Wound algebraic groups and their compactifications


  • Philippe Gille
    Université Claude Bernard, Lyon 1


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

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 Gm (resp. Gm or Ga).

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 Gc such that G(k) = Gc(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 Gc such that for any separable extension F/k we have G(F) = Gc(F) if and only if GF 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 Ga,A or Gm,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.


  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.