }{H}}_k\text{-modules}$ ${\stackrel{\sim }{H}}_k^{\lambda /\mu },$ is accomplished as in [Ram0401326, Thm. 4.1].

(c) The element $X^{{\epsilon }_1}\cdots X^{{\epsilon }_k}$ is central in ${\stackrel{\sim }{ℬ}}_k$ (it is a full twist) and hence its image is central in ${\stackrel{\sim }{H}}_k\text{.}$ The constant describing its action on ${\stackrel{\sim }{H}}_k^{\lambda /\mu }$ follows from the formula $X^{{\epsilon }_i}{v}_T=q^{2c\left(T\left(i\right)\right)}{v}_T\text{.}$

$\square$

(a) The elements $X^{{\epsilon }_i}$ commute with one another in the affine Hecke algebra (see (2.4) and the $m_j$ are by definition linear combinations of the $X^{{\epsilon }_i}$ (see 2.14), so they commute.

(b) Let $p$ be a path to $\lambda /\mu$ in ${\hat{T}}^{\mu }$ and let $T$ be the corresponding standard tableau on 2 rows. If $p^{\left(i\right)}=p^{(i-1)}-1=p^{(i-2)}-2$ or if $p^{\left(i\right)}=p^{(i-1)}+1=p^{(i-2)}+2$ then $c\left(T(i-1)\right)=c\left(T\left(i\right)\right)-1$ and, from (2.14) and Theorem 4.1(b), $$m_i{v}_T=q^{i-2}\frac{q^{-2c\left(T\left(i\right)\right)}-q^{-2}{q}^{-2c\left(T(i-1)\right)}}{q-q^{-1}}{v}_T=q^{i-2}\frac{q^{-2c\left(T\left(i\right)\right)}-q^{-2}{q}^{-2c\left(T\left(i\right)\right)+2}}{q-q^{-1}}{v}_T=0\text{.}$$ If $p^{\left(i\right)}=p^{(i-2)}=p^{(i-1)}-1$ with $T^{(i-1)}=(a,b)$ then $c\left(T\left(i\right)\right)=a$ and $c\left(T(i-1)\right)=b-2$ and $$m_i{v}_T=q^{i-2}\frac{q^{-2a}-q^{-2}{q}^{-2b+4}}{q-q^{-1}}{v}_T=q^i\frac{q^{-(a+b+1)}(q^{-(a-b+1)}-q^{(a-b+1)})}{q-q^{-1}}=-q^{-m}[a-b+1]v_T,$$ where $m=\left|\mu \right|=a+b-i+1\text{.}$ If $p^{\left(i\right)}=p^{(i-2)}=p^{(i-1)}+1$ with $T^{(i-1)}=(a,b)$ then $c\left(T(i-1)\right)=a-1$ and $c\left(T\left(i\right)\right)=b-1$ and $$m_i{v}_T=q^{i-2}\frac{q^{-2b+2}-q^{-2}{q}^{-2a+2}}{q-q^{-1}}{v}_T=q^i\frac{q^{-(a+b+1)}(q^{(a-b+1)}-q^{-(a-b+1)})}{q-q^{-1}}=q^{-m}[a-b+1]v_T,$$ where $m=\left|\mu \right|=a+b-i+1\text{.}$

(c) Let $k=|\lambda /\mu |\text{.}$ The identity $$q^{{\lambda }_1+{\lambda }_2-2}\sum _{b\in \lambda /\mu }{q}^{-2c\left(b\right)}=\left(\sum _{i={\mu }_2}^{{\lambda }_2-1}[{\lambda }_1+{\lambda }_2-2i](q-q^{-1})\right)+\left[k\right]q^{{\mu }_2-{\mu }_1+1},$$ is best visible in an example: With $\lambda =(10,6)$ and $\mu =(4,2),$ $$\begin{array}{c}{q}^{16-2}\left(\begin{array}{cccccccccc}\phantom{+}0& +0& +0& +0& +q^{-8}& +q^{-10}& +q^{-12}& +q^{-14}& +q^{-16}& +q^{-18}\\ +0& +0& +q^{-2}& +q^{-4}& +q^{-6}& +q^{-8}\end{array}\right)\\ =\begin{array}{cccccccccc}\phantom{+}0& +0& +0& +0& +q^6& +q^4& +q^2& +q^{-}& +q^{-2}& +q^{-4}\\ +0& +0& +q^{12}& +q^{10}& +q^8& +q^6\end{array}\\ \left(\begin{array}{cccccccccc}\phantom{+}0& +0& +0& +0& +0& +0& +0& +0& +0& +0\\ +0& +0& +(q^{12}-q^{-12})& +(q^{10}-q^{-10})& +(q^8-q^{-8})& +(q^6-q^{-6})\end{array}\right)\\ +\left(\begin{array}{cccccccccc}\phantom{+}0& +0& +0& +0& +q^{-4}& +q^{-2}& +q^0& +q^2& +q^4& +q^6\\ +0& +0& +q^{-12}& +q^{-10}& +q^{-8}& +q^{-6}\end{array}\right)\\ =(\sum _{i=2}^{6-1}{q}^{16-2i}-q^{-(16-2i)})+\left[10\right]q^{4-2+1}\text{.}\end{array}$$ Then Proposition 2.6 says $$X^{-{\epsilon }_1}+\cdots +X^{-{\epsilon }_k}=q^{-(k-2)}(q-q^{-1})(m_k+\left[2\right]m_{k-1}+\cdots +\left[k\right]m_1),$$ and so $m_k+\left[2\right]m_{k-1}+\cdots +\left[k\right]m_1$ acts on $T_k^{\lambda /\mu }$ by the constant $$\begin{array}{ccc}& \multicolumn{2}{c}{{(q-q^{-1})}^{-1}{q}^{k-2}\sum _{b\in \lambda /\mu }{q}^{-2c\left(b\right)}}\\ & =& {(q-q^{-1})}^{-1}{q}^{-({\mu }_1+{\mu }_2)}{q}^{{\lambda }_1+{\lambda }_2-2}\sum _{b\in \lambda /\mu }{q}^{-2c\left(b\right)}\\ & =& {(q-q^{-1})}^{-1}{q}^{-({\mu }_1+{\mu }_2)}(\left[k\right]q^{{\mu }_2-{\mu }_1+1}+\sum _{i={\mu }_2}^{{\lambda }_2-1}[{\lambda }_1+{\lambda }_2-2i](q-q^{-1}))\\ & =& \left[k\right]\frac{q^{-m-p^{\left(0\right)}+1}}{q-q^{-1}}+\sum _{i={\mu }_2}^{{\lambda }_2-1}{q}^{-m}[m+k-2i]\\ & =& \left[k\right]\frac{q^{-m-p^{\left(0\right)}+1}}{q-q^{-1}}+q^{-m}([p^{\left(k\right)}+2]+[p^{\left(k\right)}+4]+\cdots +[p^{\left(0\right)}+k-2]+[p^{\left(0\right)}+k]),\end{array}$$ since ${\mu }_1+{\mu }_2=m,$ ${\mu }_1-{\mu }_2=p^{\left(0\right)},$ ${\lambda }_1+{\lambda }_2=m+k$ and ${\lambda }_1-{\lambda }_2=p^{\left(k\right)}\text{.}$

$\square$