Cyclotomic KLR functors define an integrable quantum action on $K_0$ and $G_0$

The two normalizations agree with the actions. The normalized-action supplier [F1] gives, for $E_i$ applied to input root $\beta +{\alpha }_i$, the exponent $-1-h_i(\beta +{\alpha }_i)=1-h_i(\beta )$, exactly the target-weight normalization; the same computation with the factor on $F$ gives the Kang--Kashiwara normalization. Both satisfy the mixed relations, the full Serre families and local nilpotence, and $K_i{\mathrm{\mid }}_{\beta }=q_i^{h_i(\beta )}$ in all cases; hence all four displayed operator families are integrable weight module structures after scalar extension to $\mathbb{Q}(q)$, with no assertion of finite-dimensional full-$\mathfrak{g}$ submodules in affine or indefinite type.

Bilinearity and perfectness of $T$. Tensor products of finite modules are finite, so $T$ has Laurent-polynomial values. The quotient-cover lemma [F2] gives absolutely simple normalized heads and $T(P(S),S^{\mathrm{\prime }})={\delta }_{S,S^{\mathrm{\prime }}}$, proving weightwise perfectness in normalized bases; shifting either tensor factor multiplies by the same power of $q$, which is exactly $A$-bilinearity. Finite-projective evaluation identifies $\mathrm{HOM}(P,M)$ with $(P^{\mathrm{\#}}{)}^{\psi }\otimes M$, so $h(P,M)=T(P^{\mathrm{\#}},M)$, which is first-bar-sesquilinear rather than bilinear.

Adjointness of the two normalized actions. For the corner $A_{\beta }\to e{B}_{\beta +i}e$, associativity of the balanced tensor over the corner gives $T_B(F^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}P,M)=T_A(P,E^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}M)$: both sides are the dimension of the same balanced tensor. The generator-fixing anti-involution flips balanced tensors, so $T_B(P,F^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}M)=T_A(E^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{c}}P,M)$ as well, the corner factors becoming $eB$ and $Be$ in opposite order with multiplication giving the evaluation isomorphism. Multiplying the second identity by $q_i^{1-h_i(\beta )}$ gives the first normalized adjoint statement, and the first identity gives the other; neither the Frobenius theorem nor the highest-weight $K_0$ theorem is used.

The defect rescaling. Adding an $i$ changes the defect by $d_i(h_i(\beta )-1)$, so the two module actions of step 1.1 are conjugate by the weight rescaling $q^{{\mathrm{def}}_{\mathrm{\Lambda }}(\beta )}$ with ${\mathrm{def}}_{\mathrm{\Lambda }}(\beta )=(\mathrm{\Lambda },\beta )-(\beta ,\beta )\mathrm{/}2$; the same computation applies on projective $K_0$ with the roles of $E$ and $F$ exchanged. Hence the canonical and Kang--Kashiwara normalizations define the same quantum structure up to the stated conjugation, completing the proof.