Quantum Groups at root of unity

23 Jun 2012

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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 ${U_{\epsilon}^{{\mathrm{res}}}}(\mathfrak{g})$. Note that most authors consider the case $l=m=m_{i}$ odd. Then ${(-1)}^{{m_{i}}}=\epsilon _{i}^{{m_{i}}}=1$ and we can even restrict the study to the representations for which $\sigma$ is always 1. Indeed, all these assumptions make the expression below much simpler.

We consider $\mathfrak{g}$ a simple Lie algebra of rank $n$ and use standard notations $\alpha _{i}$ for roots, $\alpha _{i}^{\vee}$ for coroots and $P(\mathfrak{g})$ for the weight space. Let $l>2$ be an integer and $\epsilon$ a primitive $l$-root of the unity. We let $m=\frac{l}{2}$ if $l$ is even and $m=l$ otherwise. For all $1\leq i\leq n$, define $\epsilon _{i}=\epsilon^{{d_{i}}}$, $\delta _{i}=\gcd(d_{i},m)$, and $m_{i}=\frac{m}{\delta _{i}}$. We assume that no $d_{i}$ is a multiple of $m$, which is obviously true for $l$ large enough.

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

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

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

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

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

$$X_{j}^{\pm}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm\alpha _{j}}}$$

$${(X_{j}^{\pm})}^{{(m_{j})}}V_{{\sigma\lambda}}\subseteq V_{{\sigma,\lambda\pm m_{j}\alpha _{j}}}$$

Moreover if $V$ is simple, $V=\bigoplus _{{\sigma,\lambda}}V_{{\sigma,\lambda}}$.

$\square$