Highest weight modules for algebraic groups arising from quantum groups
Let \(L_p(\lambda)\) be the irreducible module for a quantum group \(U_p(\mathfrak g)\) at a \(p\)-th root of \(1\). Lusztig constructed a module \(\overline L_p(\lambda)\) for the corresponding simply connected semisimple algebraic group \(G\) of the same root system as the semisimple Lie algebra \(\mathfrak g\), and conjectured that \(\overline L_p(\lambda)\) is irreducible when \(\lambda\) is a \(p\)-restricted weight. This conjecture has been proved by \textit{H. H. Andersen, J. C. Jantzen} and \textit{W. Soergel} [Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): Independence of \(p\), Astérisque 220 (1994; Zbl 0802.17009)]. In this paper, the author follows Lusztig's conjecture to get a class of highest weight modules \(\overline L_{p^r}(\lambda)\) for the algebraic group \(G\) from invariant lattices in the irreducible modules \(L_{p^r}(\lambda)\) for the quantum group \(U_\xi (\mathfrak g)\) at a \(p^r\)-th root of \(1\). Assuming the above Lusztig conjecture at \(p\)-th roots of \(1\), the main result of this paper shows that, under certain generic conditions on \(\lambda\) in the bottom \(p^3\) alcove, \[ [\overline L_p(\lambda):L(\mu)]=[V_p(\lambda):L_p(\mu)], \] where \(V_p(\lambda)\) is the quantum Weyl module at \(p\)-th root of \(1\). The right hand side can be computed using Lusztig's quantum conjecture, which has been proved by Kashiwara and Tanisaki combined with the work of Kazhdan and Lusztig. For larger \(r\), a recursive formula is also provided.
scientific article; zbMATH DE number 1221664
scientific article