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Quantum Groups at root of unity

All the papers or books I read so far on quantum groups about Lusztig’s restricted specialization consider only primitive roots of unity of odd order and additional conditions. In most cases, it is claimed that these restrictions could be removed without too much harm but details are not given. So I have tried to do the calculation myself. Below is what I find for the finite dimensional representations of Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g}). Note that most authors consider the case l=m=mil=m=m_{i} odd. Then -1mi=ϵimi=1{(-1)}^{{m_{i}}}=\epsilon _{i}^{{m_{i}}}=1 and we can even restrict the study to the representations for which σ\sigma is always 1. Indeed, all these assumptions make the expression below much simpler.

We consider g\mathfrak{g} a simple Lie algebra of rank nn and use standard notations αi\alpha _{i} for roots, αi\alpha _{i}^{\vee} for coroots and PgP(\mathfrak{g}) for the weight space. Let l>2l>2 be an integer and ϵ\epsilon a primitive ll-root of the unity. We let m=l2m=\frac{l}{2} if ll is even and m=lm=l otherwise. For all 1in1\leq i\leq n, define ϵi=ϵdi\epsilon _{i}=\epsilon^{{d_{i}}}, δi=gcddi,m\delta _{i}=\gcd(d_{i},m), and mi=mδim_{i}=\frac{m}{\delta _{i}}. We assume that no did_{i} is a multiple of mm, which is obviously true for ll large enough.

We denote Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g}) the restricted specialization as defined in Chari and Pressley’s guide to quantum groups and keep their notations for the elements KiK_{i}, Ki;crϵi{\genfrac{[}{]}{0.0pt}{0}{{K_{i}};{c}}{r}}_{{\epsilon _{i}}}, Xi±X_{i}^{{\pm}} and Xi±r{(X_{i}^{\pm})}^{{(r)}} of Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g}). Finally, we let VV be a finite Uϵresg{U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g})-module.

For any λPg\lambda\in P(\mathfrak{g}) and σ-1,1n\sigma\in{\{-1,1\}}^{n} define

Vσ,λ=1inKerKi-σiϵiλαi1KerKi;0miϵi-Δiλλαimi1V_{{\sigma,\lambda}}=\bigcap _{{1\leq i\leq n}}{\operatorname{Ker}{\left(K_{i}-\sigma _{i}\epsilon _{i}^{{\lambda(\alpha _{i}^{\vee})}}1\right)}\cap\operatorname{Ker}{{\left({\genfrac{[}{]}{0.0pt}{0}{{K_{i}};{0}}{m_{i}}}_{{\epsilon _{i}}}-\Delta _{i}(\lambda)\left\lfloor\frac{\lambda(\alpha _{i}^{\vee})}{m_{i}}\right\rfloor 1\right)}}}

where i,Δiλ=-1mi+1σimiϵimiλαi+1\forall i,\Delta _{i}(\lambda)={(-1)}^{{m_{i}+1}}\sigma _{i}^{{m_{i}}}{(\epsilon _{i}^{{m_{i}}})}^{{\lambda(\alpha _{i}^{\vee})+1}}

For any 1jn1\leq j\leq n, we have

Xj±VσλVσ,λ±αjX_{j}^{\pm}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm\alpha _{j}}}
Xj±mjVσλVσ,λ±mjαj{(X_{j}^{\pm})}^{{(m_{j})}}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm m_{j}\alpha _{j}}}

Moreover if VV is simple, V=σ,λVσ,λV=\bigoplus _{{\sigma,\lambda}}V_{{\sigma,\lambda}}.