Tilting modules for quantum groups

introduction

• modules for $U_q(\mathfrak{g})$
• Verma modules $M_{\lambda}=U_q(\mathfrak{g})\otimes_{U_q(\mathfrak{b})}\mathbb{C}_{\lambda}$
• Weyl modules : quotients of Verma modules

$$ W_{\lambda}=M_{\lambda}/\operatorname{span}(M_{s_i\cdot \lambda}) $$

• a tilting module is a module $T$ that admies a filtration whose associated graded pieces are Weyl modules and that admits another filtration whose associated graded are dual Weyl modules