Reality Properties of Conjugacy Classes in Algebraic Groups Anupam Kumar Singh Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai 400 005, India. email : [email protected] http://www.math.tifr.res.in/»anupam 21st September 2006 1 De¯nition and Examples of Algebraic groups Let K be an algebraically closed ¯eld. De¯nition (Algebraic Group). An algebraic group G over a ¯eld K is an algebraic variety de¯ned over K which is also a group such that the maps de¯ning group structure ¹: G £ G ! G; ¹(x; y) = xy and ¡ i: G ! G; i(x) = x 1 are morphisms of varieties. The group GLn; SLn;Dn (non-singular diagonal matrices), Tn (upper triangular matrices in GLn), Un (unipotent upper triangular matrices), On, SOn, Spn, elliptic curves etc. are examples of algebraic groups. 2 An algebraic group G is called a linear algebraic group if the underlying variety of G is a±ne. Such groups can be embedded in GLn for some n, hence the name. In what follows algebraic group will always refer to linear algebraic group. Let k be a ¯eld. An algebraic group G is said to be de¯ned over k if the underlying variety of G is de¯ned over k. The notation G(k) will denote the k points of G. 3 A Question About Reality of Elements Let G be an algebraic group de¯ned over k. An element t 2 G(k) is called real if there exists g 2 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 2 G is called an involution if t2 = 1. Involutions play an important role in our investigation. 4 Strongly Real Elements De¯nition (Strongly Real). An element in G is called strongly real if it is a product of two involutions in G. Note that a strongly k-real element in G(k) is always k-real in G(k). 2 For if t = ¿1¿2 with ¿i = 1 then ¡1 ¡1 ¡1 ¡1 ¿1:t:¿1 = ¿1:¿1¿2:¿1 = ¿2¿1 = ¿2 ¿1 = t : Conversely, a real element t 2 G(k) is strongly k-real if and only if there exists a conjugating element ¿ 2 G(k) which is an involution, i.e., there exists ¿ 2 G(k) with ¿ 2 = 1 such that ¿t¿ ¡1 = t¡1. In that case, t = ¿:¿t. Let G be an algebraic group (semisimple) de¯ned over k. When a real element in G(k) strongly real in G(k)? 5 Plan of the Talk Part-I Real elements in some classical groups and in the groups of type G2 Part-II Reality in linear algebraic groups Part-III Groups of type G2 Part-IV Conclusion and some questions 6 Part-I Real Elements in Classical Groups and in G2 7 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). Ellers (1977) showed that this result does not generalise to matrix algebras over division rings. We have looked into the structure of real elements in SLn(k). Theorem. Let V be a vector space of dimension n over k. Let t 2 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 ). 8 The Groups of type A1 Any group of type A1 over k is isomorphic to SL1(Q) for some Q, a quaternion algebra over k. That is, it is a form of SL2 de¯ned over k. a;b Let Q = ³ k ´ be a quaternion algebra over k. That is, Q has a basis f1; i; j; ijg with i2 = a; j2 = b; ij = ¡ji: It is a central simple algebra over k of degree 2 with norm de¯ned by 2 2 2 2 N(x01 + x1i + x2j + x3ij) = x0 ¡ ax1 ¡ bx2 + abx3: We denote the set of norm 1 elements of Q by SL1(Q). We remark that M2(k) is a quaternion algebra with norm form given by determinant. 9 Proposition (A). Let t 2 SL2(k) be a real semisimple element. Then there 2 ¡1 ¡1 exists g 2 SL2(k) with g = ¡I such that gtg = t . In particular if t is a real semisimple element in P SL2(k) then t is strongly real. Proposition (B). Let G = P SL1(Q) and t 2 G be a semisimple element. Then, t is real in G if and only if t is strongly real. 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 P SL1(H). 10 Proof of (A): Over k¹ the element t is conjugate to the matrix ¡1 t0 = diagf®; ® g for some ® 2 k¹. If t is central then t is either I or ¡I 0 ¡1 otherwise t is regular and ®2 6= 1. We write n = 0 1 and then @ 1 0 A ¡1 ¡1 2 ¹ nt0n = t0 where n = ¡1 and n 2 SL2(k). In fact n conjugates every element of the torus T = fdiagf®; ®¡1g j ® 2 k¹g to its inverse. Hence there ¡1 ¡1 2 exist h 2 SL2(k¹) such that hth = t and h = ¡I. As t is real in SL2(k) ¡1 ¡1 there exists g 2 SL2(k) such that gtg = t . Then g 2 hZGL2(k¹)(t). We note that as t is regular we have ZGL2(k¹)(t) = T , a maximal torus. We write g = hx where x 2 T . We check that g2 = ¡I and this proves the required result. 11 Orthogonal Groups Let k be a ¯eld (of char 6= 2) and V be a vector space over k of dimension n. Let Q be a non-degenerate quadratic form on V and B be the corresponding bilinear form on V . Let O(Q) = ft 2 End(V ) j B(t(x); t(y)) = B(x; y)g be the orthogonal group. Then, it was proved by Wonenburger (1966) that every element of O(Q) is a product of two involutions hence strongly real. Knuppel and Nielsen (1987) proved that if n 6´ 2 (mod 4) then every element of SO(Q) is a product of two involutions in SO(Q). They also proved that any element of SO(Q) (for any n) is a product of three involutions. However we prove, Theorem. Let t 2 SO(Q) be a semisimple element. Then, t is real in SO(Q) if and only if t is strongly real in SO(Q). 12 Symplectic Groups Let k be a ¯eld (of char 6= 2) and V be a vector space over k of dimension 2n. Let B be a skew-symmetric bilinear form on V . We denote Sp(V; B) = ft 2 End(V ) j B(t(x); t(y)) = B(x; y)g and ESp(V; B) = ft 2 End(V ) j B(t(x); t(y)) = §B(x; y)g: The group of similitude is denoted by ¤ GSp(V; B) = ft 2 End(V ) j B(t(x); t(y)) = ¹(t)B(x; y); ¹(t) 2 k g where ¹(t) is similitude factor. The elements t 2 ESp(V; B) which satisfy B(t(x); t(y)) = ¡B(x; y) are called skew-symplectic. 13 Wonenburger (1966) proved that every element of Sp(V; B) is a product of two skew-symplectic involutions. Theorem. Let t 2 P Sp(2n; k) be a real, semisimple element. Then t is strongly real. Recently Vinroot (2004) analysed the group GSp(2n; k) and proved following extension of Wonenburger's result. Let g 2 GSp(2n; k) with similitude factor ¹(g) = ¯. Then g = t1t2, where t1 is a skew-symplectic involution and t2 is 2 such that ¹(t2) = ¡¯ with t2 = ¯I. 14 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. Let (V; h) be a hermitian space over K. Let t 2 U(V; h) be a semisimple element. Then, t is real in U(V; h) if and only if it is strongly real. Theorem. Let t 2 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. 15 Groups of type G2 It is known that for a group G of type G2 over k, there exists an octonion algebra C over k, unique up to a k-isomorphism, such that G =» Aut(C), the group of k-algebra automorphisms of C. Octonion algebras (also called Cayley algebras) are 8-dimensional non-commutative, non-associative algebras obtained by doubling a quaternion algebra. 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. 16 We determine real elements in these groups and prove that, Theorem. In addition, if char(k) 6= 3, every unipotent element in G(k) is strongly real in G(k).