On a question of Külshammer for representations of finite groups in reductive groups
aa r X i v : . [ m a t h . G R ] M a y ON A QUESTION OF K ¨ULSHAMMER FOR REPRESENTATIONS OFFINITE GROUPS IN REDUCTIVE GROUPS
MICHAEL BATE, BENJAMIN MARTIN, AND GERHARD R ¨OHRLE
To Burkhard K¨ulshammer on his sixtieth birthday
Abstract.
Let G be a simple algebraic group of type G over an algebraically closed fieldof characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ andan infinite family of pairwise non-conjugate homomorphisms ρ : Γ → G whose restrictions toΓ are all conjugate. This answers a question of Burkhard K¨ulshammer from 1995. We alsogive an action of Γ on a connected unipotent group V such that the map of 1-cohomologiesH (Γ , V ) → H (Γ p , V ) induced by restriction of 1-cocycles has an infinite fibre. Introduction
Let k be an algebraically closed field and let Γ be a finite group. By a representation ofΓ in a linear algebraic group H over k , we mean a group homomorphism from Γ to H . Wedenote by Hom(Γ , H ) the set of representations ρ of Γ in H ; this has the natural structure ofan affine variety over k (see, e.g., [11, II.2]). The group H acts on Hom(Γ , H ) by conjugationand we call the orbits H · ρ conjugacy classes .If either char( k ) = 0 or char( k ) = p > | Γ | is coprime to p , then every representationof Γ in GL n ( k ) is completely reducible and Hom(Γ , GL n ( k )) is a finite union of conjugacyclasses, by Maschke’s Theorem. Now suppose that char( k ) = p > p divides | Γ | . It is nolonger true that Hom(Γ , GL n ( k )) is a finite union of conjugacy classes—for example, this failseven for n = 2 and Γ = C p × C p (cf. the last paragraph of the proof of Theorem 1.2 below).Let Γ p be a Sylow p -subgroup of Γ. It is natural to ask instead whether representationsof Γ are controlled by their restrictions to Γ p . Burkhard K¨ulshammer raised the followingquestion in 1995 in [5, Sec. 2] (see also [11, I.5]). Question . Let G be a linear algebraic group and let σ ∈ Hom(Γ p , G ). Are there onlyfinitely many conjugacy classes of representations ρ ∈ Hom(Γ , G ) such that ρ | Γ p is conjugateto σ ?Straightforward representation-theoretic arguments show that the answer is yes if G =GL n ( k ) (see [5, Sec. 2]). On the other hand, an example of Cram with p = 2 shows that theanswer is no in general if we allow G to be non-connected and non-reductive [4].For the rest of this paper, we assume G is connected and reductive. Slodowy proved thatthe answer to Question 1.1 is yes under some extra hypotheses [11]; we briefly summarisehis results. If one embeds G in some GL n ( k ), then Hom(Γ , G ) embeds in Hom(Γ , GL n ( k )).Given ρ ∈ Hom(Γ , G ), the set (GL n ( k ) · ρ ) ∩ Hom(Γ , G ) splits into a union of G -conjugacy Mathematics Subject Classification.
Key words and phrases.
Modular representations of finite groups; reductive algebraic groups; conjugacyclasses; nonabelian 1-cohomology. lasses; in the first part of his paper, Slodowy applies a beautiful geometric argument dueto Richardson [8] to show that this union is finite when p is good for G , which allows one todeduce a positive answer to Question 1.1 for G from the positive answer for GL n ( k ) [11, I.5,Thm. 3].The second part of Slodowy’s paper gives a different criterion for Question 1.1 to havepositive answer: he shows that if σ (Γ p ) has reduced centralizer in G then there are onlyfinitely many conjugacy classes of representations ρ ∈ Hom(Γ , G ) such that ρ | Γ p is conjugateto σ [11, II.4, Cor. 1]. An important ingredient in this proof, which dates back to work ofAndr´e Weil, is that one can interpret elements of the tangent space to Hom(Γ , G ) at ρ aselements of the space of 1-cocycles Z (Γ , g ), where g denotes the Lie algebra of G and Γ actson g by γ · X = Ad( ρ ( γ ))( X ). In fact, Slodowy proved a more general finiteness criterion interms of the “inseparability defects” of ρ and ρ | Γ p [11, II.4, Thm. 2] . The case of arbitraryconnected reductive G was, however, still left open.In this note we show that the answer to Question 1.1 is no in general for connectedreductive G . We prove the following result. Theorem 1.2.
Suppose G is a simple algebraic group of type G and char( k ) = 2 . Let q > be odd, let D q denote the dihedral group of order q , let Γ = D q × C = h r, s, z | r q = s = z = 1 , srs − = r − , [ r, z ] = [ s, z ] = 1 i and let Γ = h s, z i (a Sylow -subgroup of Γ ).Then there exist representations ρ a ∈ Hom(Γ , G ) for all a ∈ k such that the ρ a are pairwisenon-conjugate and the restrictions ρ a | Γ are conjugate for all a ∈ k . In [2, Sec. 7] the authors and Tange constructed families of finite subgroups of G = G in characteristic 2 with unusual properties (note, for example, that [2, Ex. 7.15] shows thatRichardson’s argument can fail in bad characteristic). Our proof of Theorem 1.2 involves amodification of this construction.Our results can be interpreted in the language of nonabelian 1-cohomology (see Section 3).Let Γ act by group automorphisms on a unipotent group V . One can form the 1-cohomologyH (Γ , V ), and the inclusion of Γ p in Γ gives a map Θ from H (Γ , V ) to H (Γ p , V ) induced byrestriction of 1-cocycles. Theorem 1.3.
Let p = 2 , let q > be odd and let Γ = D q × C . There is an action of Γ on a connected unipotent group V such that the map Θ has an infinite fibre. This is in sharp contrast to the case when V is abelian: standard results from abeliancohomology (cf. [3, III, Prop. 10.4]) show that if V is an abelian unipotent group (e.g., afinite-dimensional vector space over k ) on which Γ acts by group automorphisms then Θis injective. In fact, Slodowy uses precisely this result in the special case when V is theΓ-module g on the way to proving [11, II.4, Thm. 2] (see [11, II.4, Lem.]).Lond gave a different example with Θ having an infinite fibre [7, Ex. 4.1], using theexample of Cram discussed above. In our case, the group V is the unipotent radical of aparabolic subgroup P of a simple group G of type G , and Γ acts on V by conjugation, via ahomomorphism σ : Γ → P . Theorem 1.3 follows quickly from the construction in Section 2(see Section 3). This result actually holds for non-reductive G as well. . Proof of Theorem 1.2
Until the end of this section we take G to be a simple algebraic group of type G andchar( k ) to be 2. We recall some notation from [2, Sec. 7]. The positive roots of G withrespect to a fixed maximal torus T and a fixed Borel subgroup containing T are α (short), β (long), α + β , 2 α + β , 3 α + β and ω := 3 α + 2 β . Given a root δ , we denote the correspondingroot group by U δ and coroot by δ ∨ . We fix a group isomorphism κ δ : k → U δ . We write G δ for h U δ ∪ U − δ i and we set s δ = κ δ (1) κ − δ (1) κ δ (1); then s δ represents the reflection correspondingto δ in the Weyl group of G (since char( k ) = 2, s δ has order 2).Fix t ∈ α ∨ ( k ∗ ) such that | t | = q . For a ∈ k , define ρ a ∈ Hom(Γ , G ) by ρ a ( r ) = t, ρ a ( s ) = s α κ ω ( a ) , ρ a ( z ) = κ ω (1) . It is easily checked that this is well-defined (note that [ G α , G ω ] = 1). Set u ( x ) = κ β ( x ) κ α + β ( x )for x ∈ k . Then u ( x ) commutes with U ω and u ( x ) s α u ( x ) − = s α κ ω ( x ) (see the first para-graph of [2, p. 4307]). It follows that u ( √ a ) · ( ρ | Γ ) = ρ a | Γ .To complete the proof of Theorem 1.2, we now need to show that the ρ a are pairwise non-conjugate. Let a, b ∈ k and suppose g · ρ a = ρ b for some g ∈ G . Then g ∈ C G ( t ). It followsfrom [2, (7.1) and (7.2)] that C G ( t ) = T G ω (this is where we need our assumption that q > g = hm with h ∈ T and m ∈ G ω . We have ( hm ) s α κ ω ( a )( hm ) − = s α κ ω ( b ), so hs α h − ( hm ) κ ω ( a )( hm ) − = s α κ ω ( b ) since m commutes with s α . Now G α ∩ G ω = 1(see the paragraph following [2, (7.8)]), so the condition hs α h − ( hm ) κ ω ( a )( hm ) − = s α κ ω ( b )forces h to commute with s α , as hs α h − ∈ G α and ( hm ) κ ω ( a )( hm ) − ∈ G ω . A simplecalculation now shows that h ∈ ker( α ) ⊆ G ω . Hence g ∈ G ω . But G ω is a simple groupof type A , so the pair ( κ ω ( a ) , κ ω (1)) is not G ω -conjugate to the pair ( κ ω ( b ) , κ ω (1)) unless a = b . We conclude that ρ a and ρ b are not conjugate if a = b , as required. Remarks . (i). Choose an embedding i of G in some GL n ( k ). Then the representations i ◦ ρ a of Γ in GL n ( k ) fall into finitely many GL n ( k )-conjugacy classes, since Question 1.1 haspositive answer for GL n ( k ). Hence there exists a ∈ k such that (GL n ( k ) · ρ a ) ∩ Hom(Γ , G )is an infinite union of G -conjugacy classes. This gives another example of the phenomenonin [2, Ex. 7.15] discussed above.(ii). It follows from Slodowy’s result [11, II.4, Thm. 2] discussed above that ρ a has greaterinseparability defect than ρ a | Γ for at least one a ∈ k . In fact, it can be shown using thecalculations in [2, Sec. 7] that if a = 0 then ρ a has inseparability defect 1 and ρ a | Γ hasinseparability defect 5. This answers a question of Slodowy [11, II.4, Rem. 2].We do not know of any analogous examples in odd characteristic; recall from the discussionin Section 1 that if such an example exists then p must be bad for G . Our construction isclosely related to the construction of a certain triple ( G, M, H ) in [2, Sec. 7], where G = G , M is a reductive subgroup of G and H is a finite subgroup of M . We guess that furtherexamples can be obtained from other triples ( G, M, H ) with similar properties, but we leavethis for future work. The mechanism for producing these triples works only in characteristic2 (see the paragraph following [15, Rem. 1.6]). Uchiyama found triples (
G, M, H ) for G oftype E [15, Sec. 3], and showed that the construction fails for several cases involving groupsof rank at most 6, including A , A , B and E [14, Thm. 3.1.1, Ch. 4].It seems an interesting problem to find examples like that of Cram [4] but in odd charac-teristic, where we allow G to be non-reductive. . Nonabelian 1-cohomology
Another approach to K¨ulshammer’s problem is via the 1-cohomology of the unipotentradical R u ( P ), where P is a proper parabolic subgroup of G . Here is a brief explanation.Recall that a closed subgroup M of G is said to be G -completely reducible if whenever M iscontained in a parabolic subgroup P of G , M is contained in some Levi subgroup of P [10],[9]. As a special case, we say that M is G -irreducible if M is not contained in any properparabolic subgroup of G at all. We say that ρ ∈ Hom(Γ , G ) is G -completely reducible (resp., G -irreducible) if its image is.Although in general Hom(Γ , G ) is an infinite union of conjugacy classes for reductive G , it was proved in [1, Cor. 3.8] that there are only finitely many conjugacy classes ofrepresentations that are G -completely reducible. This generalizes the classical result that afinite group admits only finitely many completely reducible n -dimensional representationsin any characteristic. Moreover, it follows from [1, Cor. 3.7] that the conjugacy classes of G -completely reducible representations of Γ in G are precisely the conjugacy classes thatare Zariski-closed subsets of Hom(Γ , G ). Given ρ ∈ Hom(Γ , G ), choose a minimal parabolicsubgroup P of G with ρ (Γ) ⊆ P . Let L be a Levi subgroup of P and let π : P → L bethe canonical projection. It follows from [1, Cor. 3.5] that σ := π ◦ ρ ∈ Hom(Γ , L ) is L -irreducible and G -completely reducible. Conversely, given G -irreducible σ ∈ Hom(Γ , G ), wecan consider the set C σ of all ρ ∈ Hom(Γ , P ) such that π ◦ ρ = σ . By the result described inthe first sentence of this paragraph, there are only finitely many possibilities for ( P, L, σ ) upto G -conjugacy. Hence if C ⊆ Hom(Γ , G ) is an infinite union of G -conjugacy classes thenfor some triple ( P, L, σ ), C σ must meet infinitely many G -conjugacy classes in C . Thus wehave reduced the “global” problem of considering all representations into G to the “local”problem of considering all representations into a fixed proper parabolic subgroup P .Next we study the structure of C σ for fixed ( P, L, σ ). Let V = R u ( P ). Given ρ ∈ C σ ,there is a unique function θ ρ : Γ → V defined by ρ ( γ ) = θ ρ ( γ ) σ ( γ ). It is easily checkedthat θ ρ satisfies the 1-cocycle relation θ ρ ( γ γ ) = θ ρ ( γ )( γ · θ ρ ( γ )), where Γ acts on V by γ · v = σ ( γ ) vσ ( γ ) − . The converse is also true, so we have a bijection between C σ and thespace of 1-cocycles Z (Γ , σ, V ). A simple calculation shows that ρ, µ ∈ C σ are V -conjugateif and only if the images θ ρ of θ ρ and θ µ of θ µ in H (Γ , σ, V ) are equal. Thus we have aninterpretation of V -conjugacy classes in C σ in terms of 1-cohomology (cf. the proof of [11,I.5, Lem. 1]).This idea has been used in a slightly different context to study embeddings of reductivealgebraic groups inside simple algebraic groups [6], [12], [13], [7]. In our case we have an extraingredient arising from restriction of representations. The restriction map from Hom(Γ , G )to Hom(Γ p , G ) maps C σ to C σ | Γ p . Restriction of cocycles gives a map from Z (Γ , σ, V ) toZ (Γ p , σ | Γ p , V ) which is compatible with the correspondence between representations and1-cocycles, and this descends to give a map Θ from H (Γ , σ, V ) to H (Γ p , σ | Γ p , V ). See [7,Ch. 3–4] for a fuller explanation.Now we recast our example in this language. Let G , k , Γ, Γ and the ρ a be as in Section 2.Set P = P α , L = L α and V = R u ( P α ), and define σ ∈ Hom(Γ , L ) by σ ( r ) = t , σ ( s ) = s α and σ ( z ) = 1. Then σ is L -irreducible and every ρ a belongs to C σ . Let θ a ∈ Z (Γ , σ, V )and θ ′ a ∈ Z (Γ , σ | Γ , V ) be the 1-cocycles corresponding to ρ a and ρ a | Γ , respectively. Thecalculations in Section 2 show that the ρ a | Γ are pairwise V -conjugate, so the 1-cohomology lasses θ ′ a ∈ H (Γ , σ | Γ , V ) are equal for all a ∈ k . In contrast, no two of the ρ a are V -conjugate (since no two are G -conjugate), so the 1-cohomology classes θ a ∈ H (Γ , σ, V ) areall different. Thus we have an example where the map Θ from H (Γ , σ, V ) to H (Γ , σ | Γ , V )has an infinite fibre (cf. [7, Ex. 4.1]).We do not know of any analogous examples in odd characteristic; cf. the discussion at theend of Section 2. Acknowledgments : The authors acknowledge the financial support of EPSRC GrantEP/L005328/1, Marsden Grants UOC1009 and UOA1021, and the DFG-priority programmeSPP1388 “Representation Theory”. We are grateful to the referee for helpful suggestions.
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