Anisotropic Groups over Arbitrary Fields*

Ulf Rehmann Why anisotropic groups ?

1888/9 Killing classifies semisimple groups

and introduces the types

A_{n}, B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, F_{4}, G_{2} of semisimple Lie groups.

1961 Chevalley shows: These groups have a ℤ-structur, (“Chevalley groups”) Killing’s Classification holds over algebraically closed fields.

1965ff Borel and Tits describe the internal structure of these groups insomuch they contain unipotent elements, that is, up to their “anisotropic kernel”. This reduces the classification and structure theory to the investigation of anisotropic semisimple groups.

1984 T. A. Springer writes in a survey article over

linear algebraic groups:

The most difficult part of a classification of reductive k-groups is the classification of semi-simple anisotropic k-groups … A complete classification of all anisotropic k-groups seems out of reach. (Perspectives in Math., Anniv. of Oberwolfach, 1984, p. 477)

* Colloquium Talk, Regensburg, June 6., 1997

Anisotropic Groups over Arbitrary Fields

Anisotropic groups have, by definition, no proper parabolic subgroups.

Examples of anisotropic groups:

i) Compact semisimple Lie groups (alg. groups over ℝ),

ii) SL_{1}(D), D: finite dim. central simple k-division algebra, Killing type A_{n},

iii) SO(q), Spin(q), q: anisotropic quadratic form over k, Killing type B_{n}, D_{n},

iv) All Killing types have anisotropic “twists” over suitable fields.

Some methods to investigate anisotropic groups:

1. Galois cohomological methods

2. Splitting Patterns (with J. Hurrelbrink.)

3. Excellence properties of algebraic groups (with I. Kersten.)

Anisotropic Groups over Arbitrary Fields

1. Galois cohomological methods Notation:

k :  an arbitrary field,
k_{s} :  separable closure of k,
Γ = Gal(k_{s}/k) : its Galois group.
G :  semisimple linear alg. group über k.
H^{1}(k, G) :  poset of classes of 1-cocycles z : Γ → G(k_{s})

H^{1}(k, Aut(G)) classifies all k-twists of G, i.e., all groups G′ over k with G′(k_{s}) ≅ G(k_{s}).

A Galois cohomological invariant of G is a transformation

H^{1}(k, G) ⟶ H^{i}(k, C),

functorial in field extensions of k, with

C  = Torsion-Γ module
H^{i}(k, C) = H^{i}(Γ, C) (Galois cohomology)

This invariant is trivial if the cocycle of the split twist of G in H^{1}(k, G) is mapped to 1 ∈ H^{i}(k, C). (Def. needs to be modified in more complicated situations.)

Anisotropic Groups over Arbitrary Fields

Examples: (For simiplicity: let tor C and char k be coprime.)

\underline{i = 1} : G adjoint : C := Aut(Dynkin(G)) (finite!) G ↪ Aut(G) → → C exact ⇒∃ δ_{G} : H^{1}(k, Aut(G)) → H^{1}(k, C) the discriminant invariant

\underline{i = 2} : G inner, adjoint : G̃ s.c. covering, C := center(G̃(k_{s})) (finite!) C ↪ G̃ → → G exact ⇒∃ δ_{G}^{2} : H^{1}(k, G) → H^{2}(k, C) x ∈ Hom(C, 𝔾_{m}) gives x_{∗} : H^{2}(k, C) → H^{2}(k, 𝔾_{m}) = Br(k), the Brauer invariant is obtained by: β_{G} : C^{∗} := Hom(C, 𝔾_{m}) → Br(k), x ↦ x_{∗}δ_{G}^{2}(c), where c ∈ H^{1}(k, G) is the class of the split twist of G.

\underline{i = 3} : G = Spin(q) : ∃ α_{G} : H^{1}(k, G) → H^{3}(k, Z/2Z)

the Arason invariant \underline{i ≥ 4} : G = Spin(q) : ∃ ν_{G}^{i} : H^{1}(k, G) → H^{i}(k, Z/2Z)

(according to Voevodsky’s proof of the Milnor conjecture) these maps are only “partially defined”: ν_{G}^{3} = α_{G} and ν_{G}^{i} is defined on Ker ν_{G}^{i−1} for i > 3.

Anisotropic Groups over Arbitrary Fields

Brauer invariant of inner twists (i.e. δ_{G} trivial): C^{∗} ≅ Λ/Λ^{r} = ⟨dual roots⟩/⟨roots⟩. Group structure of C^{∗}:

A_{n}: ∘ ω  {∘ \atop 2ω} ⋯ {∘ \atop nω}  C^{∗} ≅ Z_{n+1} = ⟨ω⟩
B_{n}: ∘  ∘ ⋯ ∘ > ∘ ω  C^{∗} ≅ Z_{2} = ⟨ω⟩
C_{n}: ∘ ω  {∘ \atop 2ω} ⋯ {∘ \atop (n−1)ω}< {∘ \atop nω}  C^{∗} ≅ Z_{2} = ⟨ω⟩
D_{n}: {∘ \atop ω_{0}}  {∘ \atop 2ω_{0}} ⋯ {∘ \atop (n−2)ω_{0}}[figure: ∘ ω; ∘ ω′] \{ \begin{matrix}2|̸n : & C^{∗} ≅ Z_{4} = ⟨ω′ = −ω⟩ \\ 2|n : & C^{∗} ≅ Z_{2} × Z_{2} = ⟨ω⟩ × ⟨ω′⟩ \\ ω_{0} = ω − ω′ \end{matrix}

E_{6}: ∘ ω  {∘ \atop 2ω}  \begin{matrix}∘ \\ ∘ \end{matrix}  ∘ ω  {∘ \atop 2ω}  C^{∗} ≅ Z_{3} = ⟨ω⟩
E_{7}: ∘ ω  ∘  ∘ ω  ω ; \begin{matrix}∘ \\ ∘ \end{matrix}  ∘  ∘ C^{∗} ≅ Z_{2} = ⟨ω⟩

E_{8}, F_{4}, G_{2} have trivial C^{∗}. Meaning of β_{G}: The irreducible representation G_{k_{s}} → GL_{m}(k_{s}) with highest weight ω_{α} has k structure G → SL_{1}(A), where A is a central simple k algebra of class β_{G}(ω_{α}) in Br(k).

Anisotropic Groups over Arbitrary Fields

Remarks:

1. G is of \{ \begin{matrix}outer type, if δ_{G} not trivial \\ inner type, if δ_{G} trivial \end{matrix}

2. Tits (1990) defines:

G is of strongly inner type (SI), if β_{G}, δ_{G} trivial. Discovers (among other things) (over ℝ) the series B_{4m}, B_{4m+3}, D_{4m} of SI anisotropic groups.

3. complementary concept (UR):

G is of Brauer type (BT), if G_{L} splits for every field extension L/k, for which β_{G_{L}} is trivial.

4. Consequences:

i) groups of inner type A_{n}, C_{n} are always BT ii) over p-adic fields, inner groups are BT. iii) G_{2}, F_{4}, E_{8} are always SI, E_{6}, E_{7} mixed, and anisotropic

SI types exist iv) B_{n}, D_{n} are SI ⇔ the Clifford algebra is a matrix ring.

5. Theorem (UR). For every G, there is a generic Brauer splitting field K/k (i.e., β_{G_{K}} generically trivial). Question: Under which conditions is G_{K} anisotropic ? (“Anisotropic splitting”)

Anisotropic Groups over Arbitrary Fields

6. Let char k ≠ 2 and G of type B_{n}, D_{n}: An outer group G is said to be of discriminant type, if G_{L} splits over every field extension L/k, for which δ_{G_{L}} is trivial.

For orthogonal groups of excellent quadratic forms we have (UR, based on earlier work of M. Knebusch, I. Kersten/UR, J. Hurrelbrink/UR, B. Kahn):

Theorem. Let G = SO_{q} for some anisotropic excellent quadratic form q over k with char k ≠ 2.

i) Let dim q = 8m + ρ, where 0 ≤ ρ ≤ 7. Then one has the following table for the Killing and the twist type of G:

ρ : Killing type: Twist type:
0 D_{4m}  strongly inner
1 B_{4m}  strongly inner
2 D_{4m+1}  discriminant (outer)
3 B_{4m+1}  Brauer (inner)
4 D_{4m+2}  Brauer (inner)
5 B_{4m+2}  Brauer (inner)
6 D_{4m+3}  discriminant (outer)
7 B_{4m+3}  strongly inner.

ii) Let dim q = 8m + ρ, where 0 ≤ ρ ≤ 15. Then one has the following table for the Killing and the twist type of G:

ρ : Killing type: twist type: l.c.i.
0 D_{8m}  (α_{G} trivial, strongly inner) v_{2}(16m)
1 B_{8m}  (α_{G} trivial, strongly inner) v_{2}(16m)
2 D_{8m+1}  discriminant (outer)  1
3 B_{8m+1}  Brauer (inner)  2
4 D_{8m+2}  Brauer (inner)  2
5 B_{8m+2}  Brauer (inner)  2
6 D_{8m+3}  discriminant (outer)  1
7 B_{8m+3}  Arason (strongly inner)  3
8 D_{8m+4}  Arason (strongly inner)  3
9 B_{8m+4}  Arason (strongly inner)  3
10 D_{8m+5}  discriminant (outer)  1
11 B_{8m+5}  Brauer (inner)  2
12 D_{8m+6}  Brauer (inner)  2
13 B_{8m+6}  Brauer (inner)  2
14 D_{8m+7}  discriminant (outer)  1
15 B_{8m+7}  (α_{G} trivial, strongly inner) v_{2}(16(m + 1))

G is said here to be of Arason type, if β_{G}, δ_{G} is trivial, and if G_{L} splits for every extension L/k, for which α_{G_{L}} is trivial. l.c.i. is the lowest dimension with non trivial cohomological invariant for G.

With this technique, one can determine the Galois cohomology of spin groups of excellent quadratic forms. Example: Let

q(X_{i}) = ∑_{i=1}^{1000001} X_{i}^{2}.

Then, the following list describes the possible indices and dimensions of non trivial Galois cohomology for Spin(q) over any field k:

Indices:  Dimensions of n.t.c.
0  6 7 9 10 14 15 16 20
475713  6 7 9 10 14 15 16
491520  6 7 9 10 14 15
492097  6 7 9 10 14
499712  6 7 9 10
499777  6 7 9
499968  6 7
499969  6
500000  –none–

For example, the index 475713 occurs if and only if

2^{16} ≤ s(k) < 2^{20}

for the level s(k) of k. (s(k) = lowest s, such that −1 is a sum of s squares in k. If s(k) is finite, then it is a power of 2.)

Computation of the cohomological invariants for G = Spin(q) and anisotropic excellent q: Let q = q_{0}, …, q_{h}, dim q_{h} ≤ 1 the “higher anisotropic kernels” of q. Those are defined over k and unique up to isometry (Knebusch).

Let q̃_{i} = q_{i} ⊥ ℍ with ℍ hyperbolic and dim q̃_{i} = dim q. Then Spin(q̃_{i}) is a twist of Spin(q), and if d(q) = d(q_{i}), c(q) = c(q_{i}), then q_{i} defines x_{i} ∈ H^{1}(k, Spin(q)).

These invariants give canonically

ν_{G}^{a_{i}}(x_{i}) ∈ H^{a_{i}}(k, Z/2Z), i = 1 …, h.

a_{i} is the maximal number with q − q_{i} ∈ I^{a_{i}}(k) for i ∈ {1, …, h} .

With ν_{G}^{a_{i}}(x_{i}), the groups

Spin(q), q excellent,

can be described uniquely up to isomorphy.

Conversely: All relations between the ν_{G}^{a_{i}}(x_{i}) in the cohomology ring H^{∗}(k, Z/2Z) are known, i.e., if there is a set of these elements fulfilling the relations, then there is a group of the above type with these elements as invariants.

2. Splitting Patterns (jointly with J. Hurrelbrink.)

Explained just by an example.

Def. The Splitting Pattern SP_{G} of G is the category of all Tits-Dynkin diagrams of G_{L}, where L/k ranges over all field extensions of k; morphisms are the rank increasing k-specializations.

Splitting patterns allow the distiction of anisotropic groups:

Let, e.g., G be of orthogonal type. Then, SP_{G} is given by the sequence of Witt indices of the underlying quadratic forms, which may occur over extensions of k, morphisms are given by the relation <.

Example. Let q be anisotropic, dim q = n.

q Pfister form: SP_{SO(q)} = (0, n/2);
q “generic”:  SP_{SO(q)} = (0, 1, 2, …, [n/2]).

h(q) = #SP_{q} − 1 is the height of q.

Thm. (Wadsworth/Knebusch) h(q) = 1 ⇔q is similar to an orthogonal summand of a Pfister form of codim ≤ 1.

Forms of height 2 are not completely known. However:

2.1 Thm. Let q be a form of even dimension with h(q) = 2, whose “leading form” is defined over k. Then q is of one of the following types.

i) dim q = 2^{a}−2^{b}, a−1 > b > 0, SP_{q} = (0, 2^{a−1}−2^{b}, 2^{a−1}−2^{b−1}), and for every K/k the anisotropic kernel of q_{K} is defined over k, i.e., q is excellent.

ii) dim q = 2^{a}, SP_{q} = (0, 2^{a−2}, 2^{a−1}), q not excellent. In this case there is L/k, such that q_{L} is an anisotropic Pfister-Form. (“Anisotropic splitting”)

Remark. Part i) was proved by Knebusch (1976), part ii) verifies a conjecture by Knebusch, who had proved ii) for the case a ≤ 3.

Proved 1993 by H-R under a cohomological condition, which was proved 1996 by Voevodsky (Milnor-Vermutung). Also, in 1996, D. Hoffmann gave an elementary proof without using Voevodsky’s result.

Recently (Hurrelbrink-R): Description of all quadratic forms which are linear combinations of two Pfister forms (to appear in Crelle J.)

3. Excellence properties of algebraic groups

Def. G is called excellent, if for every L/k the anisotropic kernel of G_{L} is defined over k.

There is an analogue notion for Azumaya algebas.

3.1 Theorem (Kersten-R.): For an Azumaya algebra A over k

the following statements are equivalent:

i) A is excellent. ii) The index of A is squarefree. iii) Index and exponent of A_{L} are equal for every L/k. iv) G = SL_{1}(A) is excellent.

3.2 Theorem (Kersten-R.): (char k ≠ 2)

Let q be a regular quadratic form over k. Then equivalent: i) G = SO(q) excellent ii) For every extension L/k there is a quadratic form τ over

k and a ∈ L^{∗} with aτ_{L} ≅ (q_{L})_{an}.

Hence: Excellence of orthogonal groups is more general then excellence of quadratic forms. (for B_{n} equivalent, but not for D_{n}).

Example: if q is anisotropic with dim q = 4 and d(q) ≠ 1, then SO(q) is excellent, but q is not. More general, forms from 2.1. ii) have this property.

Apparently, excellent groups have a very simple cohomological nature (G = SO(q), q excellent: see above.). G := SL_{1}(D), D/k a skew field, central, finite dimensional over k: n = ind(D) squarefree:

i) exp(D) = ind(D)  (Brauer)
ii) SK_{1}(D) := G(k)/[D^{∗}, D^{∗}] = 1  (Wang)
iii) H^{1}(k, G) ↪ H^{3}(k, μ_{n}^{⊗2}) (n, char k coprime) (Merkurjev-Suslin)
iv) G excellent  (Kersten-R)

n = ind(D) not squarefree:

• ∃ L/k with exp(D_{L}) ≠ ind(D_{L})  (K-R)
• ∃ “anisotropic splitting: exp. reduction”  (K-R)
• ∃ D/k with SK_{1}(D) ≠ 1  (Platonov, Draxl)
• H^{1}(k, G) → H^{3}(k, μ_{n}^{⊗2}) i.g. not injective (Merkurjev)
• G not excellent  (K-R)

Conjecture (Suslin, 1991):

SK_{1}(D_{k(G)}) = 1 ⇔ ind(D) squarefree

One knows (Merkurjev 1993):

G rational (i.e., k(G)/k purely transcendental.) ⇒ SK_{1}(D_{k(G)}) = 1
char k ≠ 2 und 4 | ind(D)  ⇒ SK_{1}(D_{k(G)}) ≠ 1.

Excellence of G is a sort of “rationality property” of G.

By 3.1: G excellent ⇔ ind(D) squarefree.

Question :

G rational ⇔ SK_{1}(D_{k(G)}) = 1 ⇔ G excellent
⇔ H^{1}(k, G) ↪ H^{3}(k, μ_{n}^{⊗2}) for n, char k coprime

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