Vector valued invariants of prehomogeneous vector spaces (Q914842)

Let G be a reductive group defined over a finite field \({\mathbb{F}}_ q\), and V a finite dimensional rational G-module which has a Zariski open G- orbit \(\Omega\). Let \(R: G({\mathbb{F}}_ q)\to GL(M)\) be a complex irreducible representation. Consider an M-valued function f on \(\Omega ({\mathbb{F}}_ q)\) such that \(f(gv)=R(g)f(v)\) for \(g\in G({\mathbb{F}}_ q)\) and \(v\in \Omega ({\mathbb{F}}_ q)\), and extend it by zero. In this paper, the Fourier transforms of such functions f are calculated for every irreducible representation R of \(G({\mathbb{F}}_ q)\) when (G,V) is (i) \((GL_ n\times GL_ n,\square \otimes \square)\) (split \({\mathbb{F}}_ q\)- form), or (ii) \((GL_{2n},\theta)\). Using the result for the case (i) with a principal series representation R, the Fourier transform of the function \(\prod^{n}_{i=1}\theta_ i(P_ i(x))\) \((x\in M_ n({\mathbb{F}}_ q))\) is determined, where \(\theta_ i\in Hom({\mathbb{F}}_ q^{\times},{\mathbb{C}}^{\times})\), \(\theta_ i(0)=0\) and \(P_ i\) is the i-th principal minor. Similar result is obtained from the case (ii). Analogous problem over the real or complex number field is discussed by \textit{N. Bopp} and \textit{H. Rubenthaler} [C. R. Acad. Sci., Paris, Sér. I 310, 505-508 (1990; Zbl 0702.22015)].