Celebratio Mathematica — Atiyah

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M. Atiyah and W. Schmid: “A geometric construction of the discrete series for semisimple Lie groups,” pp. 317–378 in Harmonic analysis and representations of semisimple Lie groups (Liège, 5–17 September 1977). Edited by J. A. Wolf, M. Cahen, and M. de Wilde. Mathematical Physics and Applied Mathematics 5. D. Reidel (Dordrecht), 1980. Republished from Invent. Math. 42:1 (1977). Zbl 0466.22012 incollection

Joseph Albert Wolf	Related
Wilfried Schmid	Related
Michel Cahen	Related
M. de Wilde	Related

M. Atiyah and W. Schmid: “Erratum to the article ‘A geometric construction of the discrete series for semisimple Lie groups’” in Harmonic analysis and representations of semisimple Lie groups (Liège, 5–17 September 1977). Edited by J. A. Wolf, M. Cahen, and M. de Wilde. Mathematical Physics and Applied Mathematics 5. D. Reidel (Dordrecht), 1980. Erratum for article in same volume. Republished from Invent. Math. 54:2 (1979). Zbl 0466.22013 incollection

In the above paper [Atiyah and Schmid 1977] a key role is played by a result of Borel [1963], concerning discrete subgroups \( \Gamma \) of semisimple Lie groups \( G \). He proves that if \( G \) is linear, one can find a torsion-free \( \Gamma \) with \( \Gamma\backslash G \) compact. Unfortunately we applied this result in [Atiyah and Schmid 1977] even for non-linear \( G \), in which case the existence of such \( \Gamma \) is seriously in doubt, as pointed out to us by P. Deligne and J.-P. Serre. The difficulty is that a torsion-free subgroup of the adjoint group lifts to a cocompact subgroup \( \Gamma \subset G \) which contains the (finite) center \( Z \) of \( G \), and there may be an obstruction to removing this torsion subgroup. As it stands, [Atiyah and Schmid 1977] is correct only for linear \( G \), and we shall now indicate how to extend the proof to cover all \( G \).

Joseph Albert Wolf	Related
Wilfried Schmid	Related
Michel Cahen	Related
M. de Wilde	Related