Real Elements in Algebraic Groups Anupam Singh The Institute of Mathematical Sciences C.I.T. Campus, Taramani, Chennai 600113 India email : [email protected] http://anupamk18.googlepages.com

Transformation Groups 17-22 December 2007, Moscow

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Definition and Examples of Algebraic groups

Let K be an algebraically closed field. Definition. An algebraic group G over a field K is an algebraic variety defined over K which is also a group such that the maps defining group structure µ : G × G → G, µ(x, y ) = xy and i : G → G, i(x) = x −1 are morphisms of varieties.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Definition and Examples of Algebraic groups

Examples: GLn , SLn , Dn (non-singular diagonal matrices), Tn (upper triangular matrices in GLn ), Un (unipotent upper triangular matrices), On , SOn , Spn , elliptic curves etc.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Definition and Examples of Algebraic groups

Examples: GLn , SLn , Dn (non-singular diagonal matrices), Tn (upper triangular matrices in GLn ), Un (unipotent upper triangular matrices), On , SOn , Spn , elliptic curves etc. An algebraic group G is called a linear algebraic group if the underlying variety of G is affine. In what follows algebraic group will always refer to linear algebraic group.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Definition and Examples of Algebraic groups

Examples: GLn , SLn , Dn (non-singular diagonal matrices), Tn (upper triangular matrices in GLn ), Un (unipotent upper triangular matrices), On , SOn , Spn , elliptic curves etc. An algebraic group G is called a linear algebraic group if the underlying variety of G is affine. In what follows algebraic group will always refer to linear algebraic group. Let k be a field. An algebraic group G is said to be defined over k if the underlying variety of G is defined over k and so are the maps defining the group structure on it. Notation : G(k ) denotes the k points of G.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

A Question About Reality of Elements Let G be an algebraic group defined over k . Definition. An element t ∈ G(k ) is called real if there exists g ∈ G(k ) such that gtg −1 = t −1 .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

A Question About Reality of Elements Let G be an algebraic group defined over k . Definition. An element t ∈ G(k ) is called real if there exists g ∈ G(k ) such that gtg −1 = t −1 . Our question is when an element in G(k ) real?

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

A Question About Reality of Elements Let G be an algebraic group defined over k . Definition. An element t ∈ G(k ) is called real if there exists g ∈ G(k ) such that gtg −1 = t −1 . Our question is when an element in G(k ) real? This talk is about determining real elements (semisimple, unipotent or general elements) in algebraic groups and studying its structure. I assume characteristic of k 6= 2, now onwards. An element t ∈ G is called an involution if t 2 = 1. Involutions play an important role in our investigation.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Strongly Real Elements

Definition. An element in G(k ) is called strongly real if it is a product of two involutions in G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Strongly Real Elements

Definition. An element in G(k ) is called strongly real if it is a product of two involutions in G(k ). Note that a strongly k -real element in G(k ) is always k -real in G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Strongly Real Elements

Definition. An element in G(k ) is called strongly real if it is a product of two involutions in G(k ). Note that a strongly k -real element in G(k ) is always k -real in G(k ). For if t = τ1 τ2 with τi2 = 1 then τ1 .t.τ1−1 = τ1 .τ1 τ2 .τ1 = τ2 τ1 = τ2−1 τ1−1 = t −1 .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Strongly Real Elements

Definition. An element in G(k ) is called strongly real if it is a product of two involutions in G(k ). Note that a strongly k -real element in G(k ) is always k -real in G(k ). For if t = τ1 τ2 with τi2 = 1 then τ1 .t.τ1−1 = τ1 .τ1 τ2 .τ1 = τ2 τ1 = τ2−1 τ1−1 = t −1 . Conversely, a real element t ∈ G(k ) is strongly k -real if and only if there exists τ ∈ G(k ) with τ 2 = 1 such that τ tτ −1 = t −1 . In that case, t = τ.τ t.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Real Elements vs. Strongly Real Elements

Let G be an algebraic group (semisimple) defined over k .

When a real element in G(k ) strongly real in G(k )?

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Reality vs. Strong Reality in Linear Algebraic Groups

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Reality Question for Linear Algebraic Groups

Let G be a connected semisimple adjoint group defined over k (char (k ) 6= 2 and k perfect) with w0 = −1 in the Weyl group W of G. The adjoint groups of type A1 , Bl , Cl , D2l (l > 2), E7 , E8 , F4 , G2 are the groups which satisfy above hypothesis. Most of the work presented here is done in collaboration with Maneesh Thakur.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Semisimple Elements and Reality over k¯ Richardson-Springer Theorem, Theorem 1. Let G be a simple adjoint group over an algebraically closed field and T , a maximal torus. Then any involution c ∈ W (T ) is represented by an involution n in N(T ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Semisimple Elements and Reality over k¯ Richardson-Springer Theorem, Theorem 1. Let G be a simple adjoint group over an algebraically closed field and T , a maximal torus. Then any involution c ∈ W (T ) is represented by an involution n in N(T ). As −1 ∈ W , we have n ∈ N(T ), an involution, which maps to −1 in the exact sequence 1 → T → N(T ) → W = N(T )/T → 1.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Semisimple Elements and Reality over k¯ Richardson-Springer Theorem, Theorem 1. Let G be a simple adjoint group over an algebraically closed field and T , a maximal torus. Then any involution c ∈ W (T ) is represented by an involution n in N(T ). As −1 ∈ W , we have n ∈ N(T ), an involution, which maps to −1 in the exact sequence 1 → T → N(T ) → W = N(T )/T → 1. Which means, nxn−1 = x −1 ∀x ∈ T , i.e., (nx)2 = 1. Hence nx is an involution for all x ∈ T . Hence we get that, in G (over k¯ ) every semisimple element is real. Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Strongly Regular Elements and Reality

Theorem 2. With hypothesis as before, Let t ∈ G(k ) be a strongly regular element in G(k ). Then, t is real in G(k ) if and only if t is strongly real in G(k ). Moreover, every element of a maximal torus, which contains a strongly real element, is strongly real.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Strongly Regular Elements and Reality

Theorem 2. With hypothesis as before, Let t ∈ G(k ) be a strongly regular element in G(k ). Then, t is real in G(k ) if and only if t is strongly real in G(k ). Moreover, every element of a maximal torus, which contains a strongly real element, is strongly real. Proof: We have t ∈ G(k ) a strongly regular element. Let T be the maximal torus containing t defined over k . Then ZG (t) = T . Suppose t is real in G(k ), i.e., there exists g ∈ G(k ) such that gtg −1 = t −1 . Using Richardson and Springer we have n ∈ G such that nsn−1 = s−1 . Then g ∈ nZG (t) = nT , say g = ns. We check that g 2 = nsns = s−1 s = 1, i.e., g is an involution. Hence t is a product of two involutions.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Obstruction to Reality

Let t ∈ G be real. Denote H = ZG (t), X = {x ∈ G | xtx −1 = t −1 }. Then X is an H-torsor defined over k with action h.x = xh for h ∈ H. Note that t is real in G(k ) if and only if the H-torsor X has a k -point. We define a map from X to H 1 (k , H) by x 7→ γx where γx : Γ → H is defined by γx (σ) = x −1 σ(x).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Obstruction to Reality

Let t ∈ G be real. Denote H = ZG (t), X = {x ∈ G | xtx −1 = t −1 }. Then X is an H-torsor defined over k with action h.x = xh for h ∈ H. Note that t is real in G(k ) if and only if the H-torsor X has a k -point. We define a map from X to H 1 (k , H) by x 7→ γx where γx : Γ → H is defined by γx (σ) = x −1 σ(x). Lemma 3. With notations as above, t is real in G(k ) if and only γ is a trivial cocycle in H 1 (k , H).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Elements in k split tori

Corollary 4. An element which belongs to a k -split maximal torus is real, in fact strongly real. Since H 1 (k , T ) = 0 for a k -split torus T in G.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Semisimple Elements over fields of cd(k ) ≤ 1

Theorem 5. Let G be a simple adjoint group defined over k with cd(k ) ≤ 1. Let w0 be the longest element in the Weyl group acting as −1. Then every semisimple element in G(k ) is strongly real in G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Semisimple Elements over fields of cd(k ) ≤ 1

Theorem 5. Let G be a simple adjoint group defined over k with cd(k ) ≤ 1. Let w0 be the longest element in the Weyl group acting as −1. Then every semisimple element in G(k ) is strongly real in G(k ). Sketch of Proof : Let t ∈ G(k ) be a semisimple element. Let T be a torus in G defined over k which contains t, i.e., t ∈ T (k ). Make use of, Richardson and Springer theorem, as −1 ∈ W , to get n0 ∈ N(T ) with n02 = 1 which represents −1 in W . That is, we have n0 sn0−1 = s−1 for all s ∈ T . We claim that the coset n0 T is Γ-stable. And make use of Steinberg’s Theorem H 1 (k , T ) = 0.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Real Elements in some Algebraic Groups

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

The Groups GLn and SLn

Wonenburger (1966) proved that an element of GLn (k ) is real if and only if it is strongly real in GLn (k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

The Groups GLn and SLn

Wonenburger (1966) proved that an element of GLn (k ) is real if and only if it is strongly real in GLn (k ). We have looked into the structure of real elements in SLn (k ). Theorem 6. Let V be a vector space of dimension n over k . Let t ∈ SL(V ). Suppose n 6≡ 2 (mod 4). Then, t is real in SL(V ) if and only if t is strongly real in SL(V ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

The Groups of type A1 A form of SL2 defined over k is isomorphic to SL1 (Q) for some Q, a quaternion algebra over k .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

The Groups of type A1 A form of SL2 defined over k is isomorphic to SL1 (Q) for some Q, a quaternion algebra over k . Proposition 7. Let G = PSL1 (Q) and t ∈ G be a semisimple element. Then, t is real in G if and only if t is strongly real.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

The Groups of type A1 A form of SL2 defined over k is isomorphic to SL1 (Q) for some Q, a quaternion algebra over k . Proposition 7. Let G = PSL1 (Q) and t ∈ G be a semisimple element. Then, t is real in G if and only if t is strongly real. Example : Let t ∈ SL2 (k ) be a real semisimple element. Then there exists g ∈ SL2 (k ) with g 2 = −I such that gtg −1 = t −1 . Example : Let H be the quaternion division algebra over R. Then jij −1 = i −1 , i.e., i is a real element but i is not a product of two involutions (only involutions are ±1) whereas j is an involution in PSL1 (H).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Orthogonal Groups V vector space over k of dimension n q a non-degenerate quadratic form O(q) = {t ∈ End(V ) | q(t(x)) = q(x)} Orthogonal group,

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Orthogonal Groups V vector space over k of dimension n q a non-degenerate quadratic form O(q) = {t ∈ End(V ) | q(t(x)) = q(x)} Orthogonal group, 1

(Wonenburger, 1966) Every element of O(q) is a product of two involutions hence strongly real.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Orthogonal Groups V vector space over k of dimension n q a non-degenerate quadratic form O(q) = {t ∈ End(V ) | q(t(x)) = q(x)} Orthogonal group, 1

(Wonenburger, 1966) Every element of O(q) is a product of two involutions hence strongly real.

2

(Knuppel and Nielsen, 1987) If n 6≡ 2 (mod 4) then every element of SO(q) is a product of two involutions in SO(q).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Orthogonal Groups V vector space over k of dimension n q a non-degenerate quadratic form O(q) = {t ∈ End(V ) | q(t(x)) = q(x)} Orthogonal group, 1

(Wonenburger, 1966) Every element of O(q) is a product of two involutions hence strongly real.

2

(Knuppel and Nielsen, 1987) If n 6≡ 2 (mod 4) then every element of SO(q) is a product of two involutions in SO(q).

3

Moreover, any element of SO(q) (for any n) is a product of three involutions.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Orthogonal Groups V vector space over k of dimension n q a non-degenerate quadratic form O(q) = {t ∈ End(V ) | q(t(x)) = q(x)} Orthogonal group, 1

(Wonenburger, 1966) Every element of O(q) is a product of two involutions hence strongly real.

2

(Knuppel and Nielsen, 1987) If n 6≡ 2 (mod 4) then every element of SO(q) is a product of two involutions in SO(q).

3

Moreover, any element of SO(q) (for any n) is a product of three involutions.

On closer examination we get, Theorem 8. Let t ∈ SO(q) be a semisimple element. Then, t is real in SO(q) if and only if t is strongly real in SO(q). Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Symplectic Groups

Let V be a vector space over k of dimension 2n and B a skew-symmetric bilinear form on V . We denote Sp(V , B) = {t ∈ End(V ) | B(t(x), t(y )) = B(x, y )} ESp(V , B) = {t ∈ End(V ) | B(t(x), t(y )) = ±B(x, y )}

The elements t ∈ ESp(V , B) which satisfy B(t(x), t(y )) = −B(x, y ) are called skew-symplectic.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Symplectic Groups

Wonenburger (1966) proved that every element of Sp(V , B) is a product of two skew-symplectic involutions.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Symplectic Groups

Wonenburger (1966) proved that every element of Sp(V , B) is a product of two skew-symplectic involutions. However we have, Theorem 9. Let t ∈ PSp(2n, k ) be a real, semisimple element. Then t is strongly real.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Unitary Groups Let K be a quadratic extension of k . Let V be an n-dimensional vector space over K with hermitian form h. Then we have, Theorem 10. Let (V , h) be a hermitian space over K . Let t ∈ U(V , h) be a semisimple element. Then, t is real in U(V , h) if and only if it is strongly real. Theorem 11. Let t ∈ SU(V , h) be a semisimple element. Suppose n 6≡ 2 (mod 4). Then, t is real in SU(V , h) if and only if it is strongly real.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of type G2

Let G be a group of type G2 defined over k . Then there exists an octonion algebra C over k , unique up to a k -isomorphism, such that G(k ) ∼ = Aut(C), the group of k -algebra automorphisms of C.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of type G2

Let G be a group of type G2 defined over k . Then there exists an octonion algebra C over k , unique up to a k -isomorphism, such that G(k ) ∼ = Aut(C), the group of k -algebra automorphisms of C. Jacobson (1958) studied this group and some of its subgroups and proved that every element of Aut(C) is a product of involutions. Wonenburger (1969) proved that every element is a product of three involutions.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of type G2 We determine real elements in these groups and prove that, Theorem 12. In addition, if char (k ) 6= 3, every unipotent element in G(k ) is strongly real in G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of type G2 We determine real elements in these groups and prove that, Theorem 12. In addition, if char (k ) 6= 3, every unipotent element in G(k ) is strongly real in G(k ). For a general element in G(k ), we prove, Theorem 13. Let char (k ) 6= 2, 3. Then, an element g is real in G(k ) if and only if it is strongly real in G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of type G2 We determine real elements in these groups and prove that, Theorem 12. In addition, if char (k ) 6= 3, every unipotent element in G(k ) is strongly real in G(k ). For a general element in G(k ), we prove, Theorem 13. Let char (k ) 6= 2, 3. Then, an element g is real in G(k ) if and only if it is strongly real in G(k ). Over finite fields every unipotent element as well as every semisimple element is a product of two involutions hence real. Though there are elements which are not real. Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of Type G2

1

Let G be a group of type G2 defined over a field k . Then G(k ) ∼ = Aut(C) where C is an octonion algebra over field k .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of Type G2

1

Let G be a group of type G2 defined over a field k . Then G(k ) ∼ = Aut(C) where C is an octonion algebra over field k .

2

An octonion algebra (also called Cayley algebra) is 8-dimensional algebra which is neither commutative nor associative. It is obtained by doubling a quaternion algebra.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of Type G2

1

Let G be a group of type G2 defined over a field k . Then G(k ) ∼ = Aut(C) where C is an octonion algebra over field k .

2

An octonion algebra (also called Cayley algebra) is 8-dimensional algebra which is neither commutative nor associative. It is obtained by doubling a quaternion algebra.

3

G2 over k is either anisotropic or split.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Groups of Type G2

1

Let G be a group of type G2 defined over a field k . Then G(k ) ∼ = Aut(C) where C is an octonion algebra over field k .

2

An octonion algebra (also called Cayley algebra) is 8-dimensional algebra which is neither commutative nor associative. It is obtained by doubling a quaternion algebra.

3

G2 over k is either anisotropic or split.

4

Any composition subalgebra of C is of dimension 1, 2, 4 or 8.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Some Subgroups of the Group G2

Let C be an octonion algebra over a field k of characteristic 6= 2. Let L be a composition subalgebra of C. We define G(C/L) = {t ∈ Aut(C) | t(x) = x ∀ x ∈ L} and G(C, L) = {t ∈ Aut(C) | t(x) ∈ L ∀ x ∈ L} . Jacobson studied G(C/L) in his paper titled “Composition Algebras and their Automorphisms”(1958). Let L be a two dimensional composition subalgebra of C. Then L is either a quadratic field extension of k or L ∼ = k × k.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Subgroups of Type SU(V , h) Let us assume first that L is a quadratic field extension of k and L = k (γ), where γ 2 = c.1 6= 0. Then L⊥ is a left L vector space via the octonion multiplication. Also, h : L⊥ × L⊥ −→ L h(x, y ) = N(x, y ) + γ −1 N(γx, y ), is a non-degenerate hermitian form on L⊥ over L.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Subgroups of Type SU(V , h) Let us assume first that L is a quadratic field extension of k and L = k (γ), where γ 2 = c.1 6= 0. Then L⊥ is a left L vector space via the octonion multiplication. Also, h : L⊥ × L⊥ −→ L h(x, y ) = N(x, y ) + γ −1 N(γx, y ), is a non-degenerate hermitian form on L⊥ over L. Proposition 14. In this case, the subgroup G(C/L) of G is isomorphic to the unimodular unitary group SU(L⊥ , h) of the three dimensional space L⊥ over L relative to the hermitian form h, via the isomorphism, ψ : G(C/L) −→ SU(L⊥ , h) t Anupam Singh, IMSc Chennai, India

7−→ t|L⊥ . Real Elements in Algebraic Groups

Subgroups of Type SL3 Now, let us assume that L is a split two dimensional étale subalgebra of C. Then C is necessarily split and L contains a nontrivial idempotent e. We have subspaces U = {x ∈ C | ex = 0, xe = x} and W = {x ∈ C | xe = 0, ex = x} and for η ∈ G(C/L) we have η(U) = U and η(W ) = W .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Subgroups of Type SL3 Now, let us assume that L is a split two dimensional étale subalgebra of C. Then C is necessarily split and L contains a nontrivial idempotent e. We have subspaces U = {x ∈ C | ex = 0, xe = x} and W = {x ∈ C | xe = 0, ex = x} and for η ∈ G(C/L) we have η(U) = U and η(W ) = W . Then we have, Proposition 15. In this case G(C/L) is isomorphic to the unimodular linear group SL(U), via the isomorphism given by, φ : G(C/L) −→ SL(U) η 7−→ η|U .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Subgroups of Type SL2 Let D ⊂ C be a quaternion subalgebra. Then we have, by Cayley-Dickson doubling, C = D ⊕ Da for some a ∈ D⊥ with N(a) 6= 0. Let φ ∈ Aut(C, D). Then for z = x + ya ∈ C, there exists c, p ∈ D with N(c) 6= 0 and N(p) = 1 such that φ(z) = cxc −1 + (pcyc −1 )a.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Subgroups of Type SL2 Let D ⊂ C be a quaternion subalgebra. Then we have, by Cayley-Dickson doubling, C = D ⊕ Da for some a ∈ D⊥ with N(a) 6= 0. Let φ ∈ Aut(C, D). Then for z = x + ya ∈ C, there exists c, p ∈ D with N(c) 6= 0 and N(p) = 1 such that φ(z) = cxc −1 + (pcyc −1 )a. Hence, we have, Proposition 16. The group of automorphisms of C, leaving D point-wise fixed, is isomorphic to SL1 (D), the group of norm 1 elements of D. In the above notation, G(C/D) ∼ = SL1 (D). Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Sketch of the Proof of Main Theorem Let G(k ) = Aut(C) and t0 be a semisimple element of G(k ). We write C0 for the subspace of trace 0 elements of C. We put Vt0 = ker (t0 − 1)8 . Then Vt0 is a composition subalgebra of C with norm as the restriction of the norm on C (due to Wonenburger). Let rt0 = dim(Vt0 ∩ C0 ). Then rt0 is 1, 3 or 7.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Sketch of the Proof of Main Theorem Let G(k ) = Aut(C) and t0 be a semisimple element of G(k ). We write C0 for the subspace of trace 0 elements of C. We put Vt0 = ker (t0 − 1)8 . Then Vt0 is a composition subalgebra of C with norm as the restriction of the norm on C (due to Wonenburger). Let rt0 = dim(Vt0 ∩ C0 ). Then rt0 is 1, 3 or 7. We note that if rt0 = 7, the characteristic polynomial of t0 is (X − 1)8 and t0 is unipotent. We have, Lemma 17. Let t0 ∈ G(k ) be a unipotent element. In addition, we assume char (k ) 6= 3. Then t0 is strongly real in G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Lemma 18. Let the notation be as fixed above and let t0 ∈ G(k ) be an element which is not unipotent (e.g. a semisimple element). Then, either t0 leaves a quaternion subalgebra invariant or fixes a quadratic étale subalgebra L of C pointwise. In the latter case, t0 ∈ SU(V , h) ⊂ G(k ) for a rank 3 hermitian space V over a quadratic field extension L of k or t0 ∈ SL(3) ⊂ G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Lemma 18. Let the notation be as fixed above and let t0 ∈ G(k ) be an element which is not unipotent (e.g. a semisimple element). Then, either t0 leaves a quaternion subalgebra invariant or fixes a quadratic étale subalgebra L of C pointwise. In the latter case, t0 ∈ SU(V , h) ⊂ G(k ) for a rank 3 hermitian space V over a quadratic field extension L of k or t0 ∈ SL(3) ⊂ G(k ). Wonenburger in her paper “Automorphism of Cayley Algebras”(1969) proved that , if t0 , an automorphism of C, leaves a quaternion subalgebra invariant, it is a product of two involutions and hence real in G(k ).

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Lemma 18. Let the notation be as fixed above and let t0 ∈ G(k ) be an element which is not unipotent (e.g. a semisimple element). Then, either t0 leaves a quaternion subalgebra invariant or fixes a quadratic étale subalgebra L of C pointwise. In the latter case, t0 ∈ SU(V , h) ⊂ G(k ) for a rank 3 hermitian space V over a quadratic field extension L of k or t0 ∈ SL(3) ⊂ G(k ). Wonenburger in her paper “Automorphism of Cayley Algebras”(1969) proved that , if t0 , an automorphism of C, leaves a quaternion subalgebra invariant, it is a product of two involutions and hence real in G(k ). We discuss the other cases now, i.e., t0 leaves a quadratic étale subalgebra L of C point-wise fixed. 1

2

The fixed subalgebra L is a quadratic field extension of k and The fixed subalgebra is split, i.e., L ∼ = k × k. Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Let us denote the image of t0 by A in SU(L⊥ , h) or in SL(3) as the case may be. Let us denote by χA (X ), the characteristic polynomial and by mA (X ) the minimal polynomial of A over L in the first case and over k in the second. case 1: If χA (X ) 6= mA (X ) then t0 leaves a quaternion algebra invariant and hence strongly real. case 2: Let χA (X ) = mA (X ) then we prove in the first case t0 is ¯ is conjugate to A−1 in SU(3) conjugate to t0−1 if and only if A t in SL . and in the second case A is conjugate to A 3 Combining these results with that of Neumann and Wonenburger we get the required results.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Real elements in Algebraic Groups Let t ∈ G be real. Then t = τ1 τ2 with τ12 = ±1 = τ22 if and only if there exists g ∈ G with g 2 = ±1 such that gtg −1 = t −1 .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Real elements in Algebraic Groups Let t ∈ G be real. Then t = τ1 τ2 with τ12 = ±1 = τ22 if and only if there exists g ∈ G with g 2 = ±1 such that gtg −1 = t −1 . With closer examination we can extend the earlier results as follows: Theorem 19. Let t be an element of G of one of the following type: any element in GLn (F ), SLn (F ) or groups of type G2 defined over F or a semisimple element in groups of type A1 , O(q), SO(q), or Sp(2n). Then, t is real in G if and only if t has a decomposition t = τ1 τ2 where τ1 , τ2 ∈ G and τ12 = ±1 = τ22 .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Spin Groups The following diagram defines the Spin group: 1

1

1

1

 / {±1}

 / Spin(V , q)

 / O 0 (V , q)

/1

1

 / F∗

 / Γ+ (V , q)

 / SO(V , q)

/1

1

 / (F ∗ )2

 / F∗

 / F ∗ /(F ∗ )2

/1

N

χ

Theorem 20. Suppose dim(V ) = 0, 1, 2 (mod 4). Let t ∈ Spin(V , q) be a semisimple element. Then t is real in Spin(V , q) if and only if t = τ1 τ2 where τ12 = ±1 = τ22 and τ1 , τ2 ∈ Spin(V , q). Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Something More to Think About?

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Reality in Algebraic Groups

The answer is not complete even in the classical groups.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Reality in Algebraic Groups

The answer is not complete even in the classical groups. One also needs to study other exceptional groups other than G2 .

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Reality in Algebraic Groups

The answer is not complete even in the classical groups. One also needs to study other exceptional groups other than G2 . The results obtained here explicitly for several groups indicate better general results for a semisimple algebraic groups than proved here.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Reality in Algebraic Groups

The answer is not complete even in the classical groups. One also needs to study other exceptional groups other than G2 . The results obtained here explicitly for several groups indicate better general results for a semisimple algebraic groups than proved here. One needs to understand reality of unipotent elements and the general elements as well.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Representation Theory and Real Elements Proposition 21. Let G be a finite group. The number of real irreducible characters of G is equal to the number of real conjugacy classes of G.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Representation Theory and Real Elements Proposition 21. Let G be a finite group. The number of real irreducible characters of G is equal to the number of real conjugacy classes of G. Question: Is there any relation between orthogonal representations and strongly real conjugacy classes?

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Representation Theory and Real Elements Proposition 21. Let G be a finite group. The number of real irreducible characters of G is equal to the number of real conjugacy classes of G. Question: Is there any relation between orthogonal representations and strongly real conjugacy classes? D. Prasad (1998 and 1999 on self-dual representations) studied following question for the groups of Lie type and p-adic groups. For a group G, when a self-dual irreducible representation is orthogonal?

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Representation Theory and Real Elements Proposition 21. Let G be a finite group. The number of real irreducible characters of G is equal to the number of real conjugacy classes of G. Question: Is there any relation between orthogonal representations and strongly real conjugacy classes? D. Prasad (1998 and 1999 on self-dual representations) studied following question for the groups of Lie type and p-adic groups. For a group G, when a self-dual irreducible representation is orthogonal? Recall: In the representation theory of semisimple algebraic groups there exists an element h in the center of such a group of order ≤ 2 which acts by 1 in an irreducible self-dual representation if and only if the representation is orthogonal. Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Self-dual Representations are Orthogonal Gow, D. Prasad and C. R. Vinroot have proven following results.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Self-dual Representations are Orthogonal Gow, D. Prasad and C. R. Vinroot have proven following results. 1. (Prasad) Any self-dual representation of GLn (Fq ) is orthogonal.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Self-dual Representations are Orthogonal Gow, D. Prasad and C. R. Vinroot have proven following results. 1. (Prasad) Any self-dual representation of GLn (Fq ) is orthogonal. 2. (Vinroot) Let G = GLn (Fq ), and define G+ to be the split extension of G by the transpose-inverse involution. That is, t −1 G+ =< G, τ | τ 2 = 1, τ −1 gτ = g , ∀g ∈ G > .

Then every irreducible representation of G+ is self-dual and orthogonal.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Self-dual Representations are Orthogonal Gow, D. Prasad and C. R. Vinroot have proven following results. 1. (Prasad) Any self-dual representation of GLn (Fq ) is orthogonal. 2. (Vinroot) Let G = GLn (Fq ), and define G+ to be the split extension of G by the transpose-inverse involution. That is, t −1 G+ =< G, τ | τ 2 = 1, τ −1 gτ = g , ∀g ∈ G > .

Then every irreducible representation of G+ is self-dual and orthogonal. 3. (Gow, Prasad) If q is even, or n = 2m + 1 or n = 4m, then every irreducible self-dual representation of SLn (Fq ) is orthogonal. If n = 4m + 2 and q ≡ 1 (mod 4), then an irreducible self-dual representation of SLn (Fq ) is orthogonal if and only if the element −1 acts trivially. Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Self-dual Representations are Orthogonal

4. Any irreducible self-dual generic representation of SOn (Fq ) is orthogonal.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Self-dual Representations are Orthogonal

4. Any irreducible self-dual generic representation of SOn (Fq ) is orthogonal. 5. An irreducible self-dual generic representation of Sp2n (Fq ) is orthogonal if and only if the element −1 acts trivially on the representation.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Self-dual Representations are Orthogonal

4. Any irreducible self-dual generic representation of SOn (Fq ) is orthogonal. 5. An irreducible self-dual generic representation of Sp2n (Fq ) is orthogonal if and only if the element −1 acts trivially on the representation. 6. For finite groups this question has been studied by Tiep and Zalesski for the Groups of Lie type and by Kolesnikov and Nuzhin for simple groups.

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups

Thank You. email : [email protected] http://anupamk18.googlepages.com

Anupam Singh, IMSc Chennai, India

Real Elements in Algebraic Groups