Unipotent elements forcing irreducibility in linear algebraic groups

Theorem 1.1 ([41]).

Let X be a connected reductive subgroup of the simple algebraic group G. If X contains a regular unipotent element of G, then X is G-irreducible.

Proposition 1.2.

Assume that p=2. Let G=Sp⁡(V), where dim⁡V=4, so G is simple of type C2 and has a unique conjugacy class of distinguished unipotent elements of order p. Let u∈G be a distinguished unipotent element of order p. If X<G is connected reductive and u∈X, then X is G-irreducible unless X is conjugate to X′, where X′ is described by one of the following:

1. X′ is simple of type A1 with natural module E, embedded into G via E⊥E (orthogonal direct sum).

2. X′ is simple of type A1 with natural module E, embedded into G via E⊗E.

Furthermore, subgroups X′ in (i) and (ii) exist, contain a conjugate of u, are contained in a proper parabolic subgroup, and the conditions (i) and (ii) each determine X′ up to conjugacy in G.

Theorem 1.3.

Let u∈G be a distinguished unipotent element of order p. Let X be a connected reductive subgroup of G containing u. Then X is G-irreducible, unless p=2, the group G is simple of type C2, and X is simple of type A1 as in Proposition 1.2  (i) or (ii).

Remark 1.4.

In characteristic zero, the problem of classifying all connected reductive overgroups of distinguished unipotent elements is solved in the following sense. The reductive maximal closed connected overgroups of distinguished unipotent elements can be found using [20] and [17]. Furthermore, with a theorem of Mostow [24] and the Borel–Tits theorem, it is easily seen that any connected reductive overgroup of a distinguished unipotent element of G is contained in a reductive maximal closed connected subgroup of G. Therefore in characteristic zero, we have a recursive description of the connected reductive overgroups of distinguished unipotent elements.

Definition 1.5 (Serre [33]).

Let H be a closed subgroup of G. We say that H is G-completely reducible (G-cr), if whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi factor of P. Otherwise we say that H is non-G-completely reducible (non-G-cr).

Lemma 3.1.

Let u be a unipotent linear map on the vector space V and suppose that u has order p. Suppose that V has a filtration V=W1⊇W2⊇⋯⊇Wt⊇Wt+1=0 of K⁢[u]-submodules. Then rp⁢(u)≥∑i=1trp⁢(uWi/Wi+1).

Lemma 3.2 ([29, Theorem 1]).

Let 1≤m≤n≤p. Then

where h=min⁡{m,p-n} and N=max⁡{0,m+n-p}. In particular, we have Jm⊗Jp=m⋅Jp for all 1≤m≤p.

Lemma 3.3.

Let u∈SL⁡(V) be a unipotent element. Then u is a distinguished unipotent element of SL⁡(V) if and only if V↓K⁢[u]=Jd, where d=dim⁡V.

Lemma 3.4.

Assume p≠2. Let u∈Sp⁡(V) be a unipotent element. Then u is a distinguished unipotent element of Sp⁡(V) if and only if

where di are distinct even integers.

Lemma 3.5.

Assume p≠2. Let u∈SO⁡(V) be a unipotent element. Then u is a distinguished unipotent element of SO⁡(V) if and only if

where di are distinct odd integers.

Lemma 3.6 ([19, Proposition 6.1 and Section 6.8]).

Assume p=2. Let G=Sp⁡(V) or G=SO⁡(V). Set Z=V if dim⁡V is even and Z=V/V⊥ if dim⁡V is odd. Let u∈G be a unipotent element. Then u is a distinguished unipotent element of G if and only if there is an orthogonal decomposition

where di is even for all 1≤i≤t, and each Jdi occurs with multiplicity at most two.

Lemma 3.7.

Assume p=2. Let G=Sp⁡(V) or G=SO⁡(V), where dim⁡V is even. Let u∈G be a unipotent element with Jordan form

where di are distinct even integers. Then there exists an orthogonal decomposition V↓K⁢[u]=Jd1⊥Jd2⊥⋯⊥Jdt, and u is a distinguished unipotent element of G.

Proof.

This result is an easy consequence of the distinguished normal form for unipotent elements established in [19, Chapter 6]. In the orthogonal decomposition given in [19, Lemma 6.2], there cannot be any summands of the form W⁢(m), since u acts on W⁢(m) with two Jordan blocks of size m. Thus the claim follows from [19, Proposition 6.1].∎

Lemma 3.8.

Assume p=2. Let G=SO⁡(V,Q), where dim⁡V is even and Q is a non-degenerate quadratic form on V with polarization β. Fix a vector v∈V such that Q⁢(v)≠0. Suppose that u∈StabG⁡(v) is a unipotent element such that with respect to the alternating bilinear form induced on 〈v〉⊥/〈v〉 by β, we have an orthogonal decomposition

Then

Proof.

It follows from [19, Section 6.8] that there exists an orthogonal decomposition 〈v〉⊥↓K⁢[u]=W⊥〈v〉 such that

Now W is a non-degenerate subspace, so we have an orthogonal decomposition V↓K⁢[u]=W⊥Z, where dim⁡Z=2. It is obvious that Z↓K⁢[u]=J1⊕J1 or Z↓K⁢[u]=J2. On the other hand, we have u∈SO⁡(V), so the number of Jordan blocks of u must be even [19, Proposition 6.22 (i)]. Consequently, we have Z↓K⁢[u]=J1⊕J1 if t is even, and Z↓K⁢[u]=J2 if t is odd, as claimed.∎

Lemma 4.1.

Assume p=2. Let V be an indecomposable G-module. If V admits a non-degenerate G-invariant symmetric bilinear form, then V admits a non-degenerate G-invariant alternating bilinear form.

Proof.

Let β be a non-degenerate G-invariant symmetric bilinear form on V. Since p=2, the map f:V→K defined by f⁢(v)=β⁢(v,v) is a morphism of G-modules, where G acts trivially on K. If f=0, then β is alternating and we are done.

Suppose then that f≠0. The bilinear form β is non-degenerate, so every linear map V→K has the form v↦β⁢(v,z) for some z∈V. In particular, there exists a non-zero z∈V such that f⁢(v)=β⁢(v,z) for all v∈V. Since f is a G-morphism, the vector z is a G-fixed point in V. Hence the subspace 〈z〉 must be totally isotropic with respect to β, since V is indecomposable. That is, we have β⁢(z,z)=f⁢(z)=0.

Consider the map N:V→V defined by N⁢(v)=β⁢(v,z)⁢z=f⁢(v)⁢z for all v∈V. Then N∈EndG⁡(V) since z is fixed by G, and N2=0 since f⁢(z)=0. Hence the map ψ=idV+N is an isomorphism of G-modules. This gives a non-degenerate G-invariant symmetric bilinear form γ on V via

for all v,v′∈V. Since γ⁢(v,v)=β⁢(v,v)+f⁢(v)2=0 for all v∈V, the bilinear form γ is alternating. ∎

Lemma 4.2.

Let λ∈X⁢(T)+ be a quasi-minuscule dominant weight. Suppose that VG(λ)=LG(λ)|LG(0). Then:

1. The indecomposable tilting module TG⁢(λ) is uniserial, and

   TG⁢(λ)=LG⁢(0)⁢|LG⁢(λ)|⁢LG⁢(0).

2. If p≠2, then TG⁢(λ) admits a non-degenerate G-invariant symmetric bilinear form.

3. If p=2, then TG⁢(λ) admits a non-degenerate G-invariant alternating bilinear form, unique up to a scalar multiple.

Proof.

For (i), note that from the short exact sequence

the long exact sequence in cohomology gives H1⁢(G,VG⁢(λ))≅H1⁢(G,LG⁢(λ)), since H1⁢(G,LG⁢(0))=0=H2⁢(G,LG⁢(0)) by [14, Corollary II.4.11]. On the other hand, by the assumption VG(λ)=LG(λ)|LG(0) and [14, Proposition II.2.14] we have H1⁢(G,LG⁢(λ))≅K. Therefore H1⁢(G,VG⁢(λ))≅K, and so up to isomorphism there exists a unique non-split extension

The fact that V is a non-split extension of LG⁢(0) by VG⁢(λ) implies that

Consequently, V/soc⁡V is a non-split extension of LG⁢(0) by LG⁢(λ), so it follows that V is uniserial and V/soc⁡V≅VG⁢(-w0⁢λ)*. Hence V has a filtration by dual Weyl modules, and by construction V has a filtration by Weyl modules. Thus V is a tilting module, so V≅TG⁢(λ) since V is indecomposable with highest weight λ. This establishes (i).

For claims (ii) and (iii), note first that the fact that λ is quasi-minuscule implies that -w0⁢λ=λ, where w0 is the longest element of the Weyl group of G. Thus TG⁢(λ) is self-dual, see [14, Remark II.E.6].

We suppose first that p≠2 and consider claim (ii). Since TG⁢(λ) is indecomposable, its endomorphism ring is a local ring. It follows then from a general result in representation theory [28, Lemma 2.1] (or [43, Satz 2.11 (a)]) that there exists a non-degenerate G-invariant bilinear form β on TG⁢(λ) such that β is symmetric or alternating. Since TG⁢(λ) is uniserial and TG⁢(λ)=LG⁢(0)⁢|LG⁢(λ)|⁢LG⁢(0), the form β induces a non-degenerate G-invariant bilinear form on the subquotient of TG⁢(λ) isomorphic to LG⁢(λ). Since λ≻0, it follows from [36, pp. 226–227, Lemma 79] that the form induced on the subquotient is symmetric. Because p≠2, we conclude that β must also be symmetric, which establishes (ii).

For claim (iii), suppose that p=2 and set V=TG⁢(λ). We first show that there exists a non-degenerate G-invariant alternating bilinear form on V. Fix some isomorphism f:V→V* of G-modules. This gives a non-degenerate G-invariant bilinear form β on V via β⁢(v,v′)=f⁢(v)⁢(v′) for all v,v′∈V. Consider the bilinear form γ on V defined by γ⁢(v,v′)=β⁢(v,v′)+β⁢(v′,v) for all v,v′∈V. If γ=0, then β is symmetric and Lemma 4.1 gives a non-degenerate G-invariant alternating bilinear form on V. Suppose then that γ≠0. Then γ is a non-zero G-invariant alternating bilinear form on V, and we will show that γ must be non-degenerate. Note that γ induces a non-degenerate G-invariant alternating bilinear form on V/rad⁡γ. Since V is a uniserial G-module, and since LG⁢(0) and VG(λ)*=LG(0)|LG(λ) do not admit non-degenerate G-invariant alternating bilinear forms, the only possibility is that rad⁡γ=0. In other words, the bilinear form γ is non-degenerate.

Next we show the uniqueness statement of (iii). Let β and γ be non-degenerate G-invariant alternating bilinear forms on V. It is easy to see (for example from [43, Satz 2.3]) that there exists a unique G-isomorphism φ∈EndG⁡(V) such that γ⁢(v,v′)=β⁢(φ⁢(v),v′) for all v,v′∈V.

We now describe EndG⁡(V). We have established in the proof of (i) that TG⁢(λ)=LG⁢(0)⁢|LG⁢(λ)|⁢LG⁢(0) is uniserial. Let z∈V be a vector spanning the unique 1-dimensional submodule of V. Then as in [14, E.5], by considering where a maximal vector of weight λ can be sent, one finds that EndG⁡(V) is spanned by the identity map and a map N:V→V such that N⁢(V)=〈z〉. Thus as a basis for the vector space EndG⁡(V), we can take the identity map 1:V→V and the nilpotent map N:V→V defined by N⁢(v)=β⁢(v,z)⁢z for all v∈V.

Write φ=λ⋅1+μ⋅N. Since γ and β are alternating, we have

Now since β is non-degenerate, we can choose a v∈V with β⁢(v,z)≠0, and consequently μ=0. Therefore γ=λ⋅β, which completes the proof of (iii) and the lemma.∎

Remark 4.3.

Lemma 4.2 (ii) is true much more generally. Suppose that p≠2. If λ=-w0⁢λ, then TG⁢(λ) is self-dual and one can show that TG⁢(λ) admits a non-degenerate G-invariant bilinear form β which is alternating or symmetric. Furthermore, arguing similarly to the proof of Lemma 4.2 (ii), one can show that β must be of the same type as the corresponding form on LG⁢(λ).

Remark 4.4.

In Table 1 we have given the cases where Lemma 4.2 applies. For TG⁢(λ) with quasi-minuscule highest weight λ∈X⁢(T)+, we have left untreated the case where G is simple of type Dl when p=2 and l is even. In this case we have VG(λ)=LG(λ)|LG(0)2. We omit the proof, but proceeding similarly to the proof of Lemma 4.2, one can show that TG⁢(λ)=LG⁢(0)2⁢|LG⁢(λ)|⁢LG⁢(0)2 and that TG⁢(λ) admits a non-degenerate G-invariant alternating bilinear form. In this case such a form is not unique up to a scalar multiple, but it is unique up to a G-equivariant isometry.

Theorem 5.1 (Cline, [2, Corollary 3.9]).

Let λ=∑i≥0λi⁢pi and μ=∑i≥0μi⁢pi be weights of G, where 0≤λi,μi<p. Then

if and only if there exists k≥νp⁢(λ+1) such that μi=λi for all i≠k,k+1 and μk=p-2-λk, μk+1=λk+1±1. In all other cases, we have

Theorem 5.2 ([31, Lemma 2.3], [7, p. 47, Example 2]).

Let c∈Z≥0 be a dominant weight. Then:

1. If 0≤c≤p-1, then the indecomposable tilting module TG⁢(c) is irreducible, so TG⁢(c)=LG⁢(c).

2. If p≤c≤2⁢p-2, then the indecomposable tilting module TG⁢(c) is uniserial of dimension 2⁢p, and

   TG⁢(c)=LG⁢(2⁢p-2-c)⁢|LG⁢(c)|⁢LG⁢(2⁢p-2-c).

3. (Donkin) If c>2⁢p-2, then TG⁢(c)≅TG⁢(p-1+r)⊗TG⁢(s)[1], where s≥1 and 0≤r≤p-1 are such that c=s⁢p+(p-1+r).

Lemma 5.3.

Let m∈Z≥0 and write m=q⁢p+r, where q≥0 and 0≤r<p. Then VG⁢(m)↓K⁢[u]=q⋅Jp⊕Jr+1.

Proof.

The Weyl module VG⁢(m) is isomorphic to the dual of the symmetric power Sm⁢(E), see for example [14, II.2.16]. Therefore it suffices to compute the Jordan block sizes of u acting on the symmetric power Sm⁢(E).

Fix a basis x,y of E such that u⁢x=x and u⁢y=x+y. Consider the basis xm,xm-1⁢y,…,ym induced on Sm⁢(E). Then

so with respect to this basis, the matrix of u acting on Sm⁢(E) is the upper triangular Pascal matrix

Here we define (ij)=0 if i<j. The Jordan form of the transpose of P is computed for example in [6], and from this result we find that the transpose of P has q Jordan blocks of size p and one Jordan block of size r+1. Since a matrix is similar to its transpose, the lemma follows.∎

Lemma 5.4.

Let c≥p-1. Then

where N=dim⁡TG⁢(c)/p. In particular, if p≤c≤2⁢p-2, then

Proof.

We argue similarly to [23, Proposition 5]. Without loss of generality, we can assume that u is contained in the finite subgroup G⁢(p):=SL2⁡(𝔽p) of G. We note first that for all p-1≤c≤2⁢p-2, the restriction of TG⁢(c) to G⁢(p) is projective. For c=p-1 this follows since TG⁢(c) is the Steinberg module, and for p≤c≤2⁢p-2 this follows from [31, Lemma 2.3 (c)].

By [1, p. 47, Lemma 4], a tensor product with a projective module remains projective, so we conclude from Theorem 5.2 (iii) that the restriction of TG⁢(c) to G⁢(p) is projective for all c≥p-1. Then the restriction of TG⁢(c) to the Sylow p-subgroup H=〈u〉 of G⁢(p) is also projective [1, p. 33, Theorem 6]. Since Jp is the only projective indecomposable H-module, the claim follows.∎

Proposition 5.5.

Let V be a G-module which is a non-split extension of LG⁢(λ) by LG⁢(μ). Then u acts on V with at least one Jordan block of size p. If u acts on V with precisely one Jordan block of size p, then there exist p≤c≤2⁢p-2 and l≥0 such that one of the following holds:

1. λ=c⁢pl, μ=(2⁢p-2-c)⁢pl and V≅VG⁢(c)[l].

2. λ=(2⁢p-2-c)⁢pl, μ=c⁢pl and V≅(VG⁢(c)*)[l].

Moreover, if V, λ, μ, and c are as in case (i) or case (ii), then V is a non-split extension of LG⁢(λ) by LG⁢(μ), and V↓K⁢[u]=Jp⊕Jc-p+1.

Proof.

We begin by considering the claims about VG⁢(c)[l] and (VG⁢(c)*)[l], where p≤c≤2⁢p-2 and l≥0. It is well known (and easily seen by considering the weights in VG⁢(c)) that VG⁢(c) is a non-split extension

Since the irreducible representations of G are self-dual, we see that VG⁢(c)* is a non-split extension

Let λ=c⁢pl and μ=(2⁢p-2-c)⁢pl, where l≥0. By taking a Frobenius twist, we see that for all l≥0 the module VG⁢(c)[l] is a non-split extension of LG⁢(λ) by LG⁢(μ), and its dual (VG⁢(c)*)[l] is a non-split extension of LG⁢(μ) by LG⁢(λ). Furthermore, by Lemma 5.3 the unipotent element u∈G acts on VG⁢(c) with Jordan form Jp⊕Jc-p+1. The Jordan block sizes are not changed by taking a dual or a Frobenius twist, so for all l≥0 the element u acts on both VG⁢(c)[l] and (VG⁢(c)*)[l] with Jordan form Jp⊕Jc-p+1. This completes the proof that the properties of VG⁢(c)[l] and (VG⁢(c)*)[l] are as claimed.

Now let λ,μ∈ℤ≥0 be arbitrary weights, and let V be a non-split extension

Assume that u acts on V with at most one Jordan block of size p. We note first that to prove the proposition, it will be enough to show that there exist p≤c≤2⁢p-2 and l≥0 such that λ and μ are as in case (i) or (ii) of the claim. Indeed, if this holds, then since

(Theorem 5.1), we must have V≅VG⁢(c)[l] or V≅(VG⁢(c)*)[l]. Furthermore, as seen in the first paragraph, then u acts on V with Jordan form Jp⊕Jc-p+1, in particular with exactly one Jordan block of size p.

We proceed to show that λ and μ are as claimed, which will prove the proposition. Write λ=∑i≥0λi⁢pi, where 0≤λi≤p-1 for all i. By the Steinberg tensor product theorem, we have

Consider first the case where λl=p-1 for some l. Then u acts on LG⁢(λl)[l] with a single Jordan block of size p. Now the tensor product of a Jordan block of size p with any Jordan block of size c≤p consists of c Jordan blocks of size p (Lemma 3.2), so it follows that

where N=dim⁡LG⁢(λ)/p. Since u acts on V with at most one Jordan block of size p, by Lemma 3.1 the action of u on LG⁢(λ) can have at most one Jordan block of size p. It follows then that N=1, so λ=(p-1)⁢pl. Since V is a non-split extension, by Theorem 5.1 the weight μ must be (p-2)⁢pl-1+(p-2)⁢pl or (p-1)⁢pl+(p-2)⁢pk+pk+1 for some k≠l,l-1. By Lemma 3.1 the action of u on LG⁢(μ) has no Jordan blocks of size p. With Lemma 3.2, one finds that this happens only if p=2 and μ=(p-2)⁢pl-1+(p-2)⁢pl=0. Then λ and μ are as in case (i) of the claim.