Wound algebraic groups and their compactifications

Prof. Philippe Gille (Université Claude Bernard, Lyon 1)
Wound algebraic groups and their compactifications

Abstract
This serie of lectures deals with algebraic groups defined over an arbitrary field k [4]. We will begin by revisting basics of the theory, e.g. Weil restriction, quotients,... An algebraic k-group G is anisotropic (resp. wound) if it does not carry any k–subgroup isomorphic to G_m (resp. G_m or G_a).

If G is reductive, Borel and Tits have shown that the two notions coincide; furthermore if k is perfect this is equivalent for G to admit a projective compactification G^c such that G(k) = G^c(k) [1]. A related (equivalent) condition is that G(k[[t]])=G(k((t))) and this is Prasad’s viewpoint on the result [8]. We are inter-

ested in the generalization of that statement in the following two directions.

1. The case of an imperfect field. This includes unipotent groups [2] and pseudo-reductive groups [3]. The main result is Gabber’s compactification theorem [6] constructing for an arbitrary G a G×G-equivariant compactification G^c such that for any separable extension F/k we have G(F) = G^c(F) if and only if G_F is wound.
2. Group schemes over a ring A. In the paper [5] we extended the notion of wound group schemes in that setting and this does not involve classification results. More precisely we defined a notion of index and residue for an element in G(A((t)))\ G(A[[t]]) which connects those elements with subgroup schemes isomorphic to G_a,A or G_m,A. In the case of a reductive group G over a field k it provides a kind of stratification of G(k((t)) related with the theory of affine grassmannians.

References

1. A. Borel, J. Tits, Groupes réductifs, Publications Mathématiques de l’IHÉS 27 (1965), 55-151.
2. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete 21 (1990), Springer.
3. B. Conrad, O. Gabber, G. Prasad, Pseudo-reductive groups, Cambridge University Press, second edition (2016).
4. M. Demazure, P. Gabriel, Groupes algébriques, North-Holland (1970).
5. M. Florence, P. Gille, Residues on affine grassmannians, Journal für die reine und angewandte Mathematik 776 (2021), 119-150.
6. O. Gabber, On pseudo-reductive groups and compactification theorems, Oberwolfach Reports (2012), 2371-2374.
7. O. Gabber, P. Gille, L. Moret-Bailly, Fibrés principaux sur les corps henséliens, Algebraic Geometry 5 (2014), 573-612.
8. G. Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France 110 (1982), 197–202.