Characteristic Polynomials and Fixed Spaces of Semisimple Elements
aa r X i v : . [ m a t h . G R ] D ec CHARACTERISTIC POLYNOMIALS AND FIXED SPACES OFSEMISIMPLE ELEMENTS
ROBERT GURALNICK AND GUNTER MALLE
Dedicated to Len Scott
Abstract.
Answering a question of Frank Calegari, we extend some of our earlierresults on dimension of fixed point spaces of elements in irreducible linear groups. Weconsider characteristic polynomials rather than just fixed spaces. Introduction
In [10], the authors answered a question of Peter Neumann and proved that if G is anontrivial irreducible subgroup of GL n ( k ) = GL( V ) with k a field, then there exists anelement g ∈ G with dim C V ( g ) ≤ (1 /
3) dim V (where C V ( g ) denotes the fixed space of g acting on V ). The example G = SO ( k ) with k not of characteristic 2 shows that 1 / g ∈ G such that the characteristic polynomialof g acting on V was of the form ( T − e f ( T ) where f (1) = 0 and e < n/
2. Calegariand Gee [2] are interested in the irreducibility of the Galois representations associated toself-dual cohomological automorphic forms for GL n , especially for small n . This result iseasily seen to be true for finite G (or more generally compact G ) in characteristic 0 bythe orthogonality relations (see [12, Section 3] for this simple proof). For finite groups(or more generally for algebraic groups or if the characteristic is positive), the questionreduces to finding a semisimple element g ∈ G with dim C V ( g ) < n/ k does not divide the order of the element).The authors in [10, Thm. 1.3] proved a conjecture of Peter Neumann from 1966 aboutthe minimum dimension of fixed spaces. In particular, if G is finite and the characteristicdoes not divide | G | , this gives: Theorem 1.1.
Let k be a field of characteristic p ≥ and G a nontrivial finite subgroupof GL n ( k ) = GL( V ) with p not dividing | G | . If V is an irreducible kG -module, then thereexists a semisimple element g ∈ G with dim C V ( g ) ≤ (1 /
3) dim V . We note that this result depends upon the classification of finite simple groups. How-ever, the result had previously been unknown even for solvable groups.The conclusion of the theorem no longer holds in non-coprime characteristic. The sim-plest example is to take G = A.C the semidirect product of an elementary abelian group
Date : October 10, 2018.2010
Mathematics Subject Classification.
Primary 20C20.The first author was partially supported by DMS 1001962. We thank Frank Calegari for his questionsand comments. A of order 8 with the cyclic group of order 7 acting faithfully. Then G has an irreducible7-dimensional representation over any field of characteristic not 2. In particular, in char-acteristic 7, we see that every nontrivial semisimple element has a 3-dimensional fixedspace. More generally, if p = 2 a − G = A.C with A elementaryabelian of order p + 1 = 2 a and C of order p acting faithfully on A . Then G has an irre-ducible representation of dimension p in characteristic p and every nontrivial semisimpleelement has a fixed space of dimension ( p − /
2. Moreover, taking direct products ofthis group with itself and the corresponding tensor product of representations, one getsexamples of arbitrarily large dimension.Even in characteristic 0, the Example 6.5 in [10] shows that one can do no better than(1 /
9) dim V no matter how large the dimension of V . In this note, we will prove Calegari’sinequality and show that one can do better under various circumstances.If we consider connected algebraic groups, we can obtain similar results that do notdepend upon the classification of finite simple groups. Since a compact real Lie grouphas only semisimple elements and since the minimum value of dim C V ( g ) is attained on anonempty open subset of G , we also obtain: Theorem 1.2.
Let G = 1 be a connected compact real Lie subgroup of GL n ( C ) = GL( V ) .Assume that C V ( G ) = 0 . Then the average dimension (with respect to Haar measure) of C V ( g ) is at most (1 /
3) dim V . Let ǫ >
0. It was shown in [9, Thm. 6] that if G is compact and connected and V isirreducible, then in fact the average dimension of C V ( g ) is less than ǫ dim V as long asdim V is sufficiently large.We have a similar result for Zariski dense subgroups of connected algebraic groups. Theorem 1.3.
Let G = 1 be a subgroup of GL n ( k ) = GL( V ) with k an algebraicallyclosed field and char k = p ≥ . Assume that V is completely reducible, C V ( G ) = 0 andthe Zariski closure of G is connected. Then the set of semisimple g ∈ G with dim C V ( g ) ≤ (1 /
3) dim V is open and Zariski dense in G . An easy consequence of Theorem 1.3 is:
Corollary 1.4.
Let G be an irreducible subgroup of GL n ( k ) = GL( V ) with char k = p ≥ .Assume that G is infinite. Then there exists a semisimple g ∈ G such that dim C V ( g ) ≤ (1 /
3) dim V . Thus, we are reduced to considering finite groups. For the general case, we can show:
Theorem 1.5.
Let = G ≤ GL n ( k ) = GL( V ) with char k = p ≥ . Assume that G actsirreducibly on V . (a) There exists a semisimple g ∈ G with dim C V ( g ) < (1 /
2) dim V . (b) If p > n + 2 or p = 0 , then there exists a semisimple g ∈ G with dim C V ( g ) ≤ (1 /
3) dim V . (c) If p does not divide n , then there exists a semisimple g ∈ G with dim C V ( g ) ≤ (3 /
8) dim V . (d) If n is prime and is a multiplicative generator modulo n , then there exists asemisimple g ∈ G with dim C V ( g ) ≤ (1 /
3) dim V . HARACTERISTIC POLYNOMIALS AND FIXED SPACES OF SEMISIMPLE ELEMENTS 3
In particular, this shows that if n ≤
5, there exists a semisimple element g ∈ G withdim C V ( g ) ≤
1. If n is prime, one can prove even stronger results: Theorem 1.6.
Let G be a finite irreducible subgroup of GL n ( k ) with n an odd prime.Assume that char k = p > n − (or p = 0 ). There exists a semisimple element x ∈ G with all eigenspaces of dimension at most . The paper is organized as follows. In the next section, we deal with algebraic groupsand deduce the various results on algebraic and infinite groups. We then consider variousgeneration results about finite simple groups. In particular, in Section 3 we prove thefollowing results that may be of independent interest:
Theorem 1.7.
Let G be a finite nonabelian simple group and p be a prime. Then unless ( G, p ) = ( A , , there exist p ′ -elements x, y, z ∈ G with xyz = 1 such that G = h x, y i . Theorem 1.8.
Let G be a finite nonabelian simple group and p be a prime. Then thereexist a pair of conjugate p ′ -elements that generate G . Note that an immediate consequence of Theorem 1.7 and Scott’s Lemma [18] is:
Corollary 1.9.
Let G be a finite nonabelian simple subgroup of GL n ( k ) = GL( V ) . As-sume that C V ( G ) = C V ∗ ( G ) = 0 . If G = A and char k = 5 , assume further that V has no trivial composition factors. Then there exists a semisimple element g ∈ G with dim C V ( g ) ≤ (1 /
3) dim V . It is shown [10, Thm. 6.1] that in fact if ǫ > G is a nonabelian finite simple groupand V is an irreducible C G -module, then there exists g ∈ G with dim C V ( g ) < ǫ dim V aslong as dim V is sufficiently large (or equivalently | G | is sufficiently large). This shouldbe true for any algebraically closed field.In Section 4, we consider finite groups and prove Theorem 1.5(a)–(c). We then con-sider representations of prime dimension, complete the proof of Theorem 1.5 and proveTheorem 1.6. In the final section, we give some examples relating to the divisibility ofcharacteristic polynomials of representations (a question asked by Calegari [2]).2. Infinite Groups
In this section we prove Theorems 1.2, 1.3 and Corollary 1.4. These results do notrequire the classification of finite simple groups.We first prove Theorem 1.3. Let k be an algebraically closed field of characteristic p ≥
0. Let G be a subgroup of GL n ( k ) = GL( V ). We assume that G has no trivialcomposition factors on V and that Γ, the Zariski closure of G is connected.Our assumption is that G (equivalently Γ) acts completely reducibly on V . Thus, Γ isreductive. Note that for any e ≥ { g ∈ Γ | dim C V ( g ) ≤ e } is an open subvariety of Γ.Moreover, the set of semisimple elements of Γ is also open.Let S be a rational irreducible k Γ-module. If follows by [7, Thm. 3.3] that the setof pairs of semisimple elements in Γ which generate an irreducible subgroup on S is anontrivial open subvariety of Γ . Thus, the set of pairs of semisimple elements in Γ whichhave no fixed points on V and whose product is semisimple is also an open nonemptysubvariety of Γ . Thus, this set intersects G in a nonempty open subset of G . Choose x, y ∈ G such that h x, y i has no fixed points on V with x, y and xy semisimple. ROBERT GURALNICK AND GUNTER MALLE
By Scott’s Lemma [18], dim C V ( x ) + dim C V ( y ) + dim C V ( xy ) ≤ dim V and so somesemisimple element g ∈ G satisfies dim C V ( g ) ≤ (1 /
3) dim V . This shows that the set ofsemisimple elements of Γ with dim C V ( g ) ≤ (1 /
3) dim V is an open dense subvariety ofΓ. In particular, this set must intersect G whence Theorem 1.3 holds. Theorem 1.2 nowfollows immediately.We now prove Corollary 1.4. Arguing as in [10, Thm. 5.8], it suffices to work overan algebraically closed field. Let Γ be the Zariski closure of G and Γ ◦ the connectedcomponent of the identity in Γ. Note that Γ ◦ = 1 as G is infinite. Since V is irreducible for G , hence for Γ, Γ ◦ acts completely reducibly without fixed points on V . By Theorem 1.3,the set of semisimple g ∈ Γ ◦ with dim C V ( g ) ≤ (1 /
3) dim V contains a dense open subsetof Γ ◦ and therefore intersects G ∩ Γ ◦ non-trivially, as required.We close this section by showing that often one can do even better in the case ofalgebraic groups. There is a version of the following theorem for semisimple groups aswell. Theorem 2.1.
Let G be a simple simply connected algebraic group of rank r at least over an algebraically closed field k . Let V be a completely reducible rational kG -modulewith C V ( G ) = 0 . Let g ∈ G be a regular semisimple element and assume that g is notcentral (the latter can only fail if G = SL ). Then: (a) dim C V ( g ) ≤ (1 /
3) dim V , and (b) if g is also regular, then every eigenspace of g has dimension at most (1 /
3) dim V .Proof. Let C be the conjugacy class of g . We first prove (a). Let X be the variety oftriples of elements all in C with product 1. By [11, Thms. 6.11, 6.15], this is an irreduciblevariety (of dimension 2 dim G − r ) and the set of triples in X which generate a subgroup H such that each irreducible submodule of V remains irreducible for H (and non-isomorphicirreducibles remain non-isomorphic) is a dense open subvariety of X . Now (a) follows byScott’s Lemma.If g is regular semisimple, we consider the variety Y = { ( x, y, z ) ∈ C × C × C − | xyz = 1 } . Precisely as above, we see that the subset of Y consisting of triples so that the subgroupthey generate has the same collection of irreducibles as G is dense. Apply Scott’s Lemmato the elements ( λ − x, λ − y, λ z ) to conclude that the λ -eigenspace of at least one of themhas dimension at most (1 /
3) dim V . (cid:3) Note that we do not need to assume that V is completely reducible in the previous result.The proof goes through verbatim as long as we assume that C V ( G ) = C V ∗ ( G ) = 0.3. Generation results
The purpose of this section is the proof of the following generation results:
Theorem 3.1.
Let G be a finite non-abelian simple group, p a prime. Then we have: (a) G is generated by two conjugate p ′ -elements, and (b) G is generated by three p ′ -elements x, y, z with xyz = 1 unless ( G, p ) = ( A , . HARACTERISTIC POLYNOMIALS AND FIXED SPACES OF SEMISIMPLE ELEMENTS 5
Note that ( A ,
5) is a true exception to the conclusion of Theorem 3.1(b) since the largestelement order of A prime to 5 is 3, and the triangle groups G ( l, m, n ) with l, m, n ≤ G is generatedby a certain triple of conjugate elements with product 1. Thus the remaining task is toprove (a) and (b) for primes p dividing the common order of these elements. This will beshown in the subsequent propositions. Proposition 3.2.
Theorem 3.1 holds for sporadic groups and the Tits group.Proof.
For G a sporadic simple group both parts follow from [11, Table 4], where weexhibited a second generating systems ( x, y, z ) for G with product 1, with x ∼ y and theorders of x, y, z prime to those from [10]. (cid:3) Proposition 3.3.
Theorem 3.1 holds for alternating groups.Proof.
For G = A n an alternating group, with n ≥
11 odd, we produced in [10, Lemma 4.2]a generating system ( x , y , z ) of n − x , y , z ) with x , y squares of an n − o ( z ) = n .For n ≥
12 even, we gave a generating triple of n − x , y both n − o ( z ) = n/
2. Thisgives the claim unless p = 3 divides n . In the latter case, by [1, Cor. 2.2] for n >
12 thereexists two n − n − S n − with the first two being in A n − . The n − A has a generating triple consisting of elements of order 11.For n = 5 , , , , ,
10 we gave generating triples of orders 5,5,7,7,7,7 respectively in[10, Lemma 4.4]. For n = 6 , , , ,
10 direct computation shows that there also existgenerating triples of orders 4,5,15,15,15. Note that A is generated by two 3-cycles (withproduct of order 5). (cid:3) Proposition 3.4.
Theorem 3.1 holds for exceptional simple groups of Lie type.Proof.
For G of exceptional Lie type different from D ( q ), we produced in [11, Thm. 2.2]a second generating system consisting of conjugate elements, while for D ( q ) in [11,Prop. 2.3] we gave a generating triple containing two conjugate elements. Moreover, inall cases, the element orders in these triples are coprime to those from [10]. (cid:3) Finally assume that G is of classical Lie type. For n ≥
2, we let Φ ∗ n ( q ) denote thelargest divisor of q n − q m − ≤ m < n . Proposition 3.5.
Theorem 3.1 holds for L ( q ) , q ≥ .Proof. The groups L ( q ) with q ∈ { , , } are isomorphic to alternating groups, for whichthe claim follows from Proposition 3.3. The group L (7) has generating triples consisting ofelements of order 7, respectively of order 4. For q ≥ q = 9, we gave in [10, Lemmas 3.14,3.15] generating triples for L ( q ) of orders ( q − /d , where d = gcd(2 , q − q + 1) /d , whichproves the claim. (cid:3) ROBERT GURALNICK AND GUNTER MALLE
Definition 3.6.
Let’s say that a pair (
C, D ) of conjugacy classes of a group G is generating if G = h x, y i for all ( x, y ) ∈ C × D .Note that by the result of Gow [6], if ( C, D ) is a generating pair for a finite groupof Lie type consisting of classes of regular semisimple elements, then we find generators( x, y ) ∈ C × D with product in any given (noncentral) semisimple conjugacy class, forexample in C − . Proposition 3.7.
Theorem 3.1 holds for the groups L n ( q ) , n ≥ .Proof. Let G = L n ( q ), n ≥
3. Note that we may assume that ( n, q ) = (3 , , (4 ,
2) asL (2) ∼ = L (7) and L (2) ∼ = A were already handled above. In [10, Prop. 3.13] we showedthat G is generated by a triple of elements of order Φ ∗ n ( q ) when n is odd, respectivelyof order Φ ∗ n − ( q ) when n is even. For n = 4 and ( n, q ) = (6 , (2) is generated bya triple of elements of order 7.Now consider G = SL ( q ). Let C be a conjugacy class of regular semisimple elementsof order ( q − / ( q − ( q ) which might contain an element x ∈ C are the normalizer of Ω − ( q ), of Sp ( q ) orof GL ( q ) ∩ SL ( q ). But in the first two of these groups the centralizer order of elementsof order q + 1 is not divisible by ( q + 1)( q + 1) when q >
3. Thus, any x ∈ C lies ina unique maximal subgroup of G . Let C denote a class of regular semisimple elementsin a maximal torus of order ( q − q −
1) of SL ( q ). Then this does not intersect thenormalizer of GL ( q ) ∩ SL ( q ), so ( C , C ) is a generating pair. By [6] there exist pairswith product in C − , which must generate. Now pass to the quotient of SL ( q ) by itscenter. The group L (3) has a generating triple with elements of order 5. (cid:3) Proposition 3.8.
Theorem 3.1 holds for the unitary groups U n ( q ) , n ≥ , ( n, q ) = (4 , .Proof. Let G = U n ( q ), n ≥
3. The case n = 3 was already treated in [11, Prop. 3.1]. In[10, Prop. 3.11 and 3.12] we showed that G is generated by a triple of elements of orderΦ ∗ n ( q ) when n is odd, respectively of order Φ ∗ n − ( q ) when n is even. For n ≥ n we arguein G = SU n ( q ) and then pass to the quotient by the center. For n = 7 let C containelements of type 6 + and C elements of type 5 − ⊕ + , for n = 5 let C contain elementsof type 4 + and C elements of type 3 − ⊕ + . Then any pair from C × C generates anirreducible subgroup, and by [10, Thm. 2.2] that can’t be proper, when ( n, q ) = (5 , (2) has generating triples of elements of order 5.If n = 4 or 6, the claim of Theorem 3.1(b) holds by [11, Prop. 3.6]. For Theorem 3.1(a),the group U (2) has a generating triple of elements of order 7. Else let’s take C a class ofelements of order Φ ∗ n − ( q ), C a class of elements of order Φ ∗ ( q ) when n = 4, Φ ∗ ( q )Φ ∗ ( q )when n = 6, and C one of C , C . Then the arguments in the proof of [10, Prop. 3.12]go through, with even better estimates, since the number of characters not vanishing onboth C and C is smaller, and the same for the number of maximal subgroups containingelements from both classes. It follows that there exist generating triples with product 1 HARACTERISTIC POLYNOMIALS AND FIXED SPACES OF SEMISIMPLE ELEMENTS 7 in C × C × C and in C × C × C . Since the orders in the two classes are relativelycoprime, this gives the result. (cid:3) Proposition 3.9.
Theorem 3.1 holds for the orthogonal groups O n +1 ( q ) , n ≥ .Proof. The group G = O n +1 ( q ) possesses a generating triple of elements of order Φ ∗ n ( q ),by [10, Prop. 3.7 and 3.8]. For n ≥
7, we produced in [11, Prop. 7.9] a generating pair(
C, D ) of conjugacy classes containing regular semisimple elements of orders prime toΦ ∗ n ( q ). For n = 4 , ,
6, let C contain elements of type ( n − − ⊕ − , D elements of type( n − − ⊕ + . Then the group generated by ( x, y ) ∈ C × D acts irreducibly or is containedin a 2 n -dimensional orthogonal group. By consideration of suitable Zsigmondy primes, thelatter can possibly only occur when q ≤
3, which we exclude for the moment. Otherwise,an application of [11, Cor. 3.4] shows that (
C, D ) is a generating pair, and we concludeusing [6]. The groups O (2) ∼ = S (2), O (3), O (2) ∼ = S (2), O (3), O (2) ∼ = S (2),O (3), possess generating triples with elements of orders 7, 13, 31, 41, 31, 61 respectively,Finally, for n = 3, we argued in [11, Prop. 3.8] that conjugacy classes of regular semisimpleelements of types 3 + , 2 − ⊕ + form a generating pair in O ( q ) for q ≥
5, and we producedgenerating triples of orders 7, 13 and 17 for O (2) ∼ = S (2), O (3), O (4) respectively. (cid:3) Proposition 3.10.
Theorem 3.1 holds for the symplectic groups S n ( q ) , n ≥ , ( n, q ) =(2 , .Proof. Note that we may assume that q is odd when n ≥ G = S n ( q ) possesses a generating triple of elements of order Φ ∗ n ( q ), resp.order 5 when ( n, q ) = (2 , n ≥
3, ( n, q ) = (4 , C, D ) of conjugacy classes containing regular semisimpleelements of orders prime to Φ ∗ n ( q ), and we may conclude as usual. The group S (3) hasa generating triple with elements of order 13.For S ( q ) with q ≥
3, the claim was already proved in [11, Prop. 3.1]. (cid:3)
Proposition 3.11.
Theorem 3.1 holds for the orthogonal groups O +2 n ( q ) , n ≥ .Proof. Let G = O +2 n ( q ), n ≥
4. In [10, Prop. 3.10] we showed that G is generated by atriple of elements of order dividing Φ ∗ n − ( q )( q +1). For n = 4 we showed in [11, Prop. 3.10]that there also exist generating pairs of conjugate regular semisimple elements of orderscoprime to the former ones, and we may conclude as before.For the n = 4 we may assume that q ≥
3, since for O +8 (2) there exist generating triples ofelements of order 9, and also of elements of order 7, by [10, Prop. 3.10] and [11, Prop. 3.10].Let C contain regular semisimple elements of order ( q + 1) /d , with d = gcd(2 , q − T of order ( q + 1) /d , and C the image of C under triality. Let( x, y ) ∈ C × C . Then H := h x, y i contains T up to conjugation. By [16, Table I] the onlymaximal subgroup of O +8 ( q ) with this property is M = (O − ( q ) × O − ( q )) . . Accordingto [6] there are ( x, y ) with product of order a Zsigmondy prime divisor of q + q + 1, andhence not contained in M . This gives a triple as in Theorem 3.1(b).We now show Theorem 3.1(a) for n = 4 and q ≥
3. Let C be a conjugacy class ofelements of order Φ ∗ ( q ). Let C be a conjugacy class of regular semisimple elements oforder ( q − / q − q is even) having precisely two non-trivial invariant subspaces.Let C be either C or C . Arguing as in [10, Prop. 3.10], one gets a lower bound for the ROBERT GURALNICK AND GUNTER MALLE number of triples ( x, y, z ) ∈ C × C × C with product 1 (the bound is actually muchbetter than in [10] since there will be many fewer characters not vanishing on C and C ).Similarly, one gets an upper bound for the number of non-generating such triples (againthe bound is much better than that given in [10] since there are many fewer maximalsubgroups — for example, it is clear the group generated is irreducible). It follows thattwo elements of C or C will generate. Since the orders of the elements in C and C arerelatively prime, the result follows. (cid:3) Proposition 3.12.
Theorem 3.1 holds for the groups orthogonal O − n ( q ) , n ≥ .Proof. The group G = O − n ( q ) possesses a generating triple of elements of order Φ ∗ n ( q ),by [10, Prop. 3.6]. For ( n, q ) / ∈ { (4 , , (4 , , (5 , , (6 , } , we produced in [11, Prop. 7.10]a generating pair ( C, D ) of conjugacy classes of G containing regular semisimple elementsof orders prime to Φ ∗ n ( q ). Now by [6] there exist triples in C × C × D , for example. Directcomputation shows that the groups O − (2), O − (4), O − (2), O − (2) possess generatingtriples with elements of orders 7, 13, 17, 31 respectively, which are again prime to Φ ∗ n ( q ). (cid:3) Finite Groups
Here, we prove Theorem 1.5(a)–(c). By Corollary 1.4 it suffices to consider finite groups.Fix a field k of characteristic char k = p . By Theorem 1.1, we may assume that p > = G ≤ GL n ( k ) = GL( V ) is irreducible on V . By extending scalars, wecan reduce to the case that k is algebraically closed ( V would at worst be a direct sum ofGalois conjugates of a given irreducible module).Let N be a minimal normal subgroup of G . Thus, N acts completely reducibly on V and without fixed points. We break up the argument depending upon the structure of N . Lemma 4.1. If | N | is odd, then N is an r -group for some odd prime r = p and thereexists a p ′ -element g ∈ N with dim C V ( g ) < (1 /r ) dim V ≤ (1 /
3) dim V .Proof. Since | N | is odd, N is solvable and since it is a minimal normal subgroup, it mustbe an r -group with r = p . Now apply [12, Cor. 1.3]. (cid:3) Lemma 4.2. If N is an elementary abelian -group, then p = 2 and (a) there exists an involution g ∈ N with dim C V ( g ) < (1 /
2) dim V ; (b) if p does not divide dim V , then there exists a p ′ -element g ∈ G with dim C V ( g ) ≤ (1 /
3) dim V .Proof. Clearly, p = 2. By [12, Cor. 1.3], there exists g ∈ N with dim C V ( g ) < (1 /
2) dim V .So we may assume that p does not divide dim V . If N is central, then any 1 = g ∈ N satisfies C V ( g ) = 0 and there is nothing to prove. Otherwise, V = L V i where the V i arethe distinct N -eigenspaces. Since V is irreducible, G permutes the V i transitively. By [4,Thm. 1], there exists x ∈ G of prime power order r a having no fixed points on the set of V i . Since p does not divide dim V , r = p . If r is odd, then every orbit of x has size at least3 whence dim C V ( x ) ≤ (1 /
3) dim V . So we may assume that r = 2. It follows as in theproof of [10, Thm. 5.8] that the average dimension of the fixed point spaces of elementsin the coset xN is at most (1 /
4) dim V . Since every element in xN is a 2-element, theresult follows. (cid:3) HARACTERISTIC POLYNOMIALS AND FIXED SPACES OF SEMISIMPLE ELEMENTS 9
The remaining case is when N is a direct product of t ≥ L . Let W be an irreducible kN -submodule of V . By reordering,we may write W = U ⊗ · · · ⊗ U t where each U i is an irreducible kL -module with U i ∼ = k if and only if i > m for some m >
1. First we note that the proof of [10, Cor. 5.7] gives:
Lemma 4.3.
Let L be a simple group of Lie type over a finite field of characteristic p and let E = L × · · · × L . Let k be an algebraically closed field of characteristic p , V acompletely reducible kE -module with C V ( E ) = 0 . Then there exists a semisimple element x ∈ E with dim C V ( x ) ≤ (1 /
3) dim V . Actually, the proof in [10] does not work if L ∼ = L (5) with p = 5, or L ∼ = L (7) with p = 7. A more complicated proof can be given in these cases to show that the result isstill true. The result also follows easily if L = L ( q ) using Theorem 2.1. Lemma 4.4.
Let L be a nonabelian finite simple group and let E = L × · · · × L ( t copies).Let k be an algebraically closed field of characteristic p , V = U ⊗ · · · ⊗ U t a nontrivialirreducible kE -module with dim U i = 3 for some i . There exists a p ′ -element x ∈ E (independent of V ) such that all eigenspaces of x have dimension at most (1 /
3) dim V .Proof. First assume that t = 1, so dim V = 3. By inspection, we can choose x of oddprime order with distinct eigenvalues (indeed unless p = 3, we can take x of order 3).Suppose that t >
1. Let y be the element of E with all coordinates equal to the x chosen above. Since x has all eigenspaces of dimension at most (1 /
3) dim U i on U i , thesame is true on V . (cid:3) Lemma 4.5.
Let L be a finite group with L = h x, y i and let E = L × · · · × L . Let k bean algebraically closed field of characteristic p , V = U ⊗ · · · ⊗ U t a nontrivial irreducible kE -module with dim U i ≥ for some U i . Let G be a diagonal copy of L in E . Let x , y and z be elements of G with each coordinate x, y or z = ( xy ) − respectively. Then dim C V ( x ) + dim C V ( y ) + dim C V ( z ) ≤ (9 /
8) dim V . In particular, min { dim C V ( x ) , dim C V ( y ) , dim C V ( z ) } ≤ (3 /
8) dim V. Proof.
The assumption that some U i has dimension at least 4 gives that dim C V ( G ) ≤ (1 /
16) dim V . Indeed, assume that dim U ≥
4. Then C V ( G ) ∼ = Hom G ( U ∗ , U ⊗ · · · ⊗ U t ).Since dim U ≥ C V ( G ) ≤ (1 /
4) dim( U ⊗ · · · ⊗ U t ) ≤ (1 /
16) dim V. Similarly, dim C V ∗ ( G ) ≤ (1 /
16) dim V . By Scott’s Lemma [18]dim C V ( x )+ dim C V ( y ) + dim C V ( z ) ≤ dim V + dim C V ( G ) + dim C V ∗ ( G ) ≤ (9 /
8) dim V. The result follows. (cid:3)
Corollary 4.6.
Let G be a finite group, k an algebraically closed field of characteristic p and V a faithful irreducible kG -module. Let E = L × · · · × L be a minimal normalsubgroup of G with L a nonabelian simple group. Then there exists a p ′ -element x ∈ E with dim C V ( x ) ≤ (3 /
8) dim V . Proof. If L is of Lie type in characteristic p , the result follows by Lemma 4.3. So assumethat this is not the case.Let W be any irreducible kE -submodule of V . Then W = U ⊗ · · · ⊗ U t where each U i is an irreducible kL -module.If dim U i = 2, then p = 2 and L ∼ = SL (2 f ) is of Lie type in characteristic 2, whencethe result by Lemma 4.3.If dim U i = 3 for some i , then this will be the case for every irreducible kE -submodule(since any such is a twist of W ) and Lemma 4.4 applies.In the remaining case, dim U i > U i (and similarly for every twistof W ). By Theorem 1.7, we can choose p ′ -elements x, y, z ∈ L which generate L and haveproduct 1 aside from ( L, p ) = ( A , A ∼ = L (5), so the latter is of Lie typein defining characteristic, a case we already dealt with (alternatively, it would follow thateach nontrivial U i has dimension 5 and so if x is an element of E with each coordinate oforder 3, dim C W ( x ) ≤ (9 /
25) dim
V < (3 /
8) dim W ). It follows by Lemma 4.5 that thereexists a p ′ -element g ∈ E with dim C V ( g ) ≤ (3 /
8) dim V . (cid:3) We can now prove the first three parts of Theorem 1.5.
Proof of Theorem 1.5.
Parts (a) and (c) follow from Lemmas 4.1, 4.2 and Corollary 4.6.Now assume that p > dim V + 2. By Lemma 4.1, we may assume that G has no oddorder non-trivial normal subgroups. If O ( G ) = 1, Lemma 4.2 applies since p does notdivide dim V . So we may assume that F ( G ) = 1. Let N = L × · · · × L be a minimalnormal subgroup of G with L a nonabelian simple group. By [8, Thm. B], it follows thatone of the following holds:(1) p does not divide | N | ;(2) L is a finite group of Lie type in characteristic p ; or(3) p = 11, N = J and dim V = 7.Thus by [10, Cor. 5.7] there exists x ∈ N with dim C V ( x ) ≤ (1 /
3) dim V . If x is a p ′ -element, we are done. So we may assume that p divides the order of x (and so | N | ).If N = J , p = 11 and dim V = 7, an element g of order 19 satisfies dim C V ( g ) = 1 < (1 /
3) dim V . If L has Lie type, then Lemma 4.3 yields a semisimple element g ∈ N withdim C V ( g ) ≤ (1 /
3) dim V . This completes the proof of (b). (cid:3) Prime Degree
Recall that a group is called quasi-simple if G is perfect and G/Z ( G ) is a nonabeliansimple group.Let n be an odd prime. Let k be a field with char k = p . Let G ≤ GL n ( k ) = GL( V )be irreducible and finite. If G is not absolutely irreducible, then G must be cyclic andany generator has distinct eigenvalues on V (and is semisimple). So assume that G isabsolutely irreducible and k is algebraically closed. Lemma 5.1. If p = n and the Sylow n -subgroup is not abelian, then G contains a subgroup A of order n such that V is a free kA -module.Proof. Let N be a minimal nonabelian n -subgroup of G . Then N acts irreducibly and isextraspecial, whence the result is clear. (cid:3) HARACTERISTIC POLYNOMIALS AND FIXED SPACES OF SEMISIMPLE ELEMENTS 11
Lemma 5.2.
Suppose that G acts imprimitively. (a) If n = p , then the Sylow n -subgroup S of G has order n and V is a free kS -module. (b) If n = p , then there exists an element x ∈ S with order a power of n having alleigenspaces of dimension at most .Proof. Since n is prime, G imprimitive implies that G permutes n one dimensional (linearlyindependent) subspaces. Thus, G surjects onto a transitive permutation group of degree n . In particular, n divides the order of G . Let N be the normal subgroup of G stabilizingeach of the n one dimensional spaces. If p = n , then N has order prime to p , whence S has order n and the result follows.So assume that p = n . Let x ∈ G be an n -element with x not in N . Then x n is centralin GL( V ) and its minimal polynomial is x n − a for some a ∈ k × , whence the result. (cid:3) Note that since n is prime if N is a minimal normal noncentral subgroup of G , theneither N is an elementary abelian r -group for some prime r = p or N acts irreducibly.If N acts irreducibly, then either N is an n -group and p = n , whence N contains a(semisimple) element with distinct eigenvalues or N is quasi-simple. If Z ( N ) = 1, then N contains a semsimple element x with C V ( x ) = 0. If N is simple, then Corollary 1.9implies that there exists x ∈ N semisimple with dim C V ( x ) ≤ (1 /
3) dim V (if N = A ,the result follows by inspection).If N is abelian and not a 2-group, then [12, Cor. 1.3] implies that there exists a (semisim-ple) x ∈ N with dim C V ( x ) < (1 /
3) dim V . Finally, if N is a 2-group and 2 is a mul-tiplicative generator modulo n , then | N | = 2 n − (because G permutes transitively the n eigenspaces of N and the smallest irreducible module of a cyclic group of order n incharacteristic 2 has size 2 n − ). It follows that there exists x ∈ N with − x a reflection,whence dim C V ( x ) = 1 ≤ (1 /
3) dim V .In particular, we have proved part (d) of Theorem 1.5.We next consider quasi-simple groups and first show: Theorem 5.3.
Let G be a finite quasi-simple group of Lie type in characteristic p . Let k be an algebraically closed field of characteristic p . Suppose that V is a faithful irreducible kG -module of odd prime dimension n ≤ p . Then V is a twist of a restricted module andone of the following holds: (1) G = SL ( q ) ; (2) G = G ( q ) or G ( q ) ′ , n = 7 ; (3) G = Ω n ( q ) and V is a Frobenius twist of the natural module; or (4) G = SL n ( q ) or SU n ( q ) and V is a Frobenius twist of the natural module or its dual.Moreover, either there exists a semisimple element x ∈ G with all eigenspaces of dimensionat most on V or G = SL ( p ) and p ≤ n − . In all cases, there exists a semisimpleelement x ∈ G with dim C V ( x ) ≤ .Proof. For the first part, it suffices to prove the result for algebraic groups. Let V = L ( λ )where λ is the highest weight for V . By the Steinberg tensor product theorem and thefact that n is prime, V is a Frobenius twist of some restricted irreducible. So we mayassume that λ is p -restricted. By [14], it follows that L ( λ ) is also the Weyl module whencethe Weyl dimension formula holds for L ( λ ). Thus, it suffices to work in characteristic 0.The result in that case is due to Gabber, see [15, 1.6]. Aside from the first case, any regular semisimple element of sufficiently large order willhave n distinct eigenvalues on V . Suppose that G = SL ( q ). Let x ∈ G have order q + 1.If two distinct weights for a restricted module can coincide on x , then 2 dim V − ≥ q + 1.This can only occur if q = p ≤ n −
3. Moreover, no nontrivial weight vanishes on anelement of order q + 1. Since a restricted irreducible k SL ( q )-module has distinct weights,the result follows. (cid:3) Theorem 5.4.
Let n be an odd prime. Let G ≤ GL( V ) = GL n ( k ) be a finite irreduciblequasi-simple group, where k is an algebraically closed field of characteristic p . (a) If p > max { n − , n + 2 } or p = 0 , then there exists a semisimple x ∈ G with alleigenvalues distinct. (b) If p > n + 2 , there exists a semisimple element x ∈ G with dim C V ( x ) ≤ .Proof. If p divides | G | , then by [8, Thm. B], G is a finite group of Lie type in characteristic p (or G = J , p = 11 and n = 7, where we may take x of order 19) and Theorem 5.3applies.If p does not divide | G | , then either p = 0 or V is the reduction of a characteristic0 module. The list of possible groups and modules is given in [3, Thm. 1.2]. It isstraightforward to see that the conclusion holds for these groups (most of the examplesare related to Weil representations). (cid:3) Theorem 1.6 now follows from the previous results aside from the case n = 3 andchar k = 5. In that case, any noncentral semisimple element of order greater than 2 hasdistinct eigenvalues. The next example shows that we do need some restriction on thecharacteristic. Example 5.5.
Let k be an algebraically closed field of positive characteristic p . Let G = SL p ( k ) = SL( V ). Then G acts by conjugation on W := End( V ). Since everysemisimple g ∈ G is centralized by a maximal torus, we see that dim C W ( g ) ≥ p . Notethat W is a uniserial module with two trivial composition factors and an irreduciblecomposition factor V of dimension p −
2. Clearly, dim C V ( g ) ≥ dim C W ( g ) − ≥ p − g of G (and since semisimple elements are Zariski dense, thisis true for any g ∈ G ). Note that dim V can be prime (eg, this is true for p = 5 , , G ( p a ) := SL p ( p a ) for any a ≥ Characteristic Polynomials of Representations
Let G be a group and V a finite dimensional kG -module with k a field of characteristic p ≥
0. Let ch V denote the function from G to k [ x ] defined by ch V ( g ) = det( xI − g ). Notethat two modules have the same function if and only if their composition factors are thesame. Our results on bounds for dim C V ( g ) for some g ∈ G can be phrased in asking:given a kG -module V what is the largest power of ch k that divides ch V ?Frank Calegari asked what one could say if V and W are two irreducible kG -modulesand ch V divides ch W . Calegari and Gee [2] used this information to study Galois repre-sentations in very small dimensions.While we suspect that this does impose some constraints on the representations, wegive some examples to show that it is not that rare (at least for groups of Lie type andalgebraic groups in the natural characteristic). HARACTERISTIC POLYNOMIALS AND FIXED SPACES OF SEMISIMPLE ELEMENTS 13
Example 6.1.
Let G be a simple algebraic group over an algebraically closed field ofcharacteristic p >
0. Let V be an irreducible kG -module. By Steinberg’s tensor producttheorem, V = V ⊗ · · · ⊗ V m where V i is a twist of a restricted module by the i th powerof Frobenius. If 0 is a weight for some V j , then clearly ch V is a multiple of ch V ′ j , where V ′ j is the tensor product of all the V i , i = j .The following example was shown to us by N. Wallach (in particular see [19]). Example 6.2.
Let k be an algebraically closed field of characteristic 0. Let G be a simplealgebraic group over k . Let λ and µ be dominant weights with µ in the root lattice. Thench V ( λ ) divides ch V ( λ + µ ) .It follows the same is true in positive characteristic p as long as p is sufficiently large(depending upon λ and µ ). Here are a few cases where one can compute this directly. Wegive one such case. Example 6.3.
Let k be an algebraically closed field of characteristic p ≥
0. Let G =SL n ( k ) and let V = V ( λ ) be the natural module. Then ch V (( s + n ) λ ) is a multiple ofch V ( sλ ) for p > s + n (or p = 0). References [1]
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