Irreducibility of induced supermodules for general linear supergroups (Q1680276)

Let \(k\) a field of characteristic different from 2 and \(\mathrm{GL}(m| n)\) the general supergroup with even part \(G_{\text{ev}}\). Denote by \(B\) the Borel supergroup of \(G\) corresponding to lower triangular matrices. Let \(\lambda=(\lambda_1^+,\ldots, \lambda_m^+\mid \lambda_1^-,\ldots,\lambda_n^- )\) be a dominant integral weight of \(G\) and \(K_{\lambda}\) the one-dimensional even \(B\)-supermodule associated with \(\lambda\). The last one induces \(G\)-supermodule. \(H^0_G(K_{\lambda})\). Similarly \(H_{G_{\text{ev}}}^0(K_{\lambda})\) is the \(G_{\text{ev}}\)-supermodule induced from \(B\cap G_{\text{ev}}\)-supermodule \(K_{\lambda}\). The weight \(\lambda\) is typical if none of numbers \(\lambda^++\lambda_j^-+m+1-i-j\) is a multiple of the characteristic of \(k\). The main results of the paper states that \(H_G^0(K_{\lambda})\) is irreducible if and only if \(H_{G_{\text{ev}}}^0(K_{\lambda})\) is irreducible and \(\lambda\) is typical. As a corollary it is shown that the Weyl supermodule \(V_G(\lambda)\) is irreducible if and only if \(V_{g_{\text{ev}}}(\lambda)\) is irreducible and \(\lambda\) is typical. Moreover the Kac supermodule \(K_G(\lambda)\) is irreducible if and only if\(\lambda\) is typical. These results in the case of characteristic zero were obtained by V. Kac in 1977--1978.