On modular invariants of a vector and a covector

Let GL ₂(F _q ) be the general linear group over a finite field F _q , V be the 2-dimensional natural representation of GL ₂(F _q ) and V ^* be the dual representation. We denote by \({F_{q}V\oplus V^{\ast}]^{GL_{2}(F_{q})}}\) the corresponding invariant ring of a vector and a covector for GL ₂(F _q ). In this paper, we prove that \({F_{q}V\oplus V^{\ast}]^{GL_{2}(F_{q})}}\) is a Gorenstein algebra. This result confirms a special case (n = 2) of the recent conjecture of Bonnafé and Kemper (J Algebra 335:96–112, 2011).