THE BOREL-DE SIEBENTHAL’S THEOREM
P. GILLE, NOVEMBER 2010
This is the following.
0.1. Theorem. Let F be a field of characteristic 6= 2, 3. Let G/F be a
reductive group and let H/F be a reductive subgroup of maximal rank. Then
H = ZG(Z(H))0.
The original theorem  is about compact Lie groups and the specialists
know for a long time that is generalizes as stated. Our goal is to present
here a proof of that result. The following is well known in the case of subtori
0.2. Lemma. Let G/F be a reductive group and let T/F be a maximal torus.
Let S 1/2 T be a subgroup.
(1) The F–group ZG(S) is smooth and ZG(S)0 is reductive.
(2) If T is split, let U® be the root groups associated to ©(G, T). Then
ZG(S)0 is generated by T and the root subgroups U® for the ® 2
©(G, T) 1/2 T°Ë mapping to 0 in S°Ë.
Proof. (1) Since S is of multiplicative type, the group ZG(S) is smooth [4,
XI.5.2]. We show that G/k is reductive at the end of the proof.
(2) We can assume that F is algebraically closed. The F-group ZG(S)0
is smooth so is a linear algebraic group. The group ZG(S)0 is generated
by T and root subgroups U® of G [1, 13.20]. for ® running over R :=
©(G, T) ker(T°Ë ! S°Ë). Similarly, the radical M of ZG(S)0 is normalized
by T, hence it is generated by (M T)0 and the root subgroups U® of M.
We claim that M = (M T)0. For sake of contradiction, assume that U®
is a root subgroup of M. Its conjugate U−® in ZG(S)0 is then as well a
root subgroup of ZG(S)0, hence M contain a semisimple group of rank one,
which contradicts the solvability of M. Thus M is a torus and we conclude
that ZG(S)0 is reductive. °Ë
We first look at the behaviour of Theorem 0.1 under central extensions.
0.3. Lemma. Under the hypothesis of the theorem, let S be a central subgroup
(of multiplicative type) of G and denote by f : G ! G/S the quotient
(1) S 1/2 H and H/S is a reductive subgroup of maximal rank of G/S.
(2) If H/S = ZG/S(Z(H/S))0, then H = ZG(Z(H))0.
Proof. (1) follows from the fact that the center of G is included in all maximal
tori of G.
2 P. GILLE, NOVEMBER 2010
(2) We have an exact sequence of algebraic groups
1 ! S ! ZG(Z(H)) ! ZG/S(Z(H/S))
Since S 1/2 H 1/2 ZG(Z(H))0, we have the following exact diagram
1 −−−−! S −−−−! ZG(Z(H))0 −−−−! ZG/S(Z(H/S))0
|| [ [
1 −−−−! S −−−−! H −−−−! H/S −−−−! 1.
If H/S = ZG/S(Z(H/S))0, it follows by diagram chase that H = ZG(Z(H))0.
We can now proceed to the proof of Theorem 0.1.
Proof. Reduction to the case H semisimple: Let S = Z(H)0 be the connected
center of H Then H 1/2 ZG(S) and H/S is a semisimple subgroup
of G/S. If the result is known in the semisimple case, we have H/S =
ZZG(S)/S(Z(H/S))0. Lemma 0.3 shows that H = ZZG(S)(Z(H))0, hence
H = ZG(Z(H))0.
Furthermore Lemma 0.3 applied to Z(G) permits to assume that G is
semisimple adjoint. We can assume moreover that k is algebraically closed.
We consider a maximal (split) torus T of H.
Case H maximal proper semisimple group of G: We choose compatible orderings
on the roots systems ©(H, T) ( ©(G, T) = T°Ë. Since F is of characteristic
6= 2, 6= 3, ©(H, T) is a closed subsystem of ©(G, T) [4, XXIII.6.6].
Let A be the root lattice of H, i.e. the sublattice of T°Ë generated by ©(H, T).
The center Z(H) of H is a diagonalisable group whose character group is
Z(H)°Ë = T°Ë/A.
To show that H = ZG(Z(H))0, we note first that ZG(Z(H))0 is reductive
by Lemma 0.2. Since ZG(Z(H))0 contains the semisimple group H,
ZG(Z(H))0 is necessarily semisimple. We claim that Z(H) 6= 1. For sake of
contradiction, assume that A = T°Ë, i.e. that H is adjoint. Then our basis
for ©(H, T) would be a basis for ©(G, T), contradiction. So Z(H) 6= 1 and
ZG(Z(H))0 is a proper subgroup of G. Since H is a maximal semisimple
subgroup, we conclude that H = ZG(Z(H))0.
General case: By dimension reasons, there is a chain of semisimple groups
H = H0 ( H1 · · · ( Hn−1 ( Hn = G
such that Hi is maximal in Hi+1 for i = 0, …, n − 1. By induction on n, we
can assume that
THE BOREL-DE SIEBENTHAL’S THEOREM 3
H = ZHn−1(Z(H))0
= ZG(Z(H))0 [ Z(Hn−1) 1/2 Z(H) ].
Thus H = ZG(Z(H))0 as desired. °Ë
Acknowledgements. We thank Skip Garibaldi, Simon P´epin-Lehalleur
and Gopal Prasad for their useful comments.
 A. Borel, Linear Algebraic Groups (Second enlarged edition), Graduate text in Mathematics
126 (1991), Springer.
 A. Borel et J. de Siebenthal, Les sous-groupes fermées de rang maximum des groupes
de Lie clos, Commentarii Math. Helv. 22 (1949), 200-221.
 T.A. Springer, Linear Algebraic Groups, second edition (1998), Birk¨auser.
 Séminaire de G´eom´etrie alg´ebrique de l’I.H.E.S., 1963-1964, sch´emas en groupes,
dirig´e par M. Demazure et A. Grothendieck, Lecture Notes in Math. 151-153. Springer
UMR 8552 du CNRS, DMA, Ecole Normale Sup´erieure, F-75005 Paris, France