Prehomogeneous vector spaces and ergodic theory. III (Q1267303)

[Part I, Duke Math. J. 90, 123--147 (1997; Zbl 0896.11013); Part II, with \textit{D. Witte, R. Zierau}, Trans. Am. Math. Soc. 352, No. 4, 1687--1708 (2000; Zbl 0994.11040).] Let \(k\) be a field of characteristic zero and \((G,V,X)\) be a prehomogeneous vector space. The theory of prehomogeneous vector spaces was initiated by \textit{M. Sato} and \textit{T. Shintani} [Ann. Math. (2) 100, 131--170 (1974; Zbl 0309.10014)] and \textit{T. Shintani} [J. Math. Soc. Japan 24, 132--188 (1972; Zbl 0227.10031)]. Here the prehomogeneous vector space considered is \(G= \text{GL}(5) \times \text{GL}(3)\), \(V= \Lambda^2k^5 \otimes k^3\). This produces a family of irrational cubic forms in five variables whose values at integer points are dense in \(\mathbb R\). Let \(H_1= \text{SL}(5)\), \(H_2= \text{SL}(3)\), \(H=H_1 \times H_2\) and \(\Delta (x)\) a relative invariant polynomial on \(V\). Define \(V^{ss}= \{x\in V\mid \Delta(x) \neq 0\}\). For \(x\in V_\mathbb R^{ss}\), let \(H^\circ_{x \mathbb R+}\) be the identity component in classical topology of the stabilizer \(H_{x\mathbb R}\). It is proved in this paper that if \(x\in V_\mathbb R^{ss}\) is sufficiently irrational then \(H^\circ_{x\mathbb R+} H_\mathbb Z\) is dense in \(H_\mathbb R\). The method is based on Raghunathan's topological conjecture proved by Ratner.