A note on the mean value theorem for special homogeneous spaces.

*(English)*Zbl 0870.11021Let \(G\) be a connected linear algebraic group and \(X\) an algebraic variety, both defined over \(\mathbb{Q}\), the field of rational numbers. Suppose that \(G\) acts on \(X\) transitively and the action is defined over \(\mathbb{Q}\). Suppose that the set of rational points \(X(\mathbb{Q})\) is non-empty. Choosing \(x\in X(\mathbb{Q})\) allows us to identify \(G/G_x\) and \(X\) as varieties over \(\mathbb{Q}\), where \(G_x\) is the stabilizer of \(x\). We call \(G\) special if \(G\) has the Levi-Chevalley decomposition of the form \(G=US\) over \(\mathbb{Q}\), where \(U\) is the unipotent radical of \(G\) and \(S\) is the semisimple part. We also call a homogeneous space \((G,X)\) special if \(G\) and \(G_x\), \(x\in X(\mathbb{Q})\), are both special.

After the works of C. L. Siegel [Ann. Math., II. Ser. 46, 340-347 (1945; Zbl 0063.07011)] and A. Weil [Summa Brasil. Math. 1, 21-39 (1946; Zbl 0063.08199)], T. Ono [J. Math. Soc. Japan 20, 275-288 (1968; Zbl 0185.49101)] defined the uniformity of a special homogeneous space \((G,X)\) in the context of integral geometry over the adele spaces attached to \((G,X)\) and introduced the Tamagawa number \(\tau(G,X)\) for a special and uniform homogeneous space, and gave a criterion in terms of the homotopy groups of the complex manifold \(X(\mathbb{C})\) in order that \((G,X)\) satisfies the mean value property, \(\tau(G,X)=1\).

The purpose of this paper is to show that any special homogeneous space is uniform and has a simple description of its Tamagawa number. It follows from the observation that the fundamental group of an algebraic group does not change under inner twisting, and from the fundamental theorem of R. E. Kottwitz [Ann. Math., II. Ser. 127, 629-646 (1988; Zbl 0678.22012)] on the Tamagawa number of algebraic groups and that of V. I. Chernousov [Sov. Math., Dokl. 39, 592-596 (1989); translation from Dokl. Akad. Nauk SSSR 306, 1059-1063 (1989; Zbl 0703.20040)].

After the works of C. L. Siegel [Ann. Math., II. Ser. 46, 340-347 (1945; Zbl 0063.07011)] and A. Weil [Summa Brasil. Math. 1, 21-39 (1946; Zbl 0063.08199)], T. Ono [J. Math. Soc. Japan 20, 275-288 (1968; Zbl 0185.49101)] defined the uniformity of a special homogeneous space \((G,X)\) in the context of integral geometry over the adele spaces attached to \((G,X)\) and introduced the Tamagawa number \(\tau(G,X)\) for a special and uniform homogeneous space, and gave a criterion in terms of the homotopy groups of the complex manifold \(X(\mathbb{C})\) in order that \((G,X)\) satisfies the mean value property, \(\tau(G,X)=1\).

The purpose of this paper is to show that any special homogeneous space is uniform and has a simple description of its Tamagawa number. It follows from the observation that the fundamental group of an algebraic group does not change under inner twisting, and from the fundamental theorem of R. E. Kottwitz [Ann. Math., II. Ser. 127, 629-646 (1988; Zbl 0678.22012)] on the Tamagawa number of algebraic groups and that of V. I. Chernousov [Sov. Math., Dokl. 39, 592-596 (1989); translation from Dokl. Akad. Nauk SSSR 306, 1059-1063 (1989; Zbl 0703.20040)].

Reviewer: M.Morishita (Kanazawa)

##### MSC:

11E72 | Galois cohomology of linear algebraic groups |

20G10 | Cohomology theory for linear algebraic groups |

##### Keywords:

connected linear algebraic group; algebraic variety; uniformity; mean value property; special homogeneous space; Tamagawa number; fundamental group
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\textit{M. Morishita} and \textit{T. Watanabe}, Nagoya Math. J. 143, 111--117 (1996; Zbl 0870.11021)

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##### References:

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[3] | DOI: 10.2969/jmsj/02010275 · Zbl 0185.49101 |

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[6] | Ann. Math 49 pp 340– (1945) |

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[11] | Dokl. Akad. Nauk SSSR 306 pp 1059– (1989) |

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