arXiv:0906.0563v2 [math.GR] 11 Aug 2013 THE z-CLASSES OF ISOMETRIES KRISHNENDU GONGOPADHYAY AND RAVI S. KULKARNI Abstract. Let G be a group. Two elements x, y are said to be in the same z-class if their centralizers are conjugate in G. Let V be a vector space of dimension n over a field F of characteristic different from 2. Let B be a non-degenerate symmetric, or skew-symmetric, bilinear form on V. Let I(V, B) denote the group of isometries of (V, B). We show that the number of z-classes in I(V, B) is finite when F is perfect and has the property that it has only finitely many field extensions of degree at most n. 1. Introduction Let G be a group. We define an equivalence relation ∼ on G as follows: for x, y in G, x ∼ y if the centralizers ZG (x) and ZG (y) are conjugate in G. The equivalence class of x is called the z-class of x in G. The z-classes are pairwise disjoint and give a partition of the group G. This provides important information about the internal structure of the group, see [15] for further details. The structure of each z-class can be expressed as a certain set theoretic fibration, see [15, Theorem 2.1 ]. In general, a group may be infinite and it may have infinitely many conjugacy classes, but the number of z-classes is often finite. For example, if G is a compact Lie group, then it is implicit in Weyl’s structure theory see, [21], Borel-de Siebenthal [2], that the number of z-classes in G is finite. Analogously, Steinberg [19, p.107] has remarked on the finiteness of z-classes in reductive algebraic groups over an algebraically closed field of good characteristic. In [15], Kulkarni proposed to interpret the z-classes as an internal ingredient in a group G that can be used to make precise the intuitive notion of “dynamical types” in the G-action on any set X. The Fibration Theorem, see [15, Theorem 2.1], gives a set-theoretic fibration of the z-class of x with base the homogeneous Date: November 5, 2018. 2000 Mathematics Subject Classification. Primary 20E45; Secondary 20G15, 15A63, Key words and phrases. conjugacy classes, centralizers, z-classes, orthogonal and symplectic groups. Gongopadhyay acknowledges the support of SERC-DST FAST grant SR/FTP/MS004/2010. 1 space G/N (x), where N (x) is the normalizer of ZG (x) in G, and a fiber consists of the elements y in the center of ZG (x) such that ZG (x) = ZG (y). For example, in classical geometries over R, C or H, it is observed that the “dynamical types” that our mind can perceive are just finite in number and this finiteness of “dynamical types” can be interpreted as a phenomenon related to the finiteness of the z-classes in the corresponding group of the geometry. With this motivation, the z-classes in the isometry group of the ndimensional real hyperbolic space were classified and counted in [9]. It is also an interesting problem to classify the z-classes in other linear groups that appear as isometry group in rank one symmetric spaces of non-compact type. The z-classes in the isometry group Sp(n, 1) of the n-dimensional quternionic hyperbolic space have been classified and counted in [8]. Classification of the z-classes in U(n, 1), the isometry group of the n-dimensional complex hyperbolic space, has been obtained in [5], also see [8, Appendix]. Recently, z-classes have also been used in the context of classifying the isometries in hyperbolic geometries, see, [4, 6, 7]. In addition to these, it is of independent algebraic interest to parametrize both the conjugacy and the z-classes in a group. For example, the problem can be asked for finite groups of Lie type; classical groups or exceptional groups. The conjugacy classes, z-classes and the set of operators themselves of the general linear groups and the affine groups have been parametrized by Kulkarni [14]. This has been extended to linear operators over division rings by Gouraige [11]. In an attempt to understand the z-classes in exceptional groups, Singh [17] has proved a finiteness result for the z-classes in the compact real form G2 . Let F be a field of characteristic different from 2. Let V be a vector space of dimension n over F. Let V be equipped with a non-degenerate symmetric or skew-symmetric bilinear form B. The group of isometries of (V, B) is denoted by I(V, B; F), or simply I(V, B) when the underlying field is fixed. When B is symmetric, resp. skew-symmetric, I(V, B) is the orthogonal, resp. symplectic group. In this paper we ask for the z-classes in I(V, B). Our main theorem is the following. Theorem 1.1. If F is perfect and has the property that it has only finitely many field extensions of degree at most dim V, then the number of z-classes in I(V, B) is finite. This holds for example when the field F is algebraically closed, the field of real numbers, or a local field. Along the way we parametrize the z-classes of the semisimple elements, see, Theorem 4.1. 2 A first step in understanding of the z-classes is to classify the conjugacy classes. There has been a considerable amount of work on the conjugacy problem in orthogonal and symplectic groups, see, Asai [1], Burgoyne and Cushman [3], Kiehm [13], Milnor [16], Springer-Steinberg [18], Wall [20] and Williamson [22]. A common theme of these works is to reduce the conjugacy problem to the equivalence problem for Hermitian forms. Our conjugacy classification has a similar flavor. However, a notable feature of our classification is that, in the “generic” case when the minimal polynomial of an element in I(V, B) is a prime-power, it gives an explicit parametrization of the conjugacy classes, see, Theorem 3.6. Consequently, we also obtain a parametrization of the z-classes in this case, see, Theorem 3.7. As we shall see, the z-classification depends on the equivalence problem of Hermitian forms over arbitrary fields. The equivalence problem of Hermitian forms was solved by Kiehm [13] and Wall [20]. We will not get into the equivalence problem of the Hermitian spaces in this paper. However, our classification of the z-classes is enough to prove our main result, Theorem 1.1. 2. Preliminaries 2.1. Self-dual polynomial. Let F[x] be the ring of polynomials over F. For a polynomial g(x) let ck (g) denote the coefficient of xk in g(x). Let F̄ denote the algebraic closure of F. Let f (x) be a monic polynomial of degree n over F such that 0, 1 and −1 are not its roots. Over F̄ let f (x) = (x − c1 )(x − c2 )....(x − cn ). Then the polynomial −1 f ∗ (x) = (x − c1−1 )(x − c−1 2 )....(x − cn ) is said to be the dual to f (x). It is easy to see that f ∗ (x) = f (0)−1 xn f (x−1 ). Clearly, ck (f ∗ ) = f (0)−1 cn−k . Definition 2.1. Let f (x) be a monic polynomial over F such that −1, 0, 1 are not its roots. The polynomial f (x) is called reciprocal, or self-dual, if f (x) = f ∗ (x). Thus if f (x) self-dual, then the degree n of f (x) is even, and for all k, ck (f ) = cn−k (f ). 3 2.2. Decomposition of the space relative to an isometry. Suppose T : V → V is an element in I(V, B). Let mT (x) denote the minimal polynomial of T . Suppose p1 (x), . . . , pl (x) are irreducible polynomials over F such that mT (x) = p1 (x)d1 . . . pl (x)dl , where for i 6= j, pi (x) 6= pj (x). Suppose degree of mT (x) is m. The integer di is called the exponent, or the multiplicity, of the prime factor pi (x). Let E = F[x]/(mT (x)). The image of the indeterminate x in E is denoted by t. There is a canonical algebra structure on E defined by tv = T v. The F-algebra E = F[t] is spanned by {1, t, t2 , · · · , tm−1 }. In particular, if the minimal polynomial is irreducible, then E is an extension field of F. The following lemma follows from Lemma 4.1 in [10]. Lemma 2.2. (i) The minimal polynomial of an element T in I(V, B) is self-dual. (ii)There is a unique automorphism e → ē of E over F which carries t to −1 t . Thus an irreducible factor p(x) of the minimal polynomial can be one of the following three types: (i) p(x) is self-dual. (ii) p(x) = x − 1, or, x + 1. (iii) p(x) is not self-dual. In this case there is an irreducible factor p∗ (x) of the minimal polynomial such that p∗ (x) is dual to p(x). Among the irreducible factors of mT (x), suppose pi (x) is self-dual for i = 1, 2, ..., k1 . Let the other irreducible factors be pj (x), p∗j (x) for j = 1, 2, ..., k2 with pj (x) 6= p∗j (x). For a prime-power polynomial p(x)d , let Vp = ker p(T )d . Let ⊕ denote the orthogonal sum, and + denote the usual sum of subspaces. It can be seen easily that there is a primary decomposition of V (with respect to T ) into T -invariant non-degenerate subspaces: (2.1) 1 Vi V = ⊕ki=1 M 2 Vj ⊕kj=1 where for i = 1, 2, ..., k1 , pi (x) is self-dual, Vi = Vpi , and B|Vi is nondegenerate; for j = 1, 2, ..., k2 , Vj = Vpj + Vp∗j , B|Vpj = 0 = B|Vp∗ , here j pj (x) 6= p∗j (x). Let Tl denote the restriction of T to Vl . Then mTi (x) = pi (x)di for i = 1, 2, ..., k1 , and mTj (x) = pj (x)dj p∗j (x)dj for j = 1, 2, ..., k2 . Let Z(T ) denote the centralizer of T in I(V, B). We observe that the decomposition (2.1) is in fact invariant under Z(T ). Moreover we have a canonical decomposition 2 1 Z(Tj ). Z(Ti ) × Πkj=1 Z(T ) = Πki=1 4 Thus the conjugacy classes and the z-classes of T are determined by the restriction of T to each of the primary subspaces. Hence it is enough to determine the conjugacy and the z-classes of an isometry T : V → V with minimal polynomial mT (x) = p(x)d , where p(x) is one of the types (i), (ii), (iii) above. Finally note the following lemma. For a proof of the lemma, see, [10, Lemma 4.2]. Lemma 2.3. Let T be an element in I(V, B). Suppose T : V → V is such that the minimal polynomial is one of the types (i), (ii) above. Suppose mT (x) = p(x)d . There is an orthogonal decomposition V = ⊕ki=1 Vdi , where 1 ≤ d1 < · · · < dk = d, and for each i = 1, ..., k, Vdi is free over the algebra F[x]/(p(x)di ). For each i, the summand Vdi corresponds to the elementary divisor p(x)di of T . Remark 2.4. In the above lemma, suppose deg p(x) = m. Then dimF F[x]/(p(x)di ) = mdi . Suppose Vdi has dimension li as a free module over F[x]/(p(x)di ). Thus dimF Vdi = mdi li . This gives us a secondary Pk n = i=1 di li . partition π : m We end this section with the following definition. Definition 2.5. Let R be a commutative ring with involution e 7→ ē. Let ǫ = 1 or −1. An ǫ-Hermitian form on an R-module M is a sesquilinear mapping s : M × M → R such that for all x, y ∈ M , s(x, y) = ǫs(y, x). That is for ǫ = 1, s is Hermitian; for ǫ = −1, s is skew-Hermitian. 3. The induced form and the conjugacy classes 3.1. The minimal polynomial is prime-power. Lemma 3.1. (Springer-Steinberg [18]) Let T : V → V in I(V, B) be such that mT (x) = p(x)d , where p(x) is an irreducible polynomial over F. Assume that p(x) is either self-dual, or, x − 1. If p(x) = x − 1, then assume d > 1. Consider the cyclic F-algebra ETd = F[x]/(p(x)d ). We simply denote it by ET when there is no confusion about d. The ET -module V is denoted by VT . Then we have the following. (i) There is a unique automorphism e → ē of ET over F which carries t to t−1 . (ii) There exists an F-linear function hT : ET → F such that the symmetric bilinear map h̄T : (a, b) 7→ hT (ab) on ET × ET is non-degenerate. Also 5 there exists c ∈ ET such that for all e ∈ ET , hT (ē) = hT (ce). Moreover, if p(x) 6= x − 1, we can take c = 1. If p(x) = x − 1, then c = (−1)d−1 . For a proof of the above lemma cf. Springer-Steinberg [18, p.254]. For a proof when the field extension Fd = F[x]/(p(x)) is separable, cf. Asai [1, p.329]. Applying the above lemma we have the following theorem. The theorem is implicit in the work of Springer-Steinberg [18]. Milnor [16] gave a version of the following theorem when T is semisimple. We have given a detailed proof in the general case. The proof is essentially imitating Milnor’s proof in the semisimple case. Lemma 3.2. The module V over ET admits a unique ǫ-Hermitian form H T (u, v) = ǫH T (v, u), ET -linear in the first variable, and is related to the original F-valued inner product by the identity (3.1) B(u, v) = hT (H T (u, v)). Proof. For u, v in V, consider the linear map L : ET → F given by L(e) = B(eu, v). There exists a unique e′ in ET such that hT (ee′ ) = L(e). We define H T (u, v) to be this element e′ . That is, H T (u, v) is defined as follows: for all e in ET , and for u, v in V, hT (eH T (u, v)) = B(eu, v). In particular taking e = 1 we have hT (H T (u, v)) = B(u, v) Now we see that for u1 , u2 , v in V, hT (e(H T (u1 , v) + H T (u2 , v))) = hT (eH T (u1 , v)) + hT (eH T (u2 , v)) = B(eu1 , v) + B(eu2 , v) = B(eu1 + eu2 , v) = B(e(u1 + u2 ), v) = hT (H T (u1 + u2 , v)) (3.2) ⇒ H T (u1 , v) + H T (u2 , v) = H T (u1 + u2 , v) Now for all e′ in ET we have hT (e′ eH T (u, v)) = = (3.3) B(e′ eu, v) B(e′ (eu), v) = hT (e′ H T (eu, v)) ⇒ eH T (u, v) = H T (eu, v) This shows that hT is ET -linear in the first variable. Given any Hermitian form H(u, v) satisfying (3.1) we see that hT (eH(u, v)) = hT (H(eu, v)) = B(eu, v). 6 Therefore H T (u, v) is unique. Further, for all e in ET , hT (e(H T (u, v))) = ǫhT (ēH T (u, v)), using part (ii) of Lemma 3.1 = ǫB(ēu, v) = ǫB(ev, u) = hT (eǫH T (v, u)) (3.4) ⇒ H T (u, v) = ǫH T (v, u) This proves the theorem.  Remark 3.3. Let S : V → V and T : V → V be two isometries such that mS (x) = p(x)d , mT (x) = q(x)d , where p(x), q(x) are irreducible and selfdual, deg p(x) = deg q(x) and ES and ET are F-isomorphic. Let s and t are images of S and T in ES and ET respectively. Let f : ES → ET be an F-isomorphism such that f (s) = t. Let hS : ES → F be the linear map as in Lemma 3.1. Then hT = hS ◦ f −1 is such a linear map on ET , and this map induces a Hermitian form H ′ on VT . Since such a Hermitian form is unique, hence we must have H ′ = H T . Thus for u, v in VS , hS (H S (u, v)) = B(u, v), and for u′ , v ′ in VT , hT (H T (u′ , v ′ )) = hS ◦ f −1 (H T (u′ , v ′ )). Definition 3.4. Suppose E and E′ are isomorphic modules over F, and let f : E → E′ be an isomorphism. Let H be an E-valued Hermitian form on V and let H ′ be an E′ -valued Hermitian form on V′ . Then (V, H) and (V′ , H ′ ) are equivalent if there exists an F-isomorphism T : V → V′ such that for all u, v in V and for all e in E the following conditions are satisfied. (i) T (ev) = f (e)T (v), and (ii) H ′ (T (u), T (v)) = f (H(u, v)). When E = E′ , we take f to be the identity in the definition. Theorem 3.5. Suppose S and T are isometries of (V, B). Let the minimal polynomial of both S and T be (x − 1)d or, p(x)d , where p(x) is monic, self-dual, and, irreducible over F. Let H S and H T be the Hermitian form induced by S and T respectively. (i) Then S and T are conjugate in I(V, B) if and only if H S and H T are equivalent. (ii) Let Z(T ) be the centralizer of T in I(V, B). Then an isometry C is in Z(T ) if and only if C preserves H T , i.e. Z(T ) = U (VT , H T ). Proof. Suppose S is conjugate to T in I(V, B). Let C in I(V, B) be such that T = CSC −1 . Then C : VS → VT is an F-isomorphism. For l ≥ 1, and 7 v in VS , C(sl v) = C ◦ S l (v) = T l ◦ C(v) = tl C(v) = f (sl )C(v). It follows that, for all e in ES , and v in VS , C(ev) = f (e)C(v). For u, v in VS , note that hS (f −1 (H T (C(u), C(v))) = hS ◦ f −1 (H T (C(u), C(v))) = hT (H T (C(u), C(v)) = B(C(u), C(v))) = B(u, v) = hS (H S (u, v)). Hence, by the uniqueness of H S we have, f −1 (H T (C(u), C(v))) = H S (u, v), i.e. H T (C(u), C(v)) = f (H S (u, v)). This shows that H S and H T are equivalent. Conversely, suppose H S and H T are equivalent. Let C : VS → VT be an F-isomorphism such that (i) and (ii) in Definition 3.4 hold. We have for v in V, CS(v) = C(sv) = f (s)C(v) = tC(v) = T C(v). that is, CSC −1 = T . Further, for x, y in V, B(C(x), C(y)) = hT (H T (C(x), C(y))) = hT (f (H S (x, y))) = hS (H S (x, y)) = B(x, y). Hence C : V → V is an isometry. This completes the proof of (i). (ii) Note that an invertible linear transformation C : V → V is ET -linear if and only if CT = T C. Now replacing S by T , and f by identity in the proof of (i) the theorem follows.  3.2. Conjugacy classes. Theorem 3.6. Let T be an element of I(V, B). Let the minimal polynomial of T be p(x)d , where p(x) = x − 1 or p(x) is monic, self-dual and irreducible over F. (1) The conjugacy class of T in I(V, B) is determined by the following data. (i) The elementary divisors of T . 8 (ii) The finite sequence of equivalence classes of Hermitian spaces {(VTd1 , HdT1 ), · · · , (VTdk , HdTk )}, where 1 ≤ d1 < d2 < · · · < dk = d, and for each i, HdTi takes values in the cyclic algebra Edi = F[x]/(p(x)di ). (2) The centralizer of T is the direct product U (VTd1 , HdT1 ) × · · · × U (VTdk , HdTk ). Proof. Suppose S : V → V and T : V → V are two isometries. If S and T are conjugate in I(V, B), then by the structure theory of linear operators and Theorem 3.5, it is clear that they have the same data. Conversely, suppose S and T have the same data. The elementary divisors of S and T determine orthogonal decompositions of V as (3.5) V = VSd1 ⊕ · · · ⊕ VSdk , (3.6) V = VTd1 ⊕ · · · ⊕ VTdk , where 1 ≤ d1 < · · · < dk = d, and for each i, VSdi , resp. VTdi is free when considered as a module over ESdi , resp. ETdi . Since ESdi and ETdi are isomorphic, wthout loss of generality, we identify them with Edi = F[x]/(p(x)di ). Since S and T have the same set of elementary divisors, VSdi is isomorphic to VTdi as a free module over Edi , for i = 1, 2, ..., k. For each i = 1, 2, · · · , k, since (VSdi , HdSi ) is equivalent to (VTdi , HdTi ), by Theorem 3.5, S|VSd is conjugate to i T |VTd . Hence S is conjugate to T . i The description of Z(T ) is clear from the orthogonal decomposition of V and part (2) of Theorem 3.5.  3.3. The z-classes. Theorem 3.7. Let T : V → V be an element in I(V, B) such that mT (x) = p(x)d , where p(x) is self-dual and irreducible over F. The z-class of T is determined by the following data. (i) The degree m of p(x). (ii) A non-decreasing sequence of integers (d1 , . . . , dk ) which corresponds n = Σki=1 di li . to the secondary partition π : m (iii) A sequence (Ed1 , . . . , Edk ) of isomorphism classes of cyclic algebras over F, where for each i = 1, 2, . . . , k, Edi is isomorphic to F[x]/(p(x)di ). (iv) A finite sequence of equivalence classes of Hermitian forms (Hd1 , . . . , Hdk ), where each Hdi takes values in Edi . 9 Proof. Let S and T be two isometries of (V, B) with same data (i)−(iv). We use the same notations as in the previous theorem. Let mS (x) = p(x)d , and mT (x) = q(x)d , degree of p(x) = degree of q(x) = m. For each i = 1, . . . , k, ESdi and ETdi are isomorphic. Let fi : ESdi → ETdi be one such isomorphism. For simplicity, for each i, we identify ESdi , and ETdi with Edi . Moreover, following Remark 3.3 assume hS = hT . Since the Hermitian forms HdSi and HdTi are equivalent, let Fi : (VSdi , HdSi ) → T (Vdi , HdTi ) be an equivalence of the Hermitian spaces. We see that, for u, v ∈ VSdi , B(Fi u, Fi v) = hT (HdTi (Fi u, Fi v)) = hT ◦ fi (HdSi (u, v)) = hS (HdSi (u, v)), see, Remark 3.3, = B(u, v). Thus Fi is an isometry with respect to B. Further, Fi conjugates Z(S|VSd ) = i U (VSdi , HdSi ) and Z(T |VTd ) = U (VTdi , HdTi ). Thus, F = F1 ⊕ F2 ⊕ · · · ⊕ Fk is i an isometry of (V, B) and F conjugates Z(S) and Z(T ). Conversely, suppose S and T are in the same z-class. Replacing S by its conjugate, we may assume, Z(S) = Z(T ). Hence by part (2) of Theorem 3.2 we see that S and T have isomorphic decompositions (3.5) and (3.6). After renaming the indices, if necessary, we may assume further that for i = 1, 2, ..., k, (VSdi , HdSi ) and (VTdi , HdTi ) are equivalent. In particular, ESdi and ETdi are isomorphic, and their common dimension over F is mdi . This implies deg mS (x) = deg mT (x). Consequently we attach the partition (see, n = Σki=1 di li to the z-class and it follows that S and T Remark 2.4) π : m have the same data (i) − (iv). This completes the proof.  3.4. The minimal polynomial is (x + 1)d . Note that, two isometries S and T are conjugate if and only if −S and −T are conjugate. Now, suppose T is an isometry with minimal polynomial (x − 1)d . Then −T : V → V is also an isometry, and m−T (x) = (x + 1)d . Conversely, if T is unipotent, then −T has minimal polynomial (x + 1)d . Thus this case is reduced to the unipotent case, and the parametrization of the conjugacy and the z-classes of T are similar to that of −T . 3.5. The minimal polynomial is a product of pairwise dual polynomials. Suppose T : V → V is an element in I(V, B) such that mT (x) = q(x)d q ∗ (x)d , where q(x), q ∗ (x) are irreducible polynomials over F of degree 10 m and are dual to each-other. For our purpose, it is enough to consider the case when T is semisimple. So assume, d = 1. Let Vq = ker q(T ), Vq∗ = ker q ∗ (T ). We have (3.7) V = Vq + Vq ∗ , and B|Vq = 0 = B|Vq∗ , dim Vq = dim Vq∗ . Since B is non-degenerate, we can choose a basis {e1 , ...., em , f1 , ..., fm } of V such that for each i, ei ∈ Vq , fi ∈ Vq∗ , and for all i, j = 1, . . . m, B(ei , ei ) = 0 = B(fi , fi ), B(ei , fj ) = δij or − δij . For each w∗ ∈ Vq∗ , define the linear map w∗ : v → B(v, w). These maps enable us to identify Vq∗ with the dual of Vq . Thus (V, B) is a standard space, see, [10, Section-2.1], and T = TL + TL∗ , where TL , the restriction of T to Vq , is an element of GL(Vq ). Conversely, given an element in GL(Vq ), it can be extended to an isometry of (V, B). Hence the conjugacy classes in I(V, B) are parametrized by the usual structure theory of linear maps. Define a form HT on V as follows: For u, v ∈ V, HT (u, v) = B(T u, v). Clearly, if S in I(V, B) commutes with T , then HT (Su, Sv) = B(T Su, Sv) = B(ST u, Sv) = B(T u, v) = HT (u, v). Conversely, if S preserves HT , then HT (Su, Sv) = HT (u, v) implies that for u, v ∈ V, B(ST u, Sv) = B(T Su, Sv). By the non-degeneracy of B, it follows that S commutes with T . Now, suppose E is the splitting field of q(x) (hence of q ∗ (x) also). Let α1 , · · · , αk be distinct roots of q(x) in E. There is a unique automorphism e 7→ ē which maps αi → α−1 i . Further V over E has a decomposition into eigenspaces: k M (Vαi + Vα−1 ). V= i i=1 Without loss of generality, assume V = Vα + Vα−1 . Then HT defines an E-valued Hermitian form on V: when u ∈ Vα and v ∈ Vα−1 , we have HT (u, v) = HT (v, u). Thus, Z(T ) = U(V, HT ). We have now proved the following lemma. Lemma 3.8. Let dim V be even. Let T be a semisimple element in I(V, B) such that mT (x) = q(x)q ∗ (x), where q(x), q ∗ (x) are irreducible polynomials over F and they are dual to each-other. Let E be the splitting field of the minimal polynomial of T . Then the z-class of T is determined by (i) The degree of q(x). (ii) Equivalence class of E-valued Hermitian forms HT on V. 11 4. Classification over a perfect field Let F be a perfect field of characteristic different from two. The group of isometries I(V, B) consists of F-points of a linear algebraic group defined over F. Thus each T in I(V, B) has the Jordan decomposition T = Ts Tu , where Ts is semisimple (that is, every Ts -invariant subspace has a Ts -invariant complement) and Tu is unipotent. Moreover Ts , Tu are also elements of I(V, B), they commute with each other, and they are polynomials in T (see, [12, Chapter 15]). Moreover we have Z(T ) = Z(Ts )∩Z(Tu ). To some extent, the Jordan decomposition reduces the study of conjugacy and z-classes in I(V, B) to the study of conjugacy and z-classes of semisimple and unipotent elements. Suppose T : V → V is a semisimple isometry with prime and self-dual minimal polynomial. Suppose E = F[x]/(p(x)). Then E is a finite simple field extension of F, [E : F] = degree of p(x). Thus the underlying cyclic algebras in Theorem 3.7 (or, Theorem 3.8) are isomorphic to the field E, and the Hermitian forms Hdi are E-valued. Now suppose T is an arbitrary semisimple isometry, and let its minimal polynomial be a product of pairwise distinct prime polynomials over F. Let mT (x) = (x − 1)e (x + 1)f Πki=1 pi (x) Πlj=1 qj (x)qj∗ (x), where e, f = 0 or 1, p1 (x), ..., pk (x) are self-dual, and for j = 1, 2, ..., l, qj (x) is dual to qj∗ (x). Suppose for each i, the degree of pi (x) is 2mi , and for each j, degree of qj (x) is lj . Let the characteristic polynomial of T be χT (x) = (x − 1)l (x + 1)m Πki=1 pi (x)di Πlj=1 qj (x)ej qj∗ (x)ej . The primary decomposition of V with respect to T is determined by the minimal and the characteristic polynomial of T . We get the following orthogonal decomposition of V into T -invariant subspaces: M (4.1) V = V1 ⊕ V−1 ⊕ki=1 Vi ⊕lj=1 (Wj + W∗j ), where V1 = ker (T −I)l , V−1 = ker (T +I)m , for each i = 1, ..., k, Vi = ker pi (T ), and for each j = 1, 2, ..., l, Wj = ker qj (T ), W∗j = ker qj∗ (T ). We have, dim Vi = 2mi di , and dim Wj = lj ej . Let Ei be the field isomorphic to F[x]/(pi (x)), and let Kj be the field isomorphic to F[x]/(qj (x)). As a vector space over Ei , Vi is the direct sum of di copies of Ei . The z-class of T is determined by the z-classes of the restrictions of T to each component in the primary decomposition (4.1). Note that T |V1 = I, T |V−1 = −I. Since I and −I belong to the center of the group, the z-class of T restricted to V1 or V−1 is determined by dim V1 or dim V−1 . Now, the following theorem follows from Theorem 3.7 and Lemma 3.8. 12 Theorem 4.1. Suppose F is perfect. Let T : V → V be a semisimple element in I(V, B). The z-class of T is determined by (i) A finite sequence of integers (l, m; m1 , ..., mk1 ; l1 , ..., lk2 ). 2 1 lj e j . mi di + 2Σkj=1 (ii) A partition of n, π : n = l + m + 2Σki=1 (iii) Field extensions Ei , 1 ≤ i ≤ k1 of F, [Ei : F] = 2mi , and Kj , 1 ≤ j ≤ k2 , [Kj : F] = lj . (iv) Equivalence classes of Ei -valued Hermitian forms Hi , 1 ≤ i ≤ k1 , and Kj -valued Hermitian forms Hj′ , 1 ≤ j ≤ k2 . 5. Finiteness of the z-classes: Proof of Theorem 1.1 If there are only finitely many z-classes of semisimple and unipotent elements, it follows from the Jordan decomposition that there are only finitely many z-classes. So it is enough to show the finiteness of z-classes of semisimple and unipotent elements respectively. Suppose F is a perfect field that has only finitely many field extensions of degree at most n. Then the number of distinct equivalence classes of quadratic forms of rank at most n is finite. Hence the number of equivalence classes of Hermitian forms of rank at most n over an extension field of F is finite. Combining this fact with Theorem 4.1, and the fact that there are only finitely many partitions of n, we have that there are only finitely many z-classes of semisimple elements. Similarly, it follows from Theorem 3.6 that there are only finitely many conjugacy classes of unipotent elements; hence there are only finitely many z-classes of unipotent elements. This completes the proof of Theorem 1.1. References [1] T. Asai, The conjugacy classes in the unitary, symplectic and orthogonal groups over an algebraic number field, J. Math. Kyoto. Univ, 16-2 (1976), 325 - 350. [2] A. 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Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Knowledge city, Sector 81, S.A.S. Nagar, P.O. Manauli 140306, India E-mail address: krishnendu@iisermohali.ac.in, krishnendug@gmail.com Bhaskaracharya Pratishthana, 56/14, Erandavane, Damle Path, Off Law College Road, Pune 411 004, India E-mail address: punekulk@gmail.com 14