Prehomogeneous vector spaces and ergodic theory III

1997/02/04 by Akihiko Yukie, Yukie, Akihiko
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Mathematical Dynamics and Fractals #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.math/9702220

arxiv created 1997/02/04 · openalex publication_date 1997/02/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H₁=SL(5), H₂=SL(3), H=H₁ × H₂. It is known that (G,V) is a prehomogeneous vector space (see [22], [26], [25], for the definition of prehomogeneous vector spaces). A non-constant polynomial δ(x) on V is called a relative invariant polynomial if there exists a character χsuch that δ(gx)=χ(g)δ(x). Such δ(x) exists for our case and is essentially unique. So we define V^ss=x in V such that δ(x) is not equal to 0. For x in V_R^ss, let H_{x R+}⁰ be the connected component of 1 in classical topology of the stabilizer H_{x R}. We will prove that if x in V_R^ss is "sufficiently irrational", H_{x R+}⁰ H_Z is dense in H_R.

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