Double centralizers of unipotent elements in simple algebraic groups of type G2 , F4 and E6

Journal of Algebra 382 (2013) 335–367

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Journal of Algebra www.elsevier.com/locate/jalgebra

Double centralizers of unipotent elements in simple algebraic groups of type G 2 , F 4 and E 6 Iulian Ion Simion EPFL, MATHGEOM GR-TES, MA B3 435 (Bâtiment MA), CH-1015, Lausanne, Switzerland

a r t i c l e

i n f o

Article history: Received 5 December 2012 Available online 13 March 2013 Communicated by Gunter Malle Keywords: Double centralizers Bad characteristic Exceptional groups

a b s t r a c t Let u be a unipotent element of a simple algebraic group G over a field k of characteristic p. We develop a method for computing the connected component of Z (C G (u )) in the cases where p is positive and both it and the rank of G are small enough. The method is then carried out for G of type G 2 , F 4 and E 6 in bad characteristic. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Let G be a semisimple algebraic group defined over an algebraically closed field k of characteristic p  0. By ‘algebraic group’ we shall mean ‘linear algebraic group’. The notation that we will use is described in Section 10. The analysis to follow focuses on describing Z (C G (u )) = C G (C G (u )) for a unipotent element u ∈ G: 1. Does the dimension formula given in [11] hold in bad characteristic? 2. Does u lie in Z (C G (u ))◦ as in good characteristic? 3. What is a characteristic independent description of Z (C G (u ))◦ which allows us to compute this subgroup? The motivation for these questions and the interest in Z (C G (u )) resides in establishing the existence of a connected abelian unipotent group W ⊆ G containing u satisfying certain uniqueness properties. Suppose that the characteristic is good for G and let the order of the element u be p t for some t ∈ N. If t = 1 then there exists a 1-dimensional such subgroup which satisfies several good properties (see [19]). If t > 1 there exists a t-dimensional abelian subgroup containing u (see [18] and [20]) however it is not canonically defined.

E-mail address: [email protected]. 0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.03.003

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A natural candidate for a canonically defined connected abelian overgroup for u is Z (C G (u ))◦ . Of course, it is not clear that u lies in Z (C G (u ))◦ . In good characteristic, Z (C G (u ))◦ contains u and it was studied in [17,20]. Among other things a dimension formula was established in [11], which expresses dim Z (C G (u )) in terms of the labeled diagram attached to the unipotent class. Here we extend the analysis of Z (C G (u ))◦ to the case where the characteristic is bad for G, i.e. if p = 2 and not all simple factors of G are of type A n , if p = 3 and G has a factor of type G 2 , F 4 or E n and if p = 5 and G has an E 8 factor. In bad characteristic one also has a way of attaching a labeled diagram to a unipotent class (see [12]) which, even though it is not a bijective correspondence, generalizes the diagrams in good characteristic. It is with regard to this labeling that the first question is precise. A standard argument shows that we can immediately reduce to the case where G is simple and of adjoint type. There exists an isogeny φ : G → G ad where G ad is the adjoint algebraic group with the same root system as G. Let Z (C G (u ))u denote the subgroup of unipotent elements of Z (C G (u )). Since Z (C G (u )) = Z (C G (u ))u × Z (G ) [17, Theorem 1.1(i)], we have φ( Z (C G (u ))) = φ( Z (C G (u ))u ). Now, φ restricted to the variety of unipotent elements is an isomorphism because ker φ consists of semisimple elements (see for example [25, Corollary 7.6.4(iii)]) so Z (C G (u )) ∼ = Z (C G ad (u )) × Z (G ). Moreover, since G ad is a direct product of simple adjoint algebraic groups, the centralizer of u in G ad is a direct product of centralizers in each factor of G ad and a similar statement holds for the center of C G ad (u ). In Section 2 we prove the following characteristic independent description of Z (C G (u ))◦ . We choose T ⊆ G a maximal torus and B ⊆ G a Borel subgroup containing T . The unipotent radical of B will be denoted by U . The root system is chosen with respect to T and the positive roots are with respect to B. Theorem A. Let u ∈ G be a unipotent element and suppose that B contains a Borel subgroup of C G (u ) then



◦

Z C G (u )

= C Z (C U (u )◦ )◦ ( T u , A˜ )◦

˜ is a set of coset representatives for C G (u )◦ in C G (u ). where T u is a maximal torus of C B (u ) and A In the sequel, we obtain Corollary 2.8 which describes the Lie algebra of the center of C G (u ) in characteristic 0 and which can be viewed as a justification for the difficulty of giving a proof for the dimension formula [11, Theorem 4] that is free of case-by-case analysis (even in characteristic 0). Another byproduct is a proof that in good characteristic Z (C G (u )◦ ) is connected (Proposition 2.9). Moreover we deduce a method for the explicit calculation of Z (C G (u ))◦ which we carry out for the simple adjoint algebraic groups of type G 2 , F 4 and E 6 in bad characteristic. Suppose for the rest of this section that G is such a group. Our analysis contains the following intermediate results: Proposition B. The unipotent elements in Tables 4, 5 and 6 given with respect to T ⊆ B are a complete set of unipotent class representatives u˜ such that C B (u˜ )◦ is a Borel subgroup of C G (u˜ ) with unipotent radical C U (u˜ )◦ and maximal torus C T (u˜ )◦ . Explicit representatives for the component group of C G (u ) can be found in the literature (see for example [12]) however they were given modulo a unipotent subgroup. In good characteristic explicit representatives for the component groups of stabilizers of nilpotent elements were given in [11]. We obtain Proposition C. Section 8 contains explicit component group representatives for C G (u ) where u is a unipotent class representative given in Section 3. The difficult step in applying our approach is to obtain Z (C U (u )◦ )◦ . In Sections 5 and 6 we describe this algebraic group in more detail. Finally we are able to answer all of the above questions (for our choice of G):

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Theorem D. Let u ∈ G be a unipotent element. The dimension of Z (C G (u )) is given in Tables 7, 8 and 9. The tables also indicate if u ∈ Z (C G (u ))◦ and if u ∈ Z (C G (u )◦ )◦ . The answer to the above second question was known for regular elements since such elements do not belong to C G (u )◦ (see [24]). Indeed, the answer is no whenever u ∈ / C G (u )◦ and those classes with this property are given in [12, Corollary 4]. Our calculations give a different proof of this corollary for our particular choice of G. What is striking is that even if u lies in C G (u )◦ it might not lie in Z (C G (u ))◦ . Inspecting the results in our tables, we deduce Corollary E. If the characteristic is 2 then, with the exception of the classes A 22 and A 2 A 1 in E 6 , a unipotent element u ∈ G lies in Z (C G (u ))◦ if and only if u 2 = 1. Excluding the exceptional unipotent classes, i.e. those with label X ( p ) , the dimension of Z (C G (u )) is independent of characteristic with the following exceptions (for which “the dimension of Z (C G (u )) changes” in bad characteristic): G

classes

G2 F4

G 2 (a1 ) ˜ 1 , C 3 (a1 ), F 4 (a3 ), B 3 , C 3 , F 4 (a1 ) A1 A

Corollary F. If the labeling of the unipotent classes is the same as in good characteristic, i.e. if (G , p ) is ( E 6 , 2), ( E 6 , 3), ( F 4 , 3) or (G 2 , 2), then for each class dim Z (C G (u )) is the same as in good characteristic. Remark. If the characteristic is good for G, it was shown in [20] that Z (C G (u ))◦ decomposes as a sum of Witt groups one of which contains u. From the remark following Proposition 2.10 we conclude that, even if u does not lie in Z (C G (u ))◦ , there might exist a Witt subgroup containing it. The calculations were automatized in two key points of the above analysis. The first one is the check that the unipotent elements in Section 3 are a complete list of unipotent class representatives by calculating their Jordan block structure (with [27]) in the adjoint representation and comparing it to the tables in [10]. We mostly used representatives already available in the literature. The second part is the calculation of Z (C U (u )◦ )◦ . In Sections 5 and 6 we translate the problem into elementary manipulations of polynomials with coefficients in k (in fact we will only use coefficients in F p 2 ). We use [5] for these computations. The structure constants for the simple Lie algebras are the ones in [5] and the signs in the commutator formula were deduced with [2, Proposition 6.4.3]. For aspects on computing in unipotent and reductive groups we refer the reader to [4]. The functionality for the above calculation is available in the (unofficial) GAP package [22]. 2. Dividing the problem This section contains our strategy for obtaining the center of the centralizer of a unipotent element in a simple algebraic group G of adjoint type. The notation that we use is described in Section 10. A key point to our approach is the fact that the center of C G (u )◦ is equal to that of a Borel subgroup B u ⊆ C G (u ). This is used in the proof of Proposition 2.3. The main results which will be needed in the remainder of the work are



◦

Z C G (u )

 ◦ = C Z (C G (u )◦ )◦ ( A˜ ) and Z C G (u )◦ = C Z (U u )◦ ( T u )

(1)

as pointed out in Section 2.3. The first equality is an application of Lemma 2.2 and the second one is an application of Lemma 2.1 using Lemma 2.6.

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2.1. Preliminary results Here we focus on proving Proposition 2.3, the main result of this paragraph, which will allow us to deduce Theorem A. The following lemma will be used to handle the center of a Borel subgroup of a subgroup. Lemma 2.1. Let H be a connected solvable algebraic group, T a maximal torus of H and denote by U the subgroup of unipotent elements of H . Suppose that Z ( H )◦ lies in U , then

Z ( H )◦ = C Z (U ) ( T )◦ = C Z (U )◦ ( T ). Proof. For x ∈ Z ( H )◦ ⊆ U it is clear that x ∈ Z (U ) ∩ C H ( T ) so Z ( H )◦ ⊆ C Z (U ) ( T ) and therefore Z ( H )◦ ⊆ C Z (U ) ( T )◦ . For the other inclusion fix x ∈ C Z (U ) ( T ). Since H = T  U (see [7, Theorem 19.3]) every element y ∈ H is of the form y = tu for some t ∈ T and u ∈ U . Therefore x y = xtu = x so x ∈ Z ( H ). This shows that C Z (U ) ( T ) ⊆ Z ( H ), hence C Z (U ) ( T )◦ ⊆ Z ( H )◦ . For the second equality note that Z ( H )◦ ⊆ U ⇒ Z ( H )◦ ⊆ Z (U )◦ so Z ( H )◦ ⊆ C Z (U )◦ ( T ). As in the previous paragraph, C Z (U )◦ ( T ) ⊆ Z ( H ). Now, by [7, Proposition 18.4.B], C Z (U )◦ ( T ) is connected and we are done. 2 The following lemma establishes a relation between the center of an algebraic group H and the center of the connected component H ◦ . We introduce some notation first. Denote by A ( H ) := H / H ◦ ˜ ( H ) a set of representatives for A ( H ). If it is clear from the context the component group and by A ˜ what H is, we write A and A.

˜ )◦ . Lemma 2.2. If H is an algebraic group then Z ( H )◦ = C Z ( H ◦ ) ( A Proof. We use the following general fact: if X and Y are closed subgroups of H then

◦  ◦  ( X ∩ Y )◦ = X ◦ ∩ Y = X ◦ ∩ Y ◦ .

(2)

This follows from the fact that ( X ∩ Y )◦ is a connected subgroup of X and Y so it lies in X ◦ and Y ◦ . Again, since it is connected it will be contained in the connected component of the intersection of any two of the latter four groups. The reverse inclusions are obvious. ˜ we have Z ( H ) = C H ( H ◦ ) ∩ C H ( A˜ ). Now consider the connected component: Since H = H ◦ , A







◦

˜) Z ( H )◦ = C H H ◦ ∩ C H ( A

  ◦ ◦ = C H H ◦ ∩ C H ( A˜ ) by (2)  ◦   ◦ ◦ ◦ = Z H ∩ C H ( A˜ ) since C H H ◦ ⊆ H ◦    ◦ = Z H ◦ ∩ C H ( A˜ ) by (2) = C Z ( H ◦ ) ( A˜ )◦ .

2

The following proposition establishes a relation between the center of a certain subgroup H of G, ˜ ( H ). Note that, the center of a Borel subgroup B H of H and the component group representatives A for any subgroup H ⊆ G, we can choose B = T  U ⊆ G a Borel subgroup of G containing the Borel subgroup B H = T H  U H of H , where T H ⊆ T are maximal tori and U and U H are the unipotent radicals, respectively.

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Proposition 2.3. Let G be a linear algebraic group and let H ⊆ G be a subgroup such that Z ( H ◦ )◦ is unipotent. With the above choice of B and B H we have

˜ )◦ = C Z (U H ) ( T H , A˜ )◦ Z ( H )◦ = C Z (U H )◦ ( T H , A ˜ := A˜ ( H ). where A Proof. Note that the second equality is trivial. We focus on the first one. From Lemma 2.2 we have

˜ )◦ = C Z ( H ◦ )◦ ( A˜ )◦ . Z ( H )◦ = C Z ( H ◦ ) ( A Since B H is a Borel subgroup of H it is a Borel subgroup of H ◦ . By [7, Corollary 21.4] we have Z ( H ◦ )◦ ⊆ Z ( B H ) ⊆ Z ( H ◦ ), thus Z ( H ◦ )◦ = Z ( B H )◦ and therefore

˜ )◦ . Z ( H )◦ = C Z ( B H )◦ ( A By assumption Z ( H ◦ )◦ = Z ( B H )◦ is unipotent and we can use the second part of Lemma 2.1 to end the proof:

Z ( H )◦ = C C Z (U

H)

◦ (T H )

 ◦ ( A˜ )◦ = C Z (U H )◦ ( T H ) ∩ C Z (U H )◦ ( A˜ ) = C Z (U H )◦ ( T H , A˜ )◦ .

2

Corollary 2.4. Let H ⊆ G be as in Proposition 2.3 and suppose in addition that char k = 0 then L Z ( H ) = ˜ ) where z denotes the Lie algebra center. C z(L H ) ( A

˜ )◦ . Using Proof. From the proof of the above proposition we have L Z ( H ) = L Z ( H )◦ = LC Z ( H ◦ ) ( A ˜ ˜ [7, Theorem 13.2] we deduce that L Z ( H ) = C L Z ( H ◦ ) ( A ) because A is finite. Finally, from [7, Theorem 13.4(b)] we have L Z ( H ◦ ) = z(L H ) and we are done. 2 2.2. A characterization of Z (C G (u ))◦ In order to apply Proposition 2.3 and its corollary to the centralizer of a unipotent element u ∈ G we need one further result. For this denote by H u the set of unipotent elements of a group H . The following proposition is a generalization of [3, Proposition 5.1.5]. Proposition 2.5. (See [17, Theorem 1.1(i)].) With the above notation the semisimple elements of Z (C G (u )) are in Z (G ) and so Z (C G (u )) = Z (C G (u ))u × Z (G ). In fact we need the following slight variation of the above result. Lemma 2.6. With the above notation the semisimple elements of Z (C G (u )◦ ) are in Z (G ) and so Z (C G (u )◦ ) = Z (C G (u )◦ )u × ( Z (G ) ∩ C G (u )◦ ). Proof. The proof is similar to the one given in [17] which is due to Ross Lawther. Let s ∈ Z (C G (u )◦ ). Since [s, u ] = 1 there exists a Borel subgroup B of G with unipotent radical U and a maximal torus T ⊆ B having positive weights on L(U ) such that u ∈ U and s ∈ T (Jordan decomposition). Let ΦG be the roots of G with respect to this torus. Since C B (s) is a Borel subgroup of the connected reductive group H := C G (s)◦ (see for example [26, Proposition II.4.1]), if Φ H denotes the roots + of H relative to T , then Φ H ⊆ ΦG+ where the positive roots are chosen with respect to B. Now, if / Z (G ) then there exists β ∈ ΦG+ − Φ H+ of maximal height. Let U β be the root group corresponding s∈ to β then U β ∩ C G (s) = 1.

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Write u =



α ∈Φ H+ uα (c α ) for some c α ∈ k. If

α ∈ Φ H+ then α + β ∈/ ΦG else α + β would contradict

the choice of β . Therefore [U β , u ] = 1 so U β ⊆ C G (u ). Since U β is connected it follows that U β ⊆ C G (u )◦ ⊆ C G (s), a contradiction. 2 At this point we have u a unipotent element in G (a simple adjoint algebraic group). Fix a Borel subgroup B of G which contains a Borel subgroup B u of C G (u ) (clearly this is always possible by maximality). Note that



◦

B u = B ∩ C G (u )

(3)

and if U is the unipotent radical of B then the unipotent radical of B u is



◦

U u = U ∩ C G (u ) .

(4)

Moreover a maximal torus T u of C G (u ) can be extended to a maximal torus T of B so



◦

T u = T ∩ C G (u ) .

(5)

In Section 4 we describe B u , T u and U u explicitly for a set of unipotent class representatives for G of type G 2 , F 4 and E 6 . We are now in a position to describe Z (C G (u ))◦ : Theorem (A). Let G be a simple algebraic group. With the above notation we have



◦

Z C G (u )

= C Z (U u )◦ ( T u , A˜ )◦ = C Z (U u ) ( T u , A˜ )◦

˜ = A˜ (C G (u )). where A Proof. It follows from Lemma 2.6 that Z (C G (u )◦ )◦ is unipotent. Whence, using Eqs. (3), (4) and (5), we may apply Proposition 2.3 with H = C G (u ). 2 Let us restate Lemma 2.2 and Corollary 2.4 in this setting: Corollary 2.7. With notation as above



◦

Z C G (u )

= C Z (C G (u )◦ ) ( A˜ )◦ = C Z (C G (u )◦ )◦ ( A˜ )◦

where the second equality is trivial. Corollary 2.8. With notation as above and under the assumption that the characteristic of k is 0 we have L Z (C G (u )) = C z(C LG (u )) ( A˜ ). In order to emphasize the role of Lemma 2.1 we give the following Proposition 2.9. If the characteristic is good for G of adjoint type then Z (C G (u )◦ ) is connected.

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Proof. Denote C G (u ) by H . We have







Z H◦ = Z H◦



 u

by Lemma 2.6

 ◦

      ⊆ C G H u since Z H ◦ = C H ◦ H ◦ ⊆ C G H ◦  ◦      ◦ ⊆ C G H ◦ ∩ C G H ◦ u since C G H ◦ u ⊆ C G H ◦ by [26, III 3.15]   ◦  = CG H◦ u   ◦  = C H H ◦ u since u ∈ H ◦ , again by [26, III 3.15]  ◦     = C H ◦ H ◦ u since H u = H ◦ u   ◦  = Z H◦ u  ◦ = Z H◦ by Lemma 2.6. 2

Note that the assumption on characteristic is used in the third, fifth and sixth lines above.

Remark. If the characteristic is bad then the conclusion of Proposition 2.9 need no longer be true. Several examples can be deduced from the analysis in the following sections. This is why we will use Lemma 2.1 to obtain the connected component of Z ( B u ) in bad characteristic. The role of Lemma 2.2 becomes visible with the following proposition and Remark.

˜ ) is connected and therefore equals Proposition 2.10. If the characteristic is good for G then C Z (C G (u )◦ ) ( A Z (C G (u ))◦ . In particular



◦

Z C G (u )

= C Z (C U (u )) ( T u , A˜ ).

Proof. By Proposition 2.9, Z (C G (u )◦ ) is connected. The assumption on characteristic allows us to ˜ = A˜ (C G (u )) (by [14, Corollary 13]) and [7, Theorem 18.3(b)] implies choose semisimple elements for A ˜ ˜ ). that C Z (C G (u )◦ ) ( A ) is connected. By Lemma 2.2 we have Z (C G (u ))◦ = C Z (C G (u )◦ ) ( A Now, Z (C G (u )◦ ) = C Z (C U (u )◦ ) ( T u ) by the proof of Theorem A and we claim that C U (u ) is connected. Indeed, [26, III 3.15] implies that C U (u ) ⊆ C G (u )◦ . Since C U (u ) is unipotent, it is contained in the radical U˜ of a Borel subgroup of C G (u )◦ . Since U was chosen such that C U (u )◦ is the radical of a Borel subgroup of C G (u )◦ we have C U (u )◦ ⊆ C U (u ) ⊆ U˜ which forces C U (u )◦ = U˜ , so C U (u ) is connected. The statement now follows from

˜ ) = CC C Z ( C G ( u )◦ ) ( A ( A˜ ) = C Z (U u ) ( T u ) ∩ C Z (U u ) ( A˜ ) = C Z (C U (u )) ( T u , A˜ ). Z (U u ) ( T u )

2

˜ ) is not necessarily connected even if Z ( H ◦ ) is Remark. If the characteristic is bad then C Z ( H ◦ ) ( A connected. As an example, assuming the characteristic is 2 and k algebraically closed, let u ∈ SL3 (k) be a regular unipotent and consider the centralizer C H (u ) of u in H , the automorphism group of SL3 (k). When treating the unipotent class A 2 (in E 6 (k)ad if char(k) = 2) we have A (C H (u )) = Z 2 = a

where a is a graph automorphism of A 2 . Our analysis shows that Z (C H (u )◦ ) is connected and contains / C Z (C H (u )◦ ) (a) = Z (C H (u ))◦ . u but u ∈

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2.3. Strategy The above results give a method for computing Z (C G (u ))◦ = C G (C G (u ))◦ where u is a unipotent element of G. It can be summarized roughly through the following steps: (1) Choose a representative u of a unipotent conjugacy class (Section 3). (2) Find a Borel subgroup B = T  U of G containing a Borel subgroup B u = T u  U u of C G (u ) such that Eqs. (3), (4) and (5) are satisfied (Section 4). (3) Calculate the unipotent radical U u = C U (u )◦ of B u (Section 5). (4) Calculate the connected component of the unipotent center Z (U u )◦ (Section 6). (5) Calculate the connected component of the center of the connected component of the centralizer Z (C G (u )◦ )◦ = C Z (U u )◦ ( T u ) (Section 7). ˜ )◦ for component group representatives A˜ of C G (u ) (Sec(6) Calculate Z (C G (u ))◦ = C Z (C G (u )◦ )◦ ( A tion 8). Remarks. Note that 1. Theorem A allows us to work with Z (U u ) from step (4) onwards however we will see that it will be more convenient to keep track of the connected component Z (U u )◦ . 2. The Borel subgroup B u is discussed in Section 4. For a representative u in Section 3 we find an element n ∈ G such that n B u = B u˜ ⊆ B where u˜ = n u and then Z (U u ) = Z (U u˜ )n . We don’t give n explicitly but we explain how it can be calculated – see Remark in Section 4.2. In the sections to follow we will apply the above strategy to the cases where G is a simple adjoint exceptional algebraic group of type G 2 , F 4 and E 6 . The groups of type E 7 and E 8 will be treated elsewhere. 3. Representatives In this section we give our choice of representatives for unipotent classes in simple exceptional algebraic groups of type G 2 , F 4 and E 6 . For notation see Section 10. Such representatives can be found in [10] and [15]. The method for obtaining them is by careful analysis of the Chevalley structure of the groups. Characteristic independent expressions of such representatives can be found in [12]. A different set of representatives was provided to us by Ross Lawther. Our choice of representatives depends on the characteristic. The reason for this is the explicit description of the component group  representatives which will be discussed in Section 8. In many cases uα (c α ) as representative for the unipotent class such that  we take an element the element c α e α is a representative with the same label given in [11]. In the remaining cases we indicate a reference in the last column of the tables below (RL stands for Ross Lawther). The indicated source contains either a unipotent element conjugate to u or a nilpotent element in which case u was deduced as above. A method for verifying that u belongs to a certain unipotent class is by means of the Jordan block structure in the adjoint representation available in [10]. This method does not apply to all cases because of coincidences in the Jordan blocks. However we can use the following lemma, which is a well-known consequence of the Bala–Carter–Pommerening method for classifying unipotent classes in simple algebraic groups (see for example [12, Lemma 2.13]). Lemma 3.1. Regular unipotent conjugacy classes in non-conjugate Levi subgroups of G induce distinct conjugacy classes in G. Extracting from [10, Table C] the coincidences in Jordan block sizes for the adjoint action in bad characteristic and dropping the classes where u is regular in the corresponding Levi factor, the am(2) biguity boils down to the pair of classes B 2 and C 3 (a1 ) in F 4 if the characteristic is 2. We use [12, Lemma 16.9] to separate these two classes.

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Table 1 Unipotent class representatives for F 4 . Class

p

u

F4 F 4 (a1 ) F 4 (a2 )

all all 2 = 2 all all all 2 all 2 all

u2 (1)u4 (1)u3 (1)u1 (1) u2 (1)u4 (1)u7 (1)u5 (1) u1 (1)u5 (1)u6 (1)u10 (1) u1 (1)u4 (1)u9 (1)u10 (1) u4 (1)u3 (1)u1 (1) u2 (1)u4 (1)u3 (1) u4 (1)u6 (1)u10 (1)u18 (1) u4 (1)u1 (1)u10 (1)u20 (1) u4 (1)u1 (1)u10 (1) u2 (1)u3 (1)u1 (1)u15 (1) u2 (1)u3 (1)u1 (1)

B2 B2 ˜1 A2 A ˜2 A A2 ˜1 A1 A

2 all all all all all

u4 (1)u3 (1)u22 (1) u4 (1)u3 (1) u2 (1)u4 (1)u1 (1) u3 (1)u1 (1) u2 (1)u4 (1) u2 (1)u1 (1)

[12, p. 332]

(2)

2 all all

u1 (1)u20 (1) u1 (1) u2 (1)

[12, p. 332]

C3 B3 F 4 (a3 ) C 3 (a1 )(2) C 3 (a1 ) ( A˜ 2 A 1 )(2) ˜ 2 A1 A (2)

˜ A 1 ˜1 A A1

Source [10] [10]

[10] [10] [10]

Table 2 Unipotent class representatives for G 2 . Class

p

u

Source

G2 G 2 (a1 ) ( A˜ 1 )(3) ˜1 A A1

all all 3 all all

u1 (1)u2 (1) u2 (1)u5 (1) u1 (1)u5 (1) u1 (1) u2 (1)

[10]

Table 3 Unipotent class representatives for E 6 . Class

p

u

E6 E 6 (a1 ) D5 E 6 (a3 )

all all all 2 =  2 all all all all all all all all all all all all all all

u1 (1)u2 (1)u3 (1)u4 (1)u5 (1)u6 (1) u1 (1)u2 (1)u9 (1)u10 (1)u5 (1)u6 (1) u1 (1)u3 (1)u4 (1)u2 (1)u5 (1) u1 (1)u8 (1)u9 (1)u11 (1)u14 (1)u19 (1) u13 (1)u1 (1)u15 (1)u6 (1)u14 (1)u4 (1) u1 (1)u8 (1)u9 (1)u2 (1)u5 (1) u1 (1)u3 (1)u4 (1)u5 (1)u6 (1) u1 (1)u3 (1)u4 (1)u2 (1)u6 (1) u3 (1)u4 (1)u2 (1)u5 (1) u1 (1)u3 (1)u4 (1)u2 (1) u2 (1)u3 (1)u8 (1)u10 (1) u1 (1)u3 (1)u4 (1)u6 (1) u1 (1)u3 (1)u5 (1)u6 (1)u2 (1) u1 (1)u3 (1)u4 (1) u2 (1)u4 (1)u1 (1)u6 (1) u1 (1)u3 (1)u5 (1)u6 (1) u1 (1)u3 (1)u2 (1) u1 (1)u3 (1) u1 (1)u4 (1)u6 (1)

all all

u1 (1)u6 (1) u1 (1)

D 5 (a1 ) A5 A4 A1 D4 A4 D 4 (a1 ) A3 A1 A 22 A 1 A3 A 2 A 21 A 22 A2 A1 A2 A 31 A 21 A1

Source RL [12, p. 172] RL

RL

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Remark. Each class representative is distinguished in some standard Levi subgroup L, i.e. C [ L , L ] (u )◦ is unipotent. If the label of the class is as in good characteristic the type of L is included in the name of the class and it is easily recognizable from the representative. The only exceptional classes for which (2)

L = G are B 2

˜ (2) in G of type F 4 . For B (2) the centralizer of a maximal torus h9 of C G (u ) is and A 1 2

˜ generated by T together with U ±3 , U ±4 , U ±18 and is of type B 3 . For A 1 torus h12 h2 in C G (u ) is T , U ±17 , U ±16 and has type B 2 .

(2)

the centralizer of a maximal

Each element u in Tables 1, 2 and 3 is distinguished in some standard Levi subgroup L u which can be calculated from the expression of u. So L u contains T , L u = C G ( Z ( L u )) (see for example [12, Lemma 2.3]) and Z ( L u ) ⊆ T is a maximal torus of C G (u ). We therefore have the following consequence of our choice of representatives: Corollary 3.2. In all cases the maximal torus T with respect to which the representatives u are given contains a maximal torus of the centralizer C G (u ). 4. Borel subgroups of centralizers In this section we describe our choice of a Borel subgroup Bˇ which contains a Borel subgroup of the centralizer C G (u ). In the previous section we selected a list of unipotent class representatives u. They were given with respect to a fixed Borel subgroup B and a maximal torus T ⊆ B which contains a maximal torus of C G (u ) (see Corollary 3.2). Since any two Borel subgroups are conjugate, Bˇ = B g for some g ∈ G, we find a conjugate u˜ = g u of u such that B contains a Borel subgroup of the centralizer C G (u˜ ). The main result of this section is Proposition B which we restate here: Proposition (B). The unipotent elements u˜ in Tables 4, 5 and 6 are a complete list of unipotent class representatives such that if B u˜ = C B (u˜ )◦ , T u˜ = C T (u˜ )◦ and U u˜ = C U (u˜ )◦ then B u˜ = T u˜  U u˜ is a Borel subgroup of C G (u˜ ). If g = un nb is the Bruhat expression of g with respect to T ⊂ B then g B = un n B, so we may assume b = 1. It will follow from the proof of Proposition B that for our choice of representatives we can choose g such that un = 1, i.e. such that g ∈ N G ( T ). However, this is not possible for an arbitrary choice of a representative u ∈ B (see the discussion in Example 4.3). ˇ We first introduce objects used in the description of B. 4.1. Associated objects



For a unipotent element u = uα (c α ), where  the factors are ordered according to the ordering of positive roots given in Section 10, set e u = c α eα . The map u → e u induces an injective map from unipotent classes to nilpotent orbits [12, Chapters 17, 18]. Clearly, e u depends on the choice of the Borel subgroup B and the choice of a maximal torus T ⊆ B with respect to which u is given. Let e be a nilpotent element of L(G ). To e we can associate a cocharacter τe with the properties that τe (c ).e = c 2 e and im τe ⊆ [ L , L ] for a Levi subgroup L ⊆ G such that e is distinguished in L( L ). In good characteristic the existence of such τe was essentially established in [16, Satz (3.1)]. To a cocharacter τ of G we can associate a parabolic subgroup P τ as described in [8, Section 5.1]. Definition 4.1. We call e u associated to u (with respect to B and T ), τe associated to e and P τ associated to τ . By transitivity we will call τu := τe u and P τu associated to u. In bad characteristic, for distinguished unipotent classes in exceptional groups the existence of associated cocharacters follows from [12, Theorem 13.2 and Proposition 14.14]. In Lemma 4.4 we give associated cocharacters explicitly for most unipotent classes in our setting. The remaining classes are

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Fig. 1.

(2)

B2

˜ and A 2

(2)

345

τu weights.

for which it is not difficult to check that the

τu described in Lemma 4.4 associated to

˜ 2 are also associated to B and A˜ our representatives of B 2 and A 2 2 respectively. These associated objects play an important role in the analysis of C G (u ) carried out in [12] and in our analysis. The following proposition shows how one can use associated cocharacters to find a Borel subgroup of the centralizer of a unipotent element. (2)

(2)

Proposition 4.2. Let u ∈ G be unipotent and assume the characteristic is good for G. Then 1. there exists a cocharacter τu associated to e u such that C G (u ) ⊆ P τu , 2. a Borel subgroup of C G (u ) is contained in a Borel subgroup B of P such that weights on L B.

τu acts with non-negative

Proof. The first statement follows from the nilpotent case [8, Proposition 5.9] and the existence of Springer maps. The second assertion follows from [12, Theorem 1(iii)(c)], again by using the existence of Springer maps. 2 Associated cocharacters are “good characteristic devices”. When passing to bad characteristic these associated cocharacters don’t share several properties of their good characteristic counterparts. This becomes visible in the following Example 4.3. Suppose the characteristic is 3 and let u = u4 (1)u6 (1) be a representative for the (distin˜ (3) class in G of type G 2 . It is not difficult to see that τu = h2 h2 is a cocharacter associated guished) A 1 2 1

˜ 1 and an associated cocharacter to u. Switching to good characteristic p, u will represent the class A would give the Dynkin labeling 10 of the class. Clearly for good p, τu is not associated to u since it gives the labeling 02 of the class G 2 (a1 ). Switching back to characteristic 3, one can still ask: does the parabolic P τu associated to τu contain C G (u )? This also fails since the weights of τu on root vectors are as in Fig. 1 and C G (u )◦ is R u ( B ) = U i : i ∈ {1, . . . , 6} . It turns out that the “right” cocharacter to consider is τ  = h4 in the sense that C G (u ) ⊆ P τ  . This ˜ 1 class. Of course τ  doesn’t have is the cocharacter associated to u4 (1) a representative for the A weight 2 on e u , it doesn’t even stabilize the subspace e u . Another point that this example makes visible is the following. For a unipotent class representative u ∈ B, such that T u ⊆ T ⊆ B, does there exist an element n ∈ N G ( T ) such that C n B (u ) is a Borel subgroup of C G (u )? The answer is yes for our choice of representatives but no in general. To see this we note that the assumption on the inclusion of maximal tori is trivial for distinguished elements. If ˜ (3) class given in Table 2, then another representative is u is the representative for the A 1 v = u2 (1)u3 (1) u = u1 (1)u3 (−1)u6 (−1)

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and the only Borel subgroups containing v and T are B and n2 B. With the method in Section 5, one finds that dim C B ( v ) = dim C n2 B ( v ) = 4. Since dim C G ( v ) = 6 we see that the answer to the above question is no. 4.2. Representatives If u represents a non-exceptional class then we use the cocharacters associated to u to deduce the representative u˜ with the properties of Proposition B. The associated cocharacters are given by the following Lemma 4.4. If τ is one of the cocharacters in [11, Section 6] associated to a representative e of the nilpotent class with Bala–Carter label X then τ is associated to the representative u with the same label given in Section 3. Proof. Let u be distinguished in the Levi subgroup L. That im τ ⊆ [ L , L ] is easy and for it is not difficult to check each case individually. 2

τ (t )e u = t 2 e u

The next lemma will be used to recognize Bˇ by its unipotent radical. Lemma 4.5. Let H be a closed subgroup of an algebraic group G with maximal tori and Borel subgroups T H ⊆ B H and T G ⊆ B G respectively. If T H ⊆ T G and n ∈ N G ( T G ) then n B G ⊃ B H ⇔ n R u ( B G ) ⊃ R u ( B H ). Proof. One implication is trivial and the other one follows from n B G ⊃ T G ⊃ T H .

2

Proof of Proposition B. A Borel subgroup Bˇ of G contains a Borel subgroup of C G (u ) if and only if C Bˇ (u )◦ is such a Borel subgroup of C G (u ). The latter can be verified by checking dimensions (since Borel groups are maximal, solvable, connected subgroups). These dimensions can be read-off from the tables in [12, Chapter 22] by summing up the dimension of the radical with the dimension of a Borel subgroup of the reductive quotient. For non-exceptional classes, the elements u˜ in Tables 4, 5 and 6 were obtained as follows: denote by  the fundamental system of roots corresponding to T ⊆ B. Take u as in the previous section and consider the associated cocharacter τu as in Lemma 4.4. The cocharacter τu viewed as an element of the coroot lattice belongs to the closure of some Weyl chamber and there exists a w in the Weyl group such that τu acts with non-negative weights on e w α for α ∈ , i.e. such that { w α : α ∈ } is a fundamental system corresponding to a Weyl chamber containing τu in its closure. Clearly w is not unique. There is a standard way of obtaining one such w (see for example [23, Section 3.3]).  Now, assume that w has  the above property and write w = sα as a product of simple reflections sα . We consider n = nα where nα = uα (1)uα (−1)uα (1) is a Weyl group representative for sα (see for example [2, Lemma 6.4.4]) and we choose u˜ = n u. In particular, our choice for Bˇ is B n . For exceptional classes X ( p ) we take Bˇ to be the Borel subgroup obtained for the non-exceptional class X . Since n ∈ N G ( T ), by Lemma 4.5 it suffices to check the dimension of the unipotent part C U (u˜ ), i.e. the dimension of the ‘unipotent centralizer’ which we obtain with the method described in Section 5. 2 Remark. The element n in the above proof was obtained as follows: let α ∈  be the first root (in Bourbaki order) such that α (τu ) < 0, replace τu by nα τu and repeat until all labels are positive. In this way we obtain an element n ∈ N G ( T ) with the desired properties and n is determined by τu (since we always choose the first negative integer in the labeling).

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Table 4 Unipotent class representatives for G 2 . Class

p

u˜

G2 G 2 (a1 ) ( A˜ 1 )(3) ˜1 A A1

all all 3 all all

u1 (1)u2 (1) u2 (1)u5 (1) u4 (1)u6 (1) u4 (1) u6 (1)

Table 5 Unipotent class representatives for F 4 . Class

p

u˜

F4 F 4 (a1 ) F 4 (a2 )

all all 2 = 2 all all all 2 all 2 all

u2 (1)u4 (1)u3 (1)u1 (1) u2 (1)u4 (1)u7 (1)u5 (1) u1 (1)u5 (1)u6 (1)u10 (1) u1 (1)u4 (1)u9 (1)u10 (1) u10 (1)u1 (1)u9 (1) u10 (1)u2 (1)u8 (1) u4 (1)u6 (1)u10 (1)u18 (1) u10 (1)u9 (1)u15 (1)u16 (1) u10 (1)u9 (1)u15 (1) u16 (1)u12 (1)u11 (1)u18 (1) u16 (1)u12 (1)u11 (1)

B2 B2 ˜1 A2 A ˜2 A A2 ˜1 A1 A

2 all all all all all

u15 (1)u9 (1)u16 (1) u15 (1)u9 (−1) u15 (1)u16 (1)u14 (1) u11 (−1)u12 (−1) u16 (−1)u18 (1) u20 (1)u19 (−1)

(2)

2 all all

u21 (1)u24 (1) u21 (1) u24 (−1)

C3 B3 F 4 (a3 ) C 3 (a1 )(2) C 3 (a1 ) ( A˜ 2 A 1 )(2) ˜ 2 A1 A (2)

˜ A 1 ˜1 A A1

Table 6 Unipotent class representatives for E 6 . Class

p

u˜

E6 E 6 (a1 ) D5 E 6 (a3 )

all all all 2 = 2 all all all all all all all all all all all all all all

u1 (1)u2 (1)u3 (1)u4 (1)u5 (1)u6 (1) u1 (1)u2 (1)u9 (1)u10 (1)u5 (1)u6 (1) u9 (−1)u2 (1)u10 (1)u7 (1)u6 (1) u1 (1)u8 (1)u9 (1)u11 (1)u14 (1)u19 (1) u13 (1)u1 (1)u15 (1)u6 (1)u14 (1)u4 (1) u15 (1)u12 (1)u8 (1)u7 (1)u11 (1) u14 (−1)u6 (1)u15 (−1)u1 (1)u13 (1) u14 (−1)u12 (1)u11 (1)u13 (1)u15 (−1) u12 (1)u2 (1)u15 (−1)u16 (1) u13 (−1)u18 (1)u6 (1)u14 (1) u15 (−1)u12 (1)u19 (1)u20 (1) u14 (−1)u21 (−1)u17 (−1)u18 (1) u20 (1)u18 (−1)u21 (1)u17 (1)u24 (1) u13 (−1)u23 (−1)u14 (1) u23 (1)u24 (1)u25 (1)u22 (−1) u21 (−1)u17 (−1)u20 (−1)u18 (1) u29 (1)u20 (1)u23 (1) u17 (1)u31 (−1) u31 (−1)u30 (1)u29 (−1)

all all

u32 (−1)u33 (1) u36 (1)

D 5 (a1 ) A5 A4 A1 D4 A4 D 4 (a1 ) A3 A1 A 22 A 1 A3 A 2 A 21 A 22 A2 A1 A2 A 31 A 21 A1

347

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Remark. Note that the signs in the expression of u˜ can be chosen as for u given in Section 3, in other words one can replace all −1’s by 1’s. Indeed the elements were obtained by conjugation under Weyl group representatives n so they can be thought of as ‘being the elements u’ with respect to a different fundamental system. However we will avoid doing this because after obtaining C Z (U u˜ ) ( T u˜ ) = Z (C G (u˜ )◦ ) we will conjugate this group by n−1 and use the component group representatives of C G (u ) to determine the connected component Z (C G (u ))◦ (see the remark at the end of Section 7). 5. Unipotent centralizer In this section we describe the connected component of the centralizer in a unipotent group U of an element u ∈ U . The description will be used to obtain the unipotent radical U u of a Borel subgroup of C G (u ) where u ∈ G is a unipotent element of a simple exceptional adjoint algebraic group G of type G 2 , F 4 or E 6 . We start out with some general facts in commutative algebra. The notation is standard (see Section 10 for more details). 5.1. Idempotents Throughout A will denote a commutative ring with unit and I will be an ideal of A. We are interested in the connected components of the prime spectrum Spec( Q ) where Q := A / I . One knows that the prime spectrum of a ring is connected if and only if the ring doesn’t contain any idempotents other then 0 and 1 [1, Ch. II, §4, Corollary 2]. The next lemma allows us to consider m-potents when analyzing connected components, i.e. elements q in Q such that qm = q for some m ∈ N. Lemma 5.1. With the above notation we have: (i) If a ∈ A is such that am − a ∈ I for some integer m > 0 then b¯ := a¯ m−1 ∈ Q is an idempotent. (ii) If b¯ = b + I is an idempotent of Q then b is such that bm − b ∈ I for any positive integer m. Proof. For the first part we check (am−1 )2 = a2m−2 = am am−2 = aam−2 = am−1 mod I . For the second part, since b¯ ∈ Q is an idempotent, we have b¯ 2 = b + I , so b¯ m = bm + I = b + I ⇒ bm − b ∈ I . 2 Note that, with b¯ as in (ii) above, we get a direct sum decomposition Q = Q b¯ ⊕ Q (1 − b¯ ) (see [1, Ch. II, §4, Proposition 15]). We now specialize to the case of affine varieties and choose A to be the polynomial ring k[x1 , . . . , xn ]. Denote by 0n the n-tuple (0, . . . , 0). For a subvariety X of An containing 0n denote by X ◦ the connected component containing 0n . We will use the next lemma as a criterion for identifying the connected component containing 0n once an m-potent is known. Lemma 5.2. If f ∈ A \ I is an element of I(V( I )◦ ) such that f m − f ∈ I , for an integer m > 1, then V( I )◦ ∼ = Spec( A /( f , I ))◦ as topological spaces. Proof. With b := f m−1 , by Lemma 5.1(i) we know that b¯ ∈ Q is an idempotent. It is non-zero / I and the decomposition Q = Q b¯ ⊕ Q (1 − b¯ ) gives a disjoint union Spec( Q ) ∼ since f ∈ = V( Q b¯ )  ¯ V( Q (1 − b)). By assumption the constant term of f and whence that of b is zero, therefore Q b¯ vanishes at 0n . It follows that 0n ∈ V( Q b¯ ) so V( I )◦ ∼ = Spec( Q / Q b¯ )◦ . Note that Spec( Q / Q b¯ ) is not reduced if m > 1 since ¯f is a nilpotent of Q . However, as topological spaces (see [13, Proposition 2.4.2])

Spec( Q / Q b¯ ) ∼ = Spec( Q / Q ¯f )red ∼ = Spec( Q / Q ¯f ) = Spec( Q / Q b¯ )red ∼

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349

where the isomorphism in the middle follows from the fact that ¯f is a nilpotent of Q / Q b¯ = Q / Q ¯f m−1 . Using an isomorphism theorem Q / Q ¯f = ( A / I )/(( f , I )/ I ) ∼ = A /( f , I ) we obtain the desired isomorphism. 2 5.2. Maximal point fiber Consider the fiber above a maximal point p (i.e. corresponding to a maximal ideal m) of a morphism between affine varieties ϕ

ϕ −1 ( p ) → Spec( A ) − → Spec( B ) where the first arrow is inclusion. There are two ways of looking at the structure ring of a fiber. With m the maximal ideal corresponding to p and k( p ) := B m /m B m its residue field, we have



A/

   ∼ O ϕ −1 ( p ) ∼ ϕ ∗ (m) = = A ⊗ B k( p ).

For the right-hand side we refer the reader to [6, Section II.3]. To see how the two are related note that since m is maximal k( p ) is isomorphic to B /m. Regarding A as a B-module via ϕ ∗ , by [9, Ch. XVI, §3, Proposition 2.7], we have the canonical B-module isomorphism

A ⊗ B ( B /m) ∼ = A /m A which is clearly also a ring isomorphism. We will use the first point of view and consider the sequence ϕ∗



B −→ A → A /



ϕ ∗ (m)

in a particular case for unipotent groups. 5.3. Unipotent groups Here we give additional structure to A and regard it as O(U (k)) where U (k) is a connected unipotent algebraic group defined over k. The field k is fixed and we write U for U (k) and An for An (k). A proof that U is isomorphic to An , where n = dim U , can be found in [21, Ch. VII, §1, Corollary] or can be deduced from [25, Theorem 14.2.6]. In the particular case – to which we restrict – where U is the unipotent radical of B a Borel subgroup of a simple algebraic group G, the isomorphism to affine space is obtained via the root subgroups. For this, fix a maximal torus T ⊆ B which will act with positive weights on U , fix an ordering (α1 , . . . , αn ) on the positive roots Φ + then the map

u : An → U

defined by

u(c 1 , . . . , cn ) →



uαi (c i )

αi ∈Φ +

where the product is taken in the fixed order, is an isomorphism of varieties. For a fixed element u ∈ U , the commutator [u , _] is an endomorphism of U and one knows that it induces an endomorphism [u , _]∗ of the structure ring A := O(U ) by pulling back functions:

  [u , _]∗ ( f )( g ) = f [u , g ] for any f ∈ O(U ) and g ∈ U and we denote [u , _]∗ (xi ) by P i .

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5.4. Unipotent centralizer We are now in a position to describe C U (u ) for an element u = (c 1 , . . . , cn ) ∈ U (k) with c i ∈ k. Since C U (u ) is just the fiber at 1U = 0n of [u , _] we have

  O C U (u ) ∼ = A / I where I := ( P 1 , . . . , P n ) = [u , _]∗ (m0n ) where m0n is the maximal ideal at 0n and the last equality is by the above discussion. Our setting is such that U is the unipotent radical of a Borel subgroup B of a simple algebraic group G. If the characteristic of k is good for G then one knows that the order of the component group of C G (u ) is not divisible by p (see for example [12]). In particular C U (u ) is connected and therefore O(C U (u )) ∼ = k[y1 , . . . , ym ] where m := dim C U (u ). At this point we focus attention on the unipotent class representatives u˜ of G (adjoint of type G 2 , F 4 and E 6 ) given in Section 4. For those elements we are able to calculate this isomorphism explicitly. This allows us to complete the proof of Proposition B. For this, note that until now we have no constraint on the ordering of the roots for the isomorphism u. We can therefore choose an ordering compatible with the height of roots. With this choice we observe that the P i depend only on the x j for which ht α j < ht αi . This allows us to solve the system P i = 0 recursively and to deduce the dimensions given in Tables 7, 8 and 9. By this we mean that, with one exception treated at the end of this section, there is a subset x J := {x j : j ∈ J = ( j 1 , . . . , jm )} of the indeterminates {xi } such that an ideal I ◦ with V( I ◦ ) = C G (u˜ )◦ is generated by xi − Q i where

Qi =

xi a polynomial in {x j } j ∈ J

if i ∈ J if i ∈ / J.

In particular this implies the following lemma, the proof of which is easy. Lemma 5.3. With the above, the map ψ : A / I ◦ = O(C U (u˜ )◦ ) → k[y1 , . . . , ym ] defined by ψ(xi ) = Q i (y), where y := (y1 , . . . , ym ), is an isomorphism of rings. It is also clear that ψ = φ ∗ is induced by (is the pull-back of) the isomorphism onto the image of the morphism of affine algebraic varieties

  φ : Am  y → u Q 1 (y), . . . , Q n (y) ∈ C U (u˜ )◦ ⊂ U . Where, by abuse of notation, we used y for an m-tuple in km . So ψ is an embedding of Am into U and has image C U (u˜ )◦ . Example 5.4. To illustrate this, consider the case where G is of type G 2 and char(k) = 5. Let u = u1 (1)u2 (1) be our representative for the regular unipotent class. The ideal I = [u , _]∗ (m06 ) is generated by

⎧ [u , _]∗ (x3 ) = 4x2 + x1 ⎪ ⎪ ⎪ 2 ∗ ⎪ ⎪ ⎨ [u , _] (x4 ) = 3x3 + 4x2 + x1 ∗ [u , _] (x5 ) = 2x4 + 2x3 + 4x2 + x31 ⎪ 2 2 ∗ ⎪ ⎪ ⎪ [u , _] (x6 ) = 4x5 + 2x4 + 2x3 + 4x2 + 3x3 + 2x2 x4 + 2x2 x3 + 3x2 ⎪ ⎩ + 3x1 x4 + x1 x3 + 3x1 x2 + 2x21 x3 + 4x31 + x31 x2 .

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Clearly, [u , _]∗ (x1 ) and [u , _]∗ (x2 ) are trivial since im[u , _] ⊆ [U , U ] = U i : i ∈ {3, 4, 5, 6} . Using the first generator we have x2 − x1 ∈ I and substituting we obtain recursively

I

⎧x − x 2 1 ⎪ ⎪ ⎨ x3 − (2x1 + 3x2 ) 1

x4 − (x1 + 2x21 + 2x1 ) ⎪ ⎪ ⎩ x5 − (4x21 + 2x31 + 4x41 ). 3

Therefore the isomorphism ψ = φ ∗ : O(C U (u )◦ ) → k[y1 , y2 ] is induced by

  φ(y1 , y2 ) = u y1 , y1 , 2y1 + 3y21 , y1 + 2y21 + 2y31 , 4y21 + 2y31 + 4y41 , y2 . 5.5. Bad characteristic If the characteristic of k is bad then C U (u ) might not be connected. In [24] it was shown that in bad characteristic regular unipotent elements u do not belong to C G (u )◦ . One can find all classes where u ∈ / C G (u )◦ in [12, Table 17.4]. In particular such u do not belong to C U (u )◦ . Example 5.5. Let G be of type G 2 and char(k) = 3. Consider again the regular element u = u1 (1)u2 (1). The ideal I = [u , _]∗ (m06 ) is generated by

⎧ [u , _]∗ (x3 ) = 2x2 + x1 ⎪ ⎪ ⎨ [u , _]∗ (x4 ) = x3 + 2x2 + x2 1

[u , _]∗ (x5 ) = 2x2 + x31 ⎪ ⎪ ⎩ [u , _]∗ (x6 ) = 2x5 + 2x2 + x22 + 2x31 + x31 x2 . Using the first and third generators we have x31 − x1 ∈ I . Since x21 + I is an idempotent of A / I , from the proof of Lemma 5.2 we recognize the connected component to be



C U (u )◦ = Spec A / I 

◦

where I  = (x1 , P 1 , . . . , P 6 ).

Modulo (x1 , x1 − x2 ) ⊂ I  we find that x3 , x5 ∈ I  . In other words the ideal I  equals (x1 , x2 , x3 , x5 ) and is prime, so Spec( A / I  ) = Spec( A / I  )◦ . The isomorphism ψ = φ ∗ : O(C U (u )◦ ) → k[y1 , y2 ] is induced by

φ(y1 , y2 ) = u(0, 0, 0, y1 , 0, y2 ). The component group A (C U (u )) ∼ = Z 3 and is generated by uC G (u )◦ (see for example [12, Theorem 19.1]). Using the above calculation, it is not difficult to see that u and u 2 are non-trivial representatives for A (C U (u )) since their projections onto U 1 are non-zero. 5.6. Exception: C 3 (a1 ) class in F 4 for p = 2 Denote by U >i the subgroup U j : j > i . Every element of U >i has a unique expression of the form

ui +1 (c i +1 ) · · · un (cn )

with c i +1 , . . . , cn ∈ k.

Choose the representative u˜ = u9 (1)u10 (1)u15 (1) for the unipotent class C 3 (a1 ) of F 4 and let c = (c 1 , . . . , c 24 ) be a point in U . A computation shows that

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[u˜ , c ] = u11 (c 1 )u12 (c 1 )u13 (c 2 ) mod U >13 so [u˜ , _]∗ (xi ) = 0 for i < 11,

[u˜ , _]∗ (x11 ) = [u˜ , _]∗ (x12 ) = x1 and [u˜ , _]∗ (x13 ) = x2

and we can already work modulo x1 = x2 = 0, i.e. we suppose c ∈ V( I ) where I = (x1 , x2 ). We get

[u˜ , c ] = u14 (c 5 )u16 (c 6 )u17 (c 8 ) mod U >17 so we replace I by (x1 , x2 , x5 , x6 , x8 ) and we suppose c ∈ V( I ). Continuing the computation

[u˜ , c ] = u19 (c 11 + c 12 )u21 (c 9 + c 15 )u22 (c 13 + c 18 )u23 (c 16 + c 20 ) mod U >23 we enlarge I to (x1 , x2 , x5 , x6 , x8 , x11 + x12 , x9 + x15 , x13 + x18 , x16 + x20 ) and for c in V( I ) we have

  2 [u˜ , c ] = u15 c 92 + c 11 + c9 . Now, the dimension of V( I ) is 15 and the dimension of V( I , x29 + x211 + x9 ) is 14 since x29 + x211 + x9 is algebraically independent of I . By [12, Table 22.1.4] the dimension of the unipotent radical of a Borel of C G (u˜ ) is 14 so V( I , x29 + x211 + x9 ), which equals C U (u ), is such a subgroup because of its dimension. The problem here is the last equation which doesn’t allow for a parametrization of C G (u˜ ) as described on page 350. 6. Unipotent center In this section we are interested in Z (U u˜ )◦ , the center of the unipotent radical U u˜ of a Borel subgroup of C G (u˜ ) where u˜ is as before a unipotent class representative of G given in Section 4. We will make use of the following well-known fact: Lemma 6.1. For an algebraic group H we have Z ( H ) ⊆ C H (L H ). Proof. By [25, Lemma 4.4.13] d[x, _] = Ad(x) − 1. For x ∈ Z ( H ) the left-hand side is 0 so Ad(x) = 1.

2

The reverse inclusion is not true in general. However, in our setting, that is for the group H = U u˜ , we establish equality except in characteristic 2 for the C 3 (a1 ) class in F 4 . 6.1. Strategy In order to obtain LC U (u˜ ) we continue the discussion in the previous section. If p = 2, the C 3 (a1 ) class in F 4 is treated separately at the end of this section. For all other classes we have an isomorphism φ ∗ : A / I ◦ = O(C U (u˜ ))◦ → k[ y 1 , . . . , ym ] where

  φ : Am  y → u Q 1 (y), . . . , Q n (y) ∈ C U (u˜ )◦ ⊂ U is an embedding of Am into U with image C U (u˜ )◦ which we can calculate explicitly. Hence LC U (u˜ ) is the image under d0m φ of the tangent space at 0m of Am . So, once the explicit embedding φ is known, it is easy to extract a basis {∂i }i ∈ J of the Lie algebra LC U (u˜ ) with respect to a basis of LU . We will choose ∂i = d0m φ(∂yi ) where {∂yi } is the standard bases of T 0m Am . Now, by Lemma 6.1, we have









Z C U (u˜ )◦ ⊆ C C U (u˜ )◦ LC U (u˜ )◦ =

 i∈ J

C C U (u˜ )◦ (∂i ) =: X

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353

and φ −1 C C U (u˜ )◦ (∂i ) is the fiber at 0m of the algebraic morphism





Ad ◦ φ(_) − 1 (∂i ) : Am → LC U (u˜ )◦ ∼ = Am .

As in Section 5, an ideal J i in B := k[y1 , . . . , ym ] = O(Am ) such that V( J i ) = φ −1 C C U (u˜ )◦ (∂i ) is





J i := Ad ◦ φ(_) − 1 (∂i )∗ (m0m ) B = ( R i ,1 , . . . , R i ,m )



where R i , j := (Ad ◦ φ(_) − 1)(∂i )∗ (y j ). Therefore, the ideal J generated by J i or equivalently by   { R i , j } is such that V( J ) = V( J i ) = V( J i ) = φ −1 X . We remark that there is a subset y K := {yk : k ∈ K = (k1 , . . . , kl )} of the indeterminates such that an ideal J ◦ with V( J ◦ ) = φ −1 X ◦ is generated by yi − S i where

Si =

yi a polynomial in {y j } j ∈ K

if i ∈ K if i ∈ / K.

(6)

This implies that the map Ψ : B / J ◦ = φ ∗ O( X ◦ ) → k[z1 , . . . , zl ] defined by Ψ (yi ) = S i (z), where z := (z1 , . . . , zl ), is an isomorphism of rings and that Ψ = Φ ∗ is induced by (is the pull-back of) the isomorphism onto the image of the morphism of affine algebraic varieties

  Φ : Al  z → S 1 (z), . . . , S m (z) ∈ φ −1 X ◦ ⊂ Am . At this point we have the inclusion Z (C U (u˜ )◦ )◦ ⊂ X ◦ so if [C U (u˜ )◦ , X ◦ ] = 1 then X ◦ = Z (C U (u˜ )◦ )◦ . This turns out to be true for our choices of G and u˜ (with the exception of the class C 3 (a1 )). Example 6.2. Let u˜ = u1 (1)u5 (1) be our representative for the subregular class G 2 (a1 ) in characteristic 2. Calculations show that

C U (u˜ ) = u2 (x3 )u3 (x1 )u4 (x2 )u5 (x3 )u6 (x4 ) i.e. the embedding of the connected component U u˜ = C U (u˜ )◦ is

φ : A4  y → u(0, y3 , y1 , y2 , y3 , y4 ) ∈ C U (u˜ )◦ ⊂ U . A basis of the Lie algebra L(U u˜ ) with respect to the basis {e1 , . . . , e6 } of LU is

e3 = (0, 0, 1, 0, 0, 0) = ∂1 , e4 = (0, 0, 0, 1, 0, 0) = ∂2 , e2 + e5 = (0, 1, 0, 0, 1, 0) = ∂3 , e6 = (0, 0, 0, 0, 0, 1) = ∂4 . The action of C U (u˜ )◦ on these vectors is as follows:

φ(y).∂1 = e3 + y2 e6

⇒

J 1 = (y2 ),

φ(y).∂2 = e4 + y1 e6

⇒

J 2 = (y1 ),

φ(y).∂3 = e2 + e5

⇒

J 3 = (0),

φ(y).∂4 = e6

⇒

J 4 = (0).

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So φ ∗ IC U u˜ (LU u˜ ) = (y1 , y2 ). Clearly (y1 , y2 ) is prime so X = C U u˜ (LU u˜ ) is connected and a computation shows that [U u˜ , X ] = 1 so Z (U u˜ ) ⊇ X . It follows by Lemma 6.1 that X = Z (U u˜ )◦ . Clearly, the isomorphism

   Ψ = Φ ∗ : φ ∗ O Z C G (u )◦ → k[z1 , z2 ] is induced by Φ(z1 , z2 ) = (0, 0, z1 , z2 ). Remark. When calculating the ideal I( X ) one doesn’t actually need an entire basis of L(U u˜ ). For instance, since U commutes with the highest root group U n , the tangent space en of that root group will always be fixed by U u˜ and so will always give (Ad ◦ φ(_) − 1)(en )∗ (m0n ) = (0). 6.2. Exception: C 3 (a1 ) class in F 4 for p = 2 This is the only class for which we didn’t obtain an embedding φ of C U (u )◦ as described in Section 5.4. We treat it separately here. Let u˜ be the representative given in Section 4. In Section 5.6 we have seen that I(U u˜ ) = ( I , x29 + x211 + x9 ) where

I = (x1 , x2 , x5 , x6 , x8 , x11 + x12 , x9 + x15 , x13 + x18 , x16 + x20 ). Clearly V := V( I ) is a subvariety of U containing U u˜ for which an embedding φ V : A15 → U is given by

φ V (y) = u3 (y1 )u4 (y2 )u7 (y3 )u9 (y7 )u10 (y4 )u11 (y5 )u12 (y5 )u13 (y9 ) · u14 (y6 )u15 (y7 )u16 (y11 )u17 (y8 )u18 (y9 )u19 (y10 )u20 (y11 ) · u21 (y12 )u22 (y13 )u23 (y14 )u24 (y15 ). A basis of T 015 V with respect to the basis {e1 , . . . , e24 } of LU is {e3 , e4 , e7 , e10 , e11 + e12 , e14 , e9 + e15 , e17 , e13 + e18 , e19 , e16 + e20 , e21 , e22 , e23 , e24 }. In addition V  := V( I , x9 , x11 ) is a subvariety of U u˜ with basis for the tangent space {e3 , e4 , e7 , e10 , e13 + e18 , e14 , e16 + e20 , e17 , e19 , e21 , e22 , e23 , e24 }. Now we have Z (U u˜ ) ⊆ C U u˜ (LU u˜ ) ⊆ C V (LU u˜ ) ⊆ C V ( T 013 V  ) and in order to obtain the connected component of the last object we denote m015 by m and we calculate

e3 − φ V (y).e3 = y2 e7 + y5 e14 + y2 y5 e17 + (y8 + y3 y5 + y1 y2 y5 )e19 + y11 e21 so the ideal generated by (Ad ◦ φ V (_) − 1)(e3 )∗ (m) is (y2 , y5 , y8 , y11 ). Similarly one finds

 ⎧ Ad ◦ φ V (_) − 1 (e4 )∗ (m) = (y1 , y9 , y6 , y13 ) ⎪ ⎪   ⎪ ⎪ ⎪ Ad ◦ φ V (_) − 1 (e7 )∗ (m) = (y5 , y6 , y9 ) ⎪ ⎪   ⎪ ⎪ ⎪ Ad ◦ φ V (_) − 1 (e10 )∗ (m) = (y5 , y9 , y11 ) ⎪ ⎪   ⎪ ⎪ Ad ◦ φ V (_) − 1 (e11 + e12 )∗ (m) = (y2 , y3 , y4 + y7 ) ⎪ ⎪  ⎪ ⎪ Ad ◦ φ (_) − 1(e )∗ (m) = (y , y ) ⎪ ⎪ V 14 2 3 ⎪ ⎪ ⎨ Ad ◦ φ (_) − 1(e + e )∗ (m) = (y , y + y ) V 15 1 4 7   9 ⎪ Ad ◦ φ V (_) − 1 (e17 )∗ (m) = (y1 ) ⎪ ⎪   ⎪ ⎪ ⎪ Ad ◦ φ V (_) − 1 (e19 )∗ (m) = (0) ⎪ ⎪   ⎪ ⎪ Ad ◦ φ V (_) − 1 (e21 )∗ (m) = (0) ⎪ ⎪ ⎪ ⎪ Ad ◦ φ (_) − 1(e )∗ (m) = (y ) ⎪ ⎪ V 2 ⎪  22 ⎪ ⎪ ⎪ Ad ◦ φ V (_) − 1 (e23 )∗ (m) = (0) ⎪ ⎪  ⎩ Ad ◦ φ V (_) − 1 (e24 )∗ (m) = (0).

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So the ideal J generated by all the above ideals is

J = (y1 , y2 , y3 , y4 − y7 , y5 , y6 , y8 , y9 , y11 , y13 ), and a parametrization φ X : A5 → X = φ V (V( J )) is given by

φ X (z) = u9 (z4 )u10 (z4 )u15 (z4 )u19 (z10 )u21 (z12 )u23 (z14 )u24 (z15 ). Clearly φ V V( J ) = C V ( T 013 V  ) is connected and contains Z (C U (u˜ )). A calculation shows that



   φ V (y), φ X (z) = u24 y7 z24 + y27 z4 + y25 z4 .

Since φ V∗ (x29 + x211 + x9 ) = y27 + y25 + y7 it follows that an element y ∈ φ V−1 (C V (u˜ )) is such that [φ V (y), φ X (z)] = 1 if and only if y7 z24 + y27 z4 + y25 z4 = 0 and y27 + y25 + y7 . In particular we have the condition y7 (z24 + z4 ) = 0. This means (see proof of Lemma 5.2) that φ X (z) ∈ Z (U u˜ )◦ if z4 = 0 ⇔ 0 = φ V∗ [(φ X )∗ ]−1 (z4 ) = y4 = y7 mod I . So φ V V( I , y4 ) = Z (U u˜ )◦ and a parametrization is

Φ : A4  z → u19 (z1 )u21 (z2 )u23 (z3 )u24 (z4 ) ∈ Z (U u˜ )◦ . 7. Maximal tori of the centralizers In this section we describe C Z (U u˜ )◦ ( T u˜ ) = Z (C G (u˜ )◦ )◦ where as before U u˜ is the unipotent radical of a Borel subgroup B u˜ of C G (u˜ ) and T u˜ ⊆ B u˜ a maximal torus. The key point here is that we can compute this object without an explicit description of T u˜ (see remark below). We recall that we have fixed a Borel subgroup B of G and a maximal torus T ⊂ B such that for the unipotent class representatives u˜ in Section 4 we have



B u˜ = C B (u˜ )◦ , T u˜ = C T (u˜ )◦ , U u˜ = C U (u˜ )◦ u˜ distinguished in L u˜ = C G ( T u˜ ).

and

The last part follows from the well-known Lemma 7.1. Let S be a maximal torus of C G (u ) for some unipotent u ∈ G and let L = C G ( S ). Then u is distinguished in L  . We need one more reference to the literature: Lemma 7.2. (See [26, II §4.1] and [7, 22.3].) Let S ⊆ T be a torus then



CG (S) = T , Uα :



α( S ) = 1 .

In Section 6 we showed how one can obtain (and at this point we have) an explicit isomorphism of varieties defined by

  φ ◦ Φ : Al  z → u R 1 (z), . . . , R n ( z) ∈ Z (U u˜ )◦ for each representative u˜ given in Section 4, where R i = Q i ( S 1 ( z), . . . , S m ( z)) by (6) in the previous section. Moreover the R i are elements of k[x1 , . . . , xn ] and there is a subset x K := {xk : k ∈ K = (k1 , . . . , kl )} of the indeterminates such that

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Ri =

if i ∈ K if i ∈ / K.

xi a polynomial in {x j } j ∈ K

(7)

Denote by I u˜ the set of i such that R i = 0. Proposition 7.3. Let I be the set of roots α (with respect to T ) such that α ( T u˜ ) = 1 then



Z C G (u˜ )◦

◦

=









uα R α (c ) : c = (c 1 , . . . , cl ) ∈ kl .

α ∈ I ∩ I u˜

Proof. First note that we have fixed the order on positive roots to be the one in Section 10 so the product in the statement is unambiguous. We denote the right-hand side by X . (a) We showed (see (1) in Section 2) that



Z C G (u˜ )◦

◦

= C Z (U u˜ )◦ ( T u˜ ) = Z (U u˜ )◦ ∩ C G ( T u˜ ),

and by Lemma 7.2, it equals ◦

Z (U u˜ ) ∩ L u˜ =



  uα R α (c ) : such that R α (c ) = 0 ∀α ∈ / I ⊆ X. 

α ∈ I u˜

(b) The torus T u˜ stabilizes Z (U u˜ ) so, since T u˜ is connected, it acts (by conjugation) on Z (U u˜ )◦ : choose t ∈ T u , x ∈ Z (U u˜ ) and y ∈ U u˜ then t

yx = y

t x

⇔

y

= t y which is true since t y ∈ U u˜ .





Therefore, since T u˜ ⊆ T , t

x=



t



uα R α (c ) =

α ∈ I u˜

uα





α (t ) R α (c ) =

α ∈ I u˜







uα R α (d)

α ∈ I u˜

for some c and d in kl . Since φ ◦ Φ is an isomorphism we have, by (7),



α (t ) R α (c1 , . . . , cl ) = R α αk1 (t )c1 , . . . , αkl (t )cl



for each α ∈ I u˜ .



(c) Now, choose an element x = α ∈ I ∩ I ˜ uα ( R α (c )) ∈ X . Without loss of generality we may assume u that c i = 0 if i ∈ K \ I . Suppose that x ∈ / Z (U u˜ )◦ ∩ L u˜ which is equivalent to x ∈ / Z (U u˜ )◦ . Denote by x˜ ◦ the element α ∈ I ˜ uα ( R α (c )) of Z (U u˜ ) and note that x˜ ∈ / L u˜ else x = x˜ would contradict the choice u of x. It follows that there exists α0 ∈ I u˜ \ I such that R α0 (c ) = 0. So there exists t ∈ T u˜ such that α0 (t ) = 1 and by (b) we have

0 = α0 (t ) R α0 (c 1 , . . . , cl ) = R α0





αk1 (t )c1 , . . . , αkl (t )cl .

This shows that there exists j ∈ K such that c j = 0 and αk j (t ) = 1. By assumption on the c i it follows that j ∈ I which is in contradiction with x ∈ L u˜ . We conclude that X ⊆ Z (C G (u˜ )◦ )◦ and together with the reverse inclusion from (a) we obtain the desired equality. 2

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Remark. The above result shows that Z (C G (u˜ )◦ )◦ can be obtained by ‘truncating’ Z (U u˜ )◦ and for this we only need the root system I of L u˜ as a subsystem of the root system of G (with respect to the common maximal torus T ). The root system of L u can be obtained from the expressions of our representatives u, see the last Remark of Section 3, and then I is obtained by conjugating with an element of the Weyl group – see next remark. Remark. Let u be a representative given in Section 3 and n such that u˜ = n u is the representative in Section 4. Since [ Z (C G (u˜ )◦ )◦ , T u˜ ] = 1 we have Z (C G (u˜ )◦ )◦ ⊆ L u˜ . It is not difficult to see that L u˜ = n L u and Z (C G (u˜ )◦ )◦ = n Z (C G (u )◦ )◦ . Moreover, one finds that for our choices of u and n we have Z (C G (u )◦ )◦ = ( Z (C G (u˜ )◦ )◦ )n ⊆ L u ∩ U . In this way we can switch back to u and use component group representatives with respect to u. 8. Component groups of centralizers

˜ for C G (u ) where u is a In this section we determine explicitly component group representatives A unipotent class representative given in Section 3. At this point Z (C G (u )◦ )◦ is known explicitly for all ˜ on such representatives u. After some preliminary considerations and results we give the action of A Z (C G (u )◦ )◦ from which we determine Z (C G (u ))◦ . 8.1. Preliminary results If the characteristic is good for G, a description of the component groups A (C G (u )) can be found in [14]. Explicit representatives for the component groups of stabilizers of nilpotent elements were used in [11]. In bad characteristic the component groups are known (for a recent treatment of A see [12]). Our strategy for handling each case can be summarized as follows: 1. Find a set of elements a1 , . . . , ar ∈ C G (u ) \ C G (u )◦ . 2. Show that the images of the ai in A generate the group A. 3. Give the action of ai on Z (C G (u )◦ )◦ and deduce Z (C G (u ))◦ . We will see that in many cases we use step 3 to give the argument for step 1 by using Lemma 8.1. The next lemmas will serve as criteria for deciding if certain elements are in C G (u ) \ C G (u )◦ . Lemma 8.1. If h ∈ H acts non-trivially on Z ( H ◦ ) then h represents a non-trivial element of A ( H ). Lemma 8.2. Let H be a closed subgroup of G and let t ∈ H be a semisimple element centralizing a maximal torus T H of H . If t ∈ / T H then t ∈ / H◦. Proof. If t ∈ H ◦ then t belongs to some maximal torus T˜ H of H so T H and T˜ H are maximal tori of C H (t ) (in particular of C H (t )◦ ). It follows that the two maximal tori are conjugate under C H (t ) which contradicts our assumption t ∈ / TH. 2 For what follows let i and ω be roots of x2 + 1 and x2 + x + 1 respectively. Denote the quotient map C G (u ) → C G (u )/C G (u )◦ by x → x¯ . 8.2. Type G2 Class G2 (a1 ): The component group A is Z 2 if p = 3 and S 3 otherwise. Our representative is u = u2 (1)u5 (1). It is distinguished so C Z (U u )◦ ( T u ) = Z (U u )◦ . Similar to [11, p. 49], we choose a1 = h1 (ω) and a2 = h2 (−1)u2 (1)n1 .

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If p = 3 then

ω = 1 so a1 = 1 and Z (U u )◦ = u2 (x1 )u3 (x2 )u4 (x3 )u5 (x1 )u6 (x4 ), a2





Z (U u )◦ = u2 (x1 )u3 (x3 )u4 (x2 )u5 (x1 )u6 2x4 + 2x1 + x21 .

The action being non-trivial, it follows from Lemma 8.1 (with H = C G (u )) that A = ¯a2 . The equations 2x4 + 2x1 + x21 = x4 ⇔ x4 = x1 − x21 and x2 = x3 show that Z (C G (u ))◦ = u2 (x1 )u3 (x2 )u4 (x2 )u5 (x1 ) × u6 (x1 − x21 ). For p = 2, calculating one finds that Z (U u )◦ = u2 (x1 )u5 (x1 )u6 (x2 ), that a1 Z (U u )◦ = u2 (x1 )u5 (x1 ) × u6 (x2 ) and a2 Z (U u )◦ = u2 (x1 )u5 (x1 )u6 (x2 + x1 + x21 ) Now, 1 = a¯ 1 ∈ A follows from Lemma 8.2 and for a2 we use Lemma 8.1. In order to recognize A, note that [U u , U u ] = U 6 and that U u acts trivially on U u /U 6 . Consider the action of {a1 , a2 } on U u = u2 (x1 )u3 (x3 )u4 (x4 )u5 (x1 )u6 (x2 )





a1

U u = u2 (x1 )u3

ω2 x3 u4 (ωx4 )u5 (x1 )u6 (x2 ),

a2

U u = u2 (x1 )u3 (x4 )u4 (x3 )u5 (x1 )u6 x3 x4 + x21 + x1 + x2 .





Let m be the maximal ideal at 1 ∈ U u /U 6 ∼ = A3 . The ai induce k-linear automorphisms on the cotangent space m/m2 . Choose v 1 = x¯ 3 + x¯ 4 , v 2 = ω2 x¯ 3 + ωx¯ 4 and v 3 = ωx¯ 3 + ω2 x¯ 4 in m/m2 and note that a1 and a2 permute these vectors as (1, 2, 3) and (2, 3) respectively. This gives an action of {¯a1 , a¯ 2 } on m/m2 and we conclude that ¯a1 , a¯ 2 ∼ = S3. =A∼ So Z (C G (u ))◦ is obtained from the equation x2 + x1 + x21 = x2 ⇔ x21 − x1 = 0 ⇒ Z (C G (u ))◦ = u6 (x2 ) (using the proof of Lemma 5.2). Class G2 : Here A is non-trivial only for p ∈ {2, 3} in which case A = u¯ ∼ = Z p [12, Table 22.1.5]. So Z (C G (u ))◦ = C Z (U u )◦ ( T u , u ) = Z (U u )◦ by Theorem A. 8.3. Type F4

˜ 1 : The representative is u = u1 (1). If p = 2 the component group is trivial. If p = 2 we have Class A A∼ = Z 2 . For p = 3, similar to [11, p. 49], we choose a = h3 (−1)n12 . Note that the factor group C G (u )◦ / R u (C G (u )◦ ) splits (see [12, Proposition 17.6]). So there is a subgroup M of type A 3 in C G (u ) such that C G (u )◦ = M  R u (C G (u )◦ ). In our case a choice for M is the subsystem subgroup with fundamental system {α2 , α4 , α22 }. One checks that a is a non-trivial graph automorphism (it interchanges α2 and α22 and fixes α4 ) so a represents a non-trivial element of A ⇒ A = ¯a . We have C Z (U u )◦ ( T u ) = u1 (x1 ) and one checks that it commutes with a so it is Z (C G (u ))◦ . Class A2 : A ∼ = Z 2 and u = u2 (1)u4 (1). Similar to [11, p. 49], we choose a = h2 (−1)u4 (1)n7 n13 . For p ∈ {2, 3} we have

C Z (U u )◦ ( T u ) = u2 (x1 )u4 (x1 )u6 (x2 ), a





C Z (U u )◦ ( T u ) = u2 (x1 )u4 (x1 )u6 x21 − x1 − x2 .

Since the action is non-trivial we have A = ¯a by Lemma 8.1. The above calculation shows that Z (C G (u ))◦ is obtained from the equation x21 − x1 − x2 = x2 . If p = 2 this is equivalent to x21 − x1 and, by the proof of Lemma 5.2, the connected component is obtained with x1 = 0 so Z (C G (u ))◦ = u6 (x2 ). If p = 3 we have x2 = 2(x21 − x1 ) so Z (C G (u ))◦ = u2 (x1 )u4 (x1 )u6 (2x21 + x1 ) in this case.

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˜ 2 : This component group is non-trivial only for p = 2 in which case A ∼ Class A = Z 2 and our representative is u = u3 (1)u1 (1). In this situation there is an extra graph automorphism interchanging long ˜ 2 and A 2 – the two cases are similar. Choosing and short roots and thus interchanging the classes A a = u3 (−1)n8 n10 ∈ C G (u ) we get C Z (U u )◦ ( T u ) = u3 (x1 )u1 (x1 )u5 (x2 ), a





C Z (U u )◦ ( T u ) = u3 (x1 )u1 (x1 )u5 x2 + x1 + x21 .

a acts non-trivially so A = ¯a by Lemma 8.1. Now Z (C G (u ))◦ is obtained from the equation x1 + x21 = 0. By the proof of Lemma 5.2, Z (C G (u ))◦ = u5 (x2 ). Class B2 : A ∼ = Z 2 and we have u = u4 (1)u3 (1). Similar to [11, p. 49], we choose a = h4 (−1)n9 . To see that A = ¯a , note that the factor group C G (u )◦ / R u (C G (u )◦ ) of type A 1 A 1 splits (see [12, Proposition 17.6]). So there is a subgroup M of type A 1 A 1 in C G (u ) such that C G (u )◦ = M  R u (C G (u )◦ ). In our case a choice for M is the subsystem subgroup with fundamental system {α15 , α24 }. One checks that a is a non-trivial graph automorphism (it interchanges α15 and α24 ) so a represents a non-trivial element of A ⇒ A = ¯a . If p ∈ {2, 3} then C Z (U u )◦ ( T u ) = u3 (2x1 )u4 (2x1 )u7 (2x21 + x1 )u10 (x2 ) and one checks that it commutes with a so it is Z (C G (u ))◦ . Note that for p = 2 we also have A = u¯ . Class C3 (a1 ): A ∼ = Z 2 if p = 2 (else it is trivial) and a unipotent class representative is u = u4 (1)u1 (1)u10 (1). Similar to [11, p. 50], we choose a = h1 (−1). A maximal torus of C G (u ) is 1dimensional (see [12, Table 22.1.4]) and one checks that im h4 is such a maximal torus. So a is a semisimple element which centralizes a maximal torus of C G (u ) to which it does not belong ⇒ A = ¯a by Lemma 8.2. If p = 3 we have C Z (U u )◦ ( T u ) = u1 (x1 )u4 (x1 )u10 (x1 )u12 (x1 − x21 )u15 (x2 ) and one checks that it commutes with a and therefore it equals Z (C G (u ))◦ . Class F4 (a3 ): The component group A is S 4 if p = 2 and S 3 otherwise. A representative is u = u4 (1)u6 (1)u10 (1)u18 (1). It is distinguished so C Z (U u )◦ ( T u ) = Z (U u )◦ . From [12, p. 182] we know that u is subregular in H , a D 4 subsystem subgroup of G. In our case a fundamental system for H is given by the set of roots {α4 , α2 , α10 , α15 }. The component group A is represented by the center of H (trivial for p = 2) and its graph automorphisms (see [12, Lemma 13.9]). Since for p = 3







Z (U u )◦ = u4 (x1 )u6 (x1 )u10 (x1 )u16 x21 − x1 u18 (x1 )u20 x1 − x21



  · u22 x1 − x21 u23 (x2 )u24 (x3 )

is in fact a subgroup of H , it commutes with Z ( H ). Hence we only need to consider the graph automorphisms. The subgroup of graph automorphisms S 3 is generated by n3 and n5 . This is so because conjugation by n3 fixes U 2 and U 15 and interchanges U 4 and U 10 while conjugation by n5 fixes U 2 and U 10 and interchanges U 4 and U 15 :

In order to obtain A we modify the above elements by elements of H to obtain elements which commute with u. We choose

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 −1

a1 = h2 (−1)h3 (i )h4 (−1) u2 (−1)u6 (1)



n3 ,

 −1

a2 = h1 (i )h2 (−1)h3 (i ) u−2 (−1)u2 (1)u18 (1)

n5 n1 .

A computation for p = 3 gives a1







Z (U u )◦ = u4 (x1 )u6 (x1 )u10 (x1 )u16 2x1 + x21 u18 (x1 )u20 x1 + 2x21



    · u22 x1 + 2x21 u23 (x2 )u24 2x3 + 2x2 + 2x1 + x21 ,     a2 Z (U u )◦ = u4 (x1 )u6 (x1 )u10 (x1 )u16 2x1 + x21 u18 (x1 )u20 x1 + 2x21       · u22 x1 + 2x21 u23 2x3 + 2x2 + 2x21 + x31 u24 x2 + 2x1 + 2x21 + 2x31 . Since the two elements act non-trivially, by Lemma 8.1 they represent non-trivial elements of A. If m is the maximal ideal at 1 ∈ Z (C G (u )◦ ) ∼ = A3 then a1 and a2 induce k-linear automorphisms on the cotangent space m/m2 . Choose vectors v 1 = x1 + x3 , v 2 = x2 and v 3 = −x2 − x3 in m/m2 and note that a1 and a2 permute these vectors as (1, 3) and (1, 2, 3) respectively. We conclude that A = ¯a1 , a¯ 2 . An element of Z (U u )◦ commutes with a1 and a2 if and only if the following system is satisfied:

⎧ 2 ⎪ ⎨ x3 = x1 − x1 − x2 − x3 x2 = x31 − x21 − x2 − x3 ⎪ ⎩ x3 = −x31 − x21 − x1 + x2 , and we find that the equations reduce to x1 = x31 and x2 − x3 = x21 − x1 . Using the proof of Lemma 5.2 the connected component Z (C G (u ))◦ is obtained with x1 = 0 and x3 = x2 so Z (C G (u ))◦ = u23 (x2 )u24 (x2 ). Computing for p = 2 we get

Z (U u )◦ = u16 (x1 )u17 (x2 )u19 (x3 )u20 (x1 )u21 (x4 )u22 (x1 )u23 (x5 )u24 (x6 ), a1

Z (U u )◦ = u16 (x1 )u17 (x3 )u19 (x2 )u20 (x1 )u21 (x4 )u22 (x1 )u23 (x5 )u24 (x6 + x5 + x1 ),

a2

Z (U u )◦ = u16 (x1 )u17 (x3 )u19 (x4 )u20 (x1 )u21 (x2 )u22 (x1 )u23 (x6 + x1 )u24 (x6 + x5 + x1 ).

To see that A = ¯a1 , a¯ 2 ∼ = S 3 note that the elements act non-trivially (use Lemma 8.1) and look at the action on the vectors x2 , x3 and x4 in the cotangent space m/m2 at 1. Now an element of Z (U u )◦ is in Z (C G (u ))◦ if and only if x4 = x3 = x2 , x5 = x1 and x6 = 0 thus



◦

Z C G (u )

= u16 (x1 )u17 (x2 )u19 (x2 )u20 (x1 )u21 (x2 )u22 (x1 )u23 (x1 ).

Class F4 (a2 ): For p = 3 the component group is A ∼ = Z 2 and our unipotent class representative u = u1 (1)u4 (1)u9 (1)u10 (1) is regular in a C 3 A 1 subsystem subgroup H of G as explained in [12, Lemma 13.8]. Since u is distinguished we have C Z (U u )◦ ( T u ) = Z (U u )◦ . We choose a = h4 (−1) ∈ Z ( H ) and a calculation gives

Z (U u )◦ = u15 (x1 )u19 (x1 )u21 (x1 )u23 (x2 )u24 (x3 ), a

Z (U u )◦ = u15 (x1 )u19 (x1 )u21 (x1 )u23 (−x2 )u24 (x3 ).

By Lemma 8.1 we have A = ¯a so Z (C G (u ))◦ = u15 (x1 )u19 (x1 )u21 (x1 )u24 (x3 ).

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For p = 2, by [12, p. 309], the component group is A ∼ = Dih8 , the dihedral group of order 8. A representative is u = u1 (1)u5 (1)u6 (1)u10 (1). Calculating we obtain

U u = u4 (x2 )u6 (x2 )u7 (x2 )u8 (x1 )u9 (x2 )u10 (x2 )u11 (x5 )u12 (x5 )u13 (x2 )

    · u14 (x3 )u15 (x5 )u16 x5 + x22 u17 (x4 )u18 (x5 )u19 (x5 + x4 )u20 x25   · u21 (x6 )u22 x5 + x25 u23 (x7 )u24 (x8 ).

When computing U u = C G (u )◦ with the method described in Section 5 one finds that the ideal I of the fiber [u , _]−1 (024 ) contains x1 + x21 and x5 + x25 . Using the proof of Lemma 5.2 we chose U u = V( I , x1 , x5 ). So x1 and x5 are idempotents in k[U ]/ I . All other components of C U (u ) are a1 = V( I , x1 , x5 − 1), a2 = V( I , x1 − 1, x5 ) and a3 = V( I , x1 − 1, x5 − 1), explicitly they are

a1 = u4 (x1 )u5 (1)u6 (x1 )u7 (x1 )u8 (x2 )u9 (1 + x1 )u10 (x1 )u11 (x3 )u12 (x3 )

  · u13 (1 + x1 )u14 (x4 )u15 (x3 + x1 )u16 x3 + x1 + x21 u17 (x5 )u18 (x3 )     · u19 (x5 + x3 )u20 x23 + x21 u21 (x6 )u22 x3 + x1 + x23 + x21 u23 (x7 )u24 (x8 ),

a2 = u1 (1)u4 (x1 )u6 (1 + x1 )u7 (x1 )u8 (x2 )u9 (1 + x1 )u10 (1 + x1 )u11 (x3 )

  · u12 (1 + x3 )u13 (1 + x1 )u14 (x4 )u15 (1 + x3 )u16 1 + x3 + x21 u17 (x5 )     · u18 (x3 )u19 (x5 + x3 )u20 1 + x1 + x23 u21 (x6 )u22 1 + x3 + x1 + x23

· u23 (x7 )u24 (x8 ), a3 = u1 (1)u4 (x1 )u5 (1)u6 (1 + x1 )u7 (x1 )u8 (x2 )u9 (x1 )u10 (1 + x1 )u11 (x3 )

  · u12 (x3 )u13 (x1 )u14 (x4 )u15 (x3 + x1 )u16 x3 + x1 + x21 u17 (x5 )u18 (x3 )     · u19 (x5 + x3 )u20 x1 + x23 + x21 u21 (x6 )u22 x3 + x23 + x21 u23 (x7 )u24 (x8 ).

In addition, as explained in [12, p. 310], we choose a4 = [u1 (1)u10 (1)]−1 n2 n3 . Below we give the action of a4 on Z (C G (u )◦ ) and since it is non-trivial a4 is a non-trivial representative of A by Lemma 8.1. To see that a4 represents a different component than the above ones note that a1 , a2 , a3 ⊆ U and that the Bruhat decomposition of a4 implies that a4 C G (u )◦ = a4 C U (u )◦  U . Calculations show that a21 = a22 = a23 = a24 = 1 mod C G (u )◦ and that a¯ 4 a¯ 1 = a¯ 1 a¯ 3 . In particular A is at least a quotient of Dih8 for a¯ 21 = a¯ 24 = (¯a1 a¯ 4 )4 = 1. Since we have 6 distinct elements it follows that a¯ 1 , a¯ 4 generate a Dih8 so they generate A. Now C Z (U u )◦ ( T u ) = Z (C U (u )◦ )◦ and we have



Z (U u )◦ = u17 (x1 )u19 (x1 )u21 (x2 )u23 (x3 )u24 (x4 ), Z (U u )◦

a4



a1 ◦

Z (U u )



= u17 (x1 )u19 (x1 )u21 (x2 + x1 )u23 (x3 )u24 (x4 ), = u17 (x1 )u19 (x1 )u21 (x2 + x1 )u23 (x4 )u24 (x3 ),

so Z (C G (u ))◦ is obtained with x1 = 0 and x3 = x4 . Class F4 (a1 ): The representative is u = u2 (1)u4 (1)u7 (1)u5 (1). If p = 2 then A is Z 2 ∼ = u¯ and there is nothing to check. For p = 3 we have A ∼ = Z 2 and we choose a = u3 (1)h3 (−1)h1 (−1) ∈ C G (u ) as / C G (u )◦ = C U (u )◦ as a ∈ / U we have A = ¯a . One checks that indicated in [12, pp. 181–182]. Since a ∈ a commutes with C Z (U u )◦ ( T u ) = Z (U u )◦ which is thus Z (C G (u ))◦ .

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Classes F4 , B3 , C3 : These classes have non-trivial A only in characteristic 2. In fact u ∈ / C G (u )◦ (this was established in [12]) and A is cyclic (see [12, Table 22.1.4]). Calculations show that with the rep˜ U (u˜ )◦ has the right order in C U (u˜ )/C U (u˜ )◦ so A = u¯ . For the F 4 class resentative u˜ in Section 4 uC A∼ = Z 4 and for the classes B 3 and C 3 we have A ∼ = Z 2 . Note that, as suggested by the extra graph automorphism for p = 2, the classes B 3 and C 3 are similar. 8.4. Type E6 Class A2 : The component group A is Z 2 and a representative for the unipotent class is u = u1 (1)u3 (1). We choose a = h3 (−1)u3 (−1)n15 n13 n26 in C G (u ) similar to [11, p. 51]. For p ∈ {2, 3} we have

C Z (U u )◦ ( T u ) = u1 (x1 )u3 (x1 )u7 (x2 ), a





C Z (U u )◦ ( T u ) = u1 (x1 )u3 (x1 )u7 x21 − x1 − x2 .

By Lemma 8.1, a is a non-trivial element of A and therefore A = ¯a . Now x21 − x1 − x2 = x2 shows that if p = 2 then x1 − x21 = 0 and by Lemma 5.2 the connected component is obtained with x1 = 0, i.e. Z (C G (u ))◦ = u7 (x2 ). If p = 3 then, by Theorem A, Z (C G (u ))◦ is obtained with x2 = x1 − x21 . Class D4 (a1 ): A ∼ = S 3 and the representative is u = u2 (1)u3 (1)u8 (1)u10 (1), a subregular element in a D 4 subsystem subgroup H  G. In our case a fundamental system for H is given by the set of roots {α2 , α3 , α4 , α5 }. In good characteristic A is generated by graph automorphisms of H .

We choose



a1 = h1 (−1)h2 (−1)h4 (−1)h6 (−1)[u3 (−1)u5 (1)]−1 n4 n32 n33 a2 = h2 (−1)h4 (−1)[u3 (1)u4 (1)u5 (−1)u9 (1)u10 (−1)]−1 n20 n21

∈ C G (u ).

A calculation for p = 3 gives







C Z (U u )◦ ( T u ) = u2 (x1 )u3 (x1 )u8 (x1 )u10 (x1 )u13 x1 + 2x21 u14 x1 + 2x21



  · u15 x1 + 2x21 u19 (x2 )u24 (x3 ),     a1 C Z (U u )◦ ( T u ) = u2 (x1 )u3 (x1 )u8 (x1 )u10 (x1 )u13 x1 + 2x21 u14 x1 + 2x21       · u15 x1 + 2x21 u19 2x3 + 2x21 + x31 u24 2x2 + 2x21 + x31 ,     a2 C Z (U u )◦ ( T u ) = u2 (x1 )u3 (x1 )u8 (x1 )u10 (x1 )u13 x1 + 2x21 u14 x1 + 2x21     · u15 x1 + 2x21 u19 (x2 )u24 2x3 + x2 + x1 + x21 + x31 . Since the two elements act non-trivially, by Lemma 8.1 they represent non-trivial elements of A. If m is the maximal ideal at 1 ∈ Z (C G (u )◦ ) ∼ = A3 then a1 and a2 induce k-linear automorphisms on the cotangent space m/m2 . Let L 1 = x3 − x2 k , L 2 = x2 + x1 k and L 3 = x1 − x3 k be 1-dimensional

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subspaces in m/m2 spanned by the indicated vectors. One checks that a1 and a2 permute these lines as (2, 3) and (1, 3) respectively ⇒ A = ¯a1 , a¯ 2 . Now the centralizer of {a1 , a2 } in C Z (U u )◦ ( T u ) is defined by the equations x2 = x31 − x21 − x3 and x3 = x31 + x21 + x1 + x2 − x3 . Since p = 3 these equations imply x1 − x31 = 0, thus (using the proof of Lemma 5.2) the connected component of Z (C G (u )) is obtained with x1 = 0 ⇒ x2 = −x3 . For p = 2 we have

C Z (U u )◦ ( T u ) = u13 (x1 )u15 (x1 )u14 (x1 )u19 (x2 )u24 (x3 ), a1

C Z (U u )◦ ( T u ) = u13 (x1 )u14 (x1 )u15 (x1 )u19 (x3 )u24 (x2 ),

a2

C Z (U u )◦ ( T u ) = u13 (x1 )u14 (x1 )u15 (x1 )u19 (x2 )u24 (x3 + x2 + x1 ).

The action of the two elements being non-trivial, it follows by Lemma 8.1 that a1 and a2 represent non-trivial elements of A. As above, if m is the maximal ideal at 1 ∈ Z (C G (u )◦ ) ∼ = A3 then a1 and a2 induce k-linear automorphisms on the cotangent space m/m2 . Choose v 1 = x2 , v 2 = x3 and v 3 = x1 + x2 + x3 in m/m2 and note that a1 and a2 permute these vectors as (1, 2) and (2, 3) respectively. We conclude that A = ¯a1 , a¯ 2 . The calculation shows that the centralizer of {a1 , a2 } in C Z (U u )◦ ( T u ) is defined by the equations x 1 = x2 = x3 . Class E6 (a3 ): A ∼ = Z 2 and a representative of the unipotent class is

u=

u1 (1)u8 (1)u9 (1)u11 (1)u14 (1)u19 (1) u13 (1)u1 (1)u15 (1)u6 (1)u14 (1)u4 (1)

( p = 2) ( p > 2).

Since T u = 1 we have C Z (U u )◦ ( T u ) = Z (U u )◦ . If p > 2 then u is regular in an A 5 A 1 subsystem subgroup H (see [12, Lemma 13.8]) the center of which is a non-trivial element of A [12, Lemma 13.9]. Similar to [11, p. 56], we can choose a = h4 (−1) ∈ Z ( H ). Another way of seeing that A = ¯a is by using Lemma 8.1. For p = 3 we have

Z (U u )◦ = u23 (x1 )u29 (x1 )u31 (2x1 )u32 (x2 )u33 (2x2 )u35 (x3 )u36 (x4 ) and a calculation shows that a Z (U u )◦ has the same expression except for u35 where we have 2x3 . Therefore Z (C G (u ))◦ is obtained with x3 = 0. For p = 2, we choose a = u−3 (1)u−2 (1)u5 (1)u6 (1)u7 (1)u8 (1)u13 (1)u14 (1)· u19 (1)u20 (1)u25 (1) ∈ C G (u ) (similar to [12, p. 193]). In order to see that A = ¯a we use Lemma 8.1. Since

Z (U u )◦ = u12 (x1 )u20 (x1 )u21 (x1 )u22 (x1 )u24 (x1 )u28 (x1 )u29 (x1 )u30 (x2 )



Z ( U u )◦

a

    · u31 (x1 )u33 x1 + x21 u34 x2 + x21 u35 (x3 )u36 (x4 ),

= u12 (x1 )u20 (x1 )u21 (x1 )u22 (x1 )u24 (x1 )u28 (x1 )u29 (x1 )u30 (x2 )       · u31 (x1 )u33 x1 + x21 u34 x2 + x21 u35 x4 + x3 + x21 u36 (x4 )

it follows that an element of Z (U u )◦ is in Z (C G (u ))◦ if and only if x4 = x21 .

/ C G (u )◦ Classes E6 , D5 , D4 : These classes have non-trivial A only when A = u¯ ∼ = Z p , i.e. only if u ∈ which happens if and only if p = 2 in all three cases and for p = 3 in the case of E 6 (see [12, Table 22.1.3]). In all these situations Z (C G (u ))◦ = C Z (U u )◦ ( T u ) by Theorem A.

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Table 7 Center of centralizer G 2 . Class

dim U u

A

dim Z (C ◦ )

dim Z (C )

2

3

G2 G 2 (a1 )

2 4

Z (6, p ) S3 : Z2

2 2:4

2 1:2

◦ ∗∗

◦

A1 ˜1 A A1

6 4:6 6

1 1 1

2 1 1

2 1 1

–

˜ (3)

u∈ / Z (C )◦

Table 8 Center of centralizer F 4 . Class

F4 F 4 (a1 ) F 4 (a2 ) C3 B3 F 4 (a3 ) C 3 (a1 )(2) C 3 (a1 )

dim U u

dim Z (C ◦ )

A

dim Z (C )

4 6 8 8 8 12 14 14 : 12

Z (12, p 2 ) Z2 Dih8 : Z 2 Z2 : 1 Z2 : 1 S3 : S4 1 1 : Z2

4 4:3 4:3 3:2 3:2 6:3 4 3:2

4 4:3 2 3:2 3:2 2:1 4 3:2

16 16 : 14

1 1

2 1

2 1

B2 B2 ˜1 A2 A ˜2 A A2 ˜1 A1 A

14 14 : 12 16 17 : 14 17 20

1 Z2 1 Z2 : 1 Z2 1

(2)

3 2 1 2:1 2 2:1

3 2 1 1 1 2:1

24 24 : 21 24

1 1 : Z2 1

2 1 1

2 1 1

˜2A A 1 ˜ 2 A1 A

(2)

(2)

˜ A 1 ˜1 A A1

u∈ / Z (C )◦ 2

3

◦ ◦ ◦ ◦ ◦ ∗ ∗ ∗ ∗ ∗ ∗ ◦ ∗ ∗∗ ∗∗

◦ ∗ ∗ ∗ ∗ ∗∗ – – –

–

Table 9 Center of centralizer E 6 . Class

E6 E 6 (a1 ) D5 E 6 (a3 ) D 5 (a1 ) A5 A4 A1 D4 A4 D 4 (a1 ) A3 A1 A 22 A 1 A3 A 2 A 21 A 22 A2 A1 A2 A 31 A 21 A1

dim U u

A

dim Z (C ◦ )

dim Z (C )

6 8 9 12 13 12 15 13 15 18 19 22 19 25 22 26 26 31

Z (6, p ) 1 Z (2, p ) Z2 1 1 1 Z (2, p ) 1 S3 1 1 1 1 1 1 Z2 1

6 5 4 4 3 3 2 2 3 3 2 1 2 1 2 2 2 1

6 5 4 3 3 3 2 2 3 1 2 1 2 1 2 2 1 1

33 36

1 1

1 1

1 1

u∈ / Z (C )◦ 2

3

◦ ∗ ◦ ∗ ∗ ∗ ∗ ◦ ∗ ∗ ∗ ∗ ∗ ∗

◦ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗∗

∗∗

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9. Tables Here we give the results of our calculations in Tables 7, 8 and 9. The component groups are given in the third column. They were determined by several authors. The most recent description was given in [12]. For a fixed unipotent class representative u selected in Section 3 we denote by C its centralizer C G (u ) and by U u the unipotent radical of a Borel subgroup of C . In the second column we give the dimensions of U u . These can be deduced from the tables in [12, Section 22]. In the fourth and fifth column we give the dimension of Z (C ◦ ) and that of the center of C respectively. In the sixth column we mark with ◦ those cases where u ∈ / C G (u )◦ , with ∗ those cases where u ∈ C G (u )◦ \ Z (C G (u )◦ )◦ and with ∗∗ those cases where u ∈ Z (C G (u )◦ )◦ \ Z (C G (u ))◦ . Since the entries in the columns depend on the characteristic p we separate them by a colon whenever they are distinct. An entry of the form A : B means ‘read A if p = 2 and B for p = 3’. 10. Notation In this section we list the notation that we use in the text:

αn is the n-th root in the ordering given below. We will write n for αn and −n for −αn . uα is a fixed isomorphism from k to the root group U α . sα is the reflection corresponding to the root α . nα is the Weyl group representative uα (1)u−α (−1)uα (1) of sα . hα is the isomorphism from k× to the 1-dimensional torus corresponding to α given in [2, Lemma 6.4.4]. We sometimes write hα for its image.   uα (xα ) stands for the set { uα (c α ): c α ∈ k}. The following are standard algebraic geometry notations (see [25, Chapter 1]):

O( X ) is the ring of regular functions of the algebraic variety X . V( I ) for an ideal I in some polynomial ring, is the zero locus of the ideal. I( V ) for a subset V of An is the ideal of polynomials vanishing on V . m denotes a maximal ideal, for example m0n is the ideal at 0n ∈ An (k). T p V for a subvariety V of An is the tangent space at the point p. For algebraic groups we use

L(G ) is the Lie algebra of G. Ad is the adjoint representation of an algebraic group. B u = T u  U u for a unipotent element u ∈ G, denotes a Borel subgroup of C G (u ) with maximal torus T u and unipotent radical U u . g H = g H g −1 and H g = g −1 H g for g ∈ G and H a subset of G. For the finite groups appearing as component groups we use Z n denotes the cyclic group of order n. S n the symmetric group on n elements. Dihn the dihedral group of order n. i and ω are roots of x2 + 1 and x2 + x + 1 respectively. The chosen numbering is the one used in [5]. We use the Bourbaki notation to indicate it. For G 2 the ordering is α1 = 10, α2 = 01, α3 = 11, α4 = 21, α5 = 31, α6 = 32. For F 4 and E 6 the ordering is given in Figs. 2 and 3.

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Fig. 2. Positive roots for F 4 .

Fig. 3. Positive roots for E 6 .

Acknowledgments I would like to express my thanks to Prof. Donna Testerman for careful reading of the manuscript. Her remarks and comments led to considerable improvements. I would also like to thank the referee for suggesting Corollary F as well as other improvements to the first version.

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