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Weil's conjecture, proved by Kottwitz, states that the Tamagawa number of a semisimple, simply connected algebraic group (over a number field) is 1. For example, $SL(n)$ and induced tori. Is the Tamagawa number of $GL(n)$ known?

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    $\begingroup$ Have you looked at: Tamagawa number of reductive algebraic groups. Lai, K. F. Compositio Mathematica, Volume 41 (1980) no. 2, p. 153-188? $\endgroup$
    – user130903
    Commented Nov 20, 2018 at 11:04
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    $\begingroup$ I think the answer is just 1. Since $GL_n = SL_n \times \mathbb{G}_m$, $\tau(GL_n)=\tau(SL_n) \tau(\mathbb{G}_m)=1 \times 1 = 1$. $\endgroup$
    – Not a grad student
    Commented Nov 20, 2018 at 21:38
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    $\begingroup$ There is quite a range of literature by now on Tamagawa numbers, including Weil's informal lectures published more formally as a 1982 volume in the PM series Adeles and Algebraic Groups. See especially Chapter III in the long paper by J. Oesterle in Invent. Math. 76 (`984), which confirms that the Tamagawa number of a general linear group is 1 (over a relevant field). Note by the way that Kottwitz finished off the Tamagawa conjejcture by dealing with a tricky exceptional type. Most of the literature is aimed at such cases: simply connected semisimple groups over number fields. $\endgroup$
    – Jim Humphreys
    Commented Nov 20, 2018 at 22:31
  • $\begingroup$ General linear groups are certainly connected. How do you define this notion? $\endgroup$
    – Jim Humphreys
    Commented Nov 22, 2018 at 23:56

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