A tensor product theorem for quantum linear groups at even roots of unity (Q1327058)

The author proves a tensor product theorem for \(\text{GL}_ q(n)\), where \(q\) is a primitive \(k\)-th root of unity, \(k\) even (over a field). If \(k\) is divisible by 4, the result is that \[ L^ q \Bigl(\lambda_ 0 +\bigl( {\textstyle {k\over 2}} \bigr) \lambda_ 1 \Bigl)\cong L^ q (\lambda_ 0)+ L(\lambda_ 1 )^{(1)}, \] where \(L^ q(\lambda)\) is the irreducible \(\text{GL}_ q(n)\)-module of highest weight \(\lambda\), and \(L(\lambda_ 1 )^{(1)}\) is the \(\text{GL}_ q (n)\)-module given by the Frobenius \(k'= k/2\) twist of the irreducible module \(L(\lambda_ 1)\) for the non-quantized \(\text{GL}(n)\) of highest weight \(\lambda_ 1\), if \(\lambda_ 0\) is \(k'\)-restricted. For \(k\) odd, an analogous result was proved by \textit{B. Parshall} and \textit{J. Wang} [Mem. Am. Math. Soc. 439 (1991; Zbl 0724.17011)], except that they used \(k\) in the Frobenius twist. The proof here in the even case is different, since the \(X_{ij}^{k'}\) are not central in \(\text{GL}_ q (n)\). If \(k\) is even but not divisible by 4, there is a similar result, with \(L(\lambda_ 1 )^{(1)}\) replaced by the Frobenius \(k'\)-twist \(L^{-1} (\lambda_ 1 )^{(1)}\) of the irreducible module \(L^{-1} (\lambda_ 1)\) for \(\text{GL}_{-1} (n)\).