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ON THE CONJUGACY CLASSES IN MAXIMAL UNIPOTENT SUBGROUPS OF SIMPLE ALGEBRAIC GROUPS SIMON M. GOODWIN Abstract. Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U . Using relative Springer isomorphisms we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0 assume G is defined and split over the finite field of p elements Fp . Let q be a power of p and let G(q) be the finite group of Fq -rational points of G. Let F be the Frobenius morphism such that G(q) = GF . Assume B is F -stable, so that U is also F -stable and U (q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U (q) are in correspondence with the F -stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U (q).

Introduction Let G be a simple algebraic group over the algebraically closed field k. Let B be a Borel subgroup of G and let T ⊆ B be a maximal torus of G. We write U for the unipotent radical of B and u for the Lie algebra of U . In this paper we are interested in the adjoint orbits of U in u. We study the adjoint U -orbits by choosing an enumeration β1 , . . . , βN of the positive roots of G relative to T such that ht(βi ) ≤ ht(βi+1 ) for i = 1, . . . , N − 1. Then we define B-submodules mi of u P by mi = N j=i+1 gβj , where gβj denotes the root subspace of g = Lie G corresponding to βj . We consider the action of U on successive quotients ui = u/mi . Using relative Springer isomorphisms (as discussed below) we show that one can determine the orbits of U in ui from the orbits of U in ui−1 . In particular, we show that any adjoint orbit contains a unique so-called minimal representative. For our analysis of the adjoint orbits of U we require the existence of a B-equivariant isomorphism of varieties U/M → u/m for certain unipotent normal subgroups M of B. We define relative Springer isomorphisms which are such isomorphisms that are induced from Springer isomorphisms, i.e. G-equivariant isomorphisms of varieties U → N , where U denotes the unipotent variety of G and N the nilpotent variety of g. In Theorem 3.9 we show that relative Springer isomorphisms exist for so-called NT-subgroups of U . Suppose char k = p > 0 and assume that G is defined and split over the finite field with p elements Fp . For a power q of p, we write G(q) for the finite group of Fq -rational points of G. We write F for the Frobenius morphism such that G(q) = GF = {g ∈ G : F (g) = g}. Assume that B is F -stable, so that U is F -stable and U (q) is a Sylow p-subgroup of G. Date: April 1, 2005. 2000 Mathematics Subject Classification. 20D20, 20G40. Accepted for publication in Transformation Groups. 1

In this situation we show that the conjugacy classes of U (q) are in correspondence with the F -stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U (q) from our results on the adjoint orbits of U . There has been a lot of interest in the conjugacy classes of the unitriangular group Un (q) = {(gij ) ∈ GLn (q) : gij = 0 for i > j and gii = 1}, where GLn (q) is the group of non-singular n × n matrices over Fq . Both G. Higman and J. Thompson have been interested in the number, k(Un (q)), of conjugacy classes of Un (q). For instance, see the paper of Higman [14] and the preprint of Thompson [33]. In particular, it is conjectured that k(Un (q)) is a polynomial in q with integer coefficients. The conjugacy classes of Un (q) have also been considered by A. Vera-L´opez and J.M. Arregi, see [35] and the references therein. In particular, they have shown that k(Un (q)) is a polynomial in q with integer coefficients for n ≤ 13. Some of the results of this paper are generalizations of results of Vera-L´opez and Arregi to arbitrary simple algebraic groups. G.R. Robinson also considered the conjugacy classes of Un (q) and certain subgroups of Un (q) in [23]. The main result in loc. cit. implies that the zeta-function   X k(Un (ps )) ζUn (t) = exp  ts  s s∈Z ≥0

(in C[[t]]) is a rational function in t whose numerator and denominator may be assumed to be elements of 1 + tZ[t]. This implies that once k(Un (ps )) is known for a certain finite number of values of s, it can be calculated for all s. Further, I.M. Isaacs and D. Karagueuzian considered the conjugacy classes and irreducible complex characters of Un (q). In [16] they showed that not all elements of Un (2) are conjugate to their inverses, implying that not all characters of Un (2) are real valued. They also discussed analogous phenomena for odd primes. Studying the conjugacy classes of U (q) generalizes the study of the conjugacy classes of Un (q). In case G = SLn (k) and F (gij ) = (gijq ), we have U (q) = Un (q). For example, one could make an analogue of the above conjecture, that k(Un (q)) is a polynomial in q, for an arbitrary simple algebraic group. The problem of determining the B-orbits (and U -orbits) in u has already been addressed by H. B¨ urgstein and W.H. Hesselink in [7]. They were motivated by the problem of describing the component configuration of the variety BX = {B ∈ B : X ∈ Lie B} where B denotes the variety of Borel subgroups of G and X ∈ g is nilpotent, see 1.4 of loc. cit. This variety seems important for the representation theory of the Weyl group of G and its associated Hecke algebra, see [18, 6.3]. In [7] B¨ urgstein and Hesselink devised an algorithm for calculating the B-orbits in u and used it for G of rank at most 4 and for G of type A5 , A6 , A7 and D5 . Their algorithm does not always give a complete classification of the B-orbits in u. The methods of this paper could be used to write a computer program to calculate all the U orbits and B-orbits in u. In [7] the coadjoint orbits of B (and U ) in u∗ were also considered; the algorithm of B¨ urgstein and Hesselink also worked in this situation. The author plans to consider the coadjoint orbits of U in u∗ in future work. There has been other interest in the coadjoint orbits of B in u∗ . For example, C. Andr´e has considered the coadjoint orbits of Un (q) and their relationship with the complex characters of Un (q), see the references in the recent paper [1]. It is known that, if char k is not less than 2

the Coxeter number of G, then the irreducible characters of U (q) are in correspondence with the F -stable coadjoint orbits of U in u∗ and there is a method for calculating a character from a given coadjoint orbit; this is the version of Kirillov’s method over finite fields, see [19] and [17]. In [10], the author considered the coadjoint action of B on u∗ and showed that B always acts on u∗ with a dense orbit. Further, there has been a lot of recent interest in the conjugacy classes of a parabolic subgroup P of G in its unipotent radical Pu . These conjugacy classes are in correspondence with the adjoint orbits of P in pu = Lie Pu and have been studied extensively. See the survey [24] and the references therein. For example, there is a classification of all instances when P acts on pu with a finite number of orbits. In [11], the author considered the question of when B acts on a B-submodule n of u with a dense orbit. By programming an algorithm in the computer algebra language GAP4, all instances when n is a prehomogeneous space for B for rank(G) ≤ 8 were determined. The methods of this paper are an extension of those in loc. cit. As a general reference for the theory of algebraic groups we cite the books of Borel [4] and Springer [29]. Acknowledgements I would like to thank G. R¨ohrle for reading through earlier versions of this paper and making many useful suggestions. I am also grateful to G. McNinch for two helpful discussions. I thank the referees for many useful suggestions. Lastly, I acknowledge the financial support of EPSRC. 1. Preliminaries Let R be a linear algebraic group over the algebraically closed field k of characteristic p ≥ 0. We write r = Lie R for the Lie algebra of R and we denote the identity component of R by R0 . Let V be an R-variety. For r ∈ R and v ∈ V , we write r · v for the image of v under r, R · v = {s · v : s ∈ R} for the R-orbit of v in V and CR (v) = {s ∈ R : s · v = v} for the stabilizer of v in R. The following well-known formula linking the dimensions of R · v and CR (v) follows from [4, Thm. AG.10.1]: (1.1)

dim R · v + dim CR (v) = dim R.

The subvarieties Vi (i ∈ Z≥0 ) of V are defined by Vi = {v ∈ V : dim R · v = i}. The irreducible components of the Vi are called the sheets of R on V . There are finitely many sheets of R on V and if R acts on V with finitely many orbits, then the sheets coincide with the orbits. For information about sheets we refer the reader to [5]. Now suppose V is a rational R-module. Then V is also a module for r. For Y ∈ r and v ∈ V , we write Y · v for the image of v under Y , r · v = {Y · v : Y ∈ r} and cr (v) = {Y ∈ r : Y · v = 0}. Let G be a simple algebraic group over k. The Lie algebra of G is denoted by g = Lie G; likewise for closed subgroups of G. In this paper we use lower case Roman letters for elements of G and upper case Roman letters for elements of g. We recall that G acts on itself by conjugation and on g via the adjoint action. For g, x ∈ G and X ∈ g we write 3

g · x = gxg −1 and g · X for the image of X under g in the adjoint action. The rank of G is denoted by r = rank(G) and we write h for the Coxeter number of G. Fix a maximal torus T of G and let Ψ be the root system of G with respect to T . We recall that a closed subgroup of G is said to be (T -)regular if it is normalized by T . For a regular subgroup H of G we write Ψ(H) = Ψ(h) ⊆ Ψ for the roots of H relative to T . For a root α ∈ Ψ we choose a parametrization uα : k → Uα of the root subgroup Uα , then eα = duα (1) is a generator for the corresponding root subspace gα of g. Let B ⊇ T be a Borel subgroup of G and let U be the unipotent radical of B. Then + Ψ = Ψ(B) = Ψ(U ) is a system of positive roots of Ψ; we write Π for the corresponding set of simple roots. P + PFor a root β = α∈Π aα eα ∈ Ψ we recall that the height of β is defined by ht(β) = α∈Π aα . P Let ρ = α∈Π bα α be the highest root in Ψ+ . We recall that p > 0 is said to be good for G provided p does not divide bα for any α. We shall assume throughout this paper that p is zero or good for G. The descending central series of U is defined as usual by U (0) = U and U (l+1) = (U (l) , U ) for l ≥ 0 where (U (l) , U ) denotes the subgroup of U generated by commutators (x, y) = xyx−1 y −1 with x ∈ U (l) and y ∈ U . By [2, Lem. 4], Y

U (l) =

Uβ ,

ht(β)≥l+1

so that we may identify U (l) /U (l+1) as a variety with Y

Uβ .

ht(β)=l+1

The descending central series of u is defined similarly and its terms have an analogous description. The unipotent variety of G is denoted by U and we write N for the nilpotent variety of g. In the following paragraphs we discuss the action of G on U and the action of B on U . There are analogous definitions and results for the action of G on N and the action of B on u. R. Richardson proved in [21] that U splits up into finitely many G-orbits. Since U is irreducible, it follows from general theory of algebraic groups that one of the G-orbits is open in U. This open orbit U r is called the regular unipotent orbit and x ∈ U r is called regular unipotent. It follows from the results in [22] that U r ∩ U is a single B-orbit which is open in U . For x ∈ U r we have that dim CG (x) = r and dim CB (x) = r, thus CG (x)0 ⊆ B. In fact CG (x)0 ⊆ U and Q if G is of adjoint type, then CG (x) = CU (x) is connected, see [30, III, 1.14]. An element x = β∈Ψ+ uβ (λβ ) ∈ U is regular if and only if λα 6= 0 for all α ∈ Π, see [31, §3] or [30, III, 1.13]. 4

2. Springer isomorphisms For char k zero or greater than 2h − 2 we have a logarithm map from U to N , i.e. we may identify U and N with their images under the respective adjoint representations of G and g and log : U → N can be defined formally by its power series expansion. This logarithm map and its inverse exp : N → U are inverse G-equivariant isomorphisms of varieties, see for example [20, §5.7]. For char k ≤ 2h − 2 the situation is not so straightforward. For char k ≥ h a “logarithm map” still exists. A statement of this result can be found in [25, Prop. 5.2] the proof is attributed to J.-P. Serre. In general, for char k zero or good for G, one has the following result which is a slightly strengthened version of a theorem of T.A. Springer (see [30, III, 3.12] and [3, Cor. 9.3.4]). Theorem 2.1. If G is of type A assume the covering map SLn (k) → G is separable. There exists a G-equivariant isomorphism φ : U → N . Such an isomorphism φ : U → N is called a Springer isomorphism and gives a substitute for the logarithm map for char k ≤ 2h − 2. A Springer isomorphism is far from unique; in fact the Springer isomorphisms form an algebraic variety of dimension r (see [26]). However, in loc. cit., J.-P. Serre showed that the induced bijection between the orbits of G in U and N is independent of the choice of Springer isomorphism. Suppose x ∈ U is regular unipotent and let φ be a Springer isomorphism. By Gequivariance of φ we have that φ(x) is a regular nilpotent element. Now CG (x)0 ⊆ B, ˜ for therefore, by G-equivariance of φ, we have that CG (φ(x))0 ⊆ B. Clearly, φ(x) ∈ Lie B ˜ of G. Then we have that CG (φ(x))0 ⊆ B. ˜ But by [32, 3.7] we some Borel subgroup B 0 ˜ Thus, know that CG (φ(x)) is contained in a unique Borel subgroup of G so that B = B. φ(x) ∈ u and so φ(B · x) = B · φ(x), i.e. φ(U ) = u. Therefore, we have the following standard corollary of Theorem 2.1. Corollary 2.2. There exists a B-equivariant isomorphism φ : U → u. Proof. The above discussion implies that a Springer isomorphism φ : U → N induces an isomorphism φ : U → u. We note that we can remove the assumption on G of type A because the covering map SLn (k) → G induces an isomorphism from Un (k) onto its image.  A map as in Corollary 2.2 is also referred to as a Springer isomorphism. This is justified because one can check that any B-equivariant isomorphism U → u is the restriction of a Springer isomorphism U → N . As mentioned above, the bijection induced between the G-orbits in U and N is independent of the choice of Springer isomorphism φ : U → N . In the appendix we discuss the situation for Springer isomorphisms of the form φ : U → u. 3. Relative Springer isomorphisms In this section we discuss the existence of a B-equivariant isomorphism U/M → u/m for a unipotent normal subgroup M of B. In Theorem 3.9 below we show that such a map exists for so-called NT-subgroups M (see Definition 3.1 below). To prove Theorem 3.9, we follow similar steps to those used by Springer to prove that Springer isomorphisms exist. 5

Definition 3.1. Pick an enumeration β1 , . . . , βN of Ψ+ such that ht(βi ) ≤ ht(βi+1 ) and define N Y Mi = Uβj . j=i+1

Then Mi is a unipotent normal subgroup of B and we have U = M0 ⊇ · · · ⊇ MN = {1}. A sequence of subgroups as above is called an NT-sequence of subgroups of U and a subgroup in such a sequence is called an NT-subgroup. Given an NT-sequence of subgroups as above we get a sequence of B-submodules of u, u = m0 ⊇ · · · ⊇ mN = {0}; and we have mi =

N M

gβj .

j=i+1

We also call such a sequence an NT-sequence and submodules in such a sequence NTsubmodules. In the case G = SLn (k) and U = Un (k), an N T -submodule “looks” close to being a triangle. The terminology NT was chosen to be an abbreviation for near triangular. Let x ∈ G be a regular unipotent element. We begin by showing that the connected centralizer CB/M (xM )0 is abelian when M is any unipotent normal subgroup of B. This is achieved in the same way as Springer shows that CG (x)0 is abelian in [28] and follows from the following proposition. Proposition 3.2. Let M be a unipotent normal subgroup of B and let y ∈ B. Then CB/M (yM ) contains an abelian subgroup of dimension r = rank(G). Proof. Since CG (T ) = T , by [4, 13.17, Cor. 2], thus CB (T ) = T . Therefore, by S G is simple, −1 [4, 11.10], b∈B bT b is dense in B, i.e. the semisimple elements of B are dense in B. Consider the natural map πM : B → B/M . It is clear that, if z ∈ B is semisimple, then zM ∈ B/M is semisimple. It follows that the semisimple elements of B/M are dense in B/M . Any semisimple element zM ∈ B/M lies in some maximal torus S of B/M . Therefore, CB/M (zM ) contains an abelian subgroup of dimension r – namely S. Now one can adapt Springer’s proof that CG (y) contains an abelian subgroup of dimension r from [28] (see also [15, Thm. 1.14]) to give the result.  Corollary 3.3. Let M be a unipotent normal subgroup of B and let x ∈ U be regular unipotent. Then CB/M (xM )0 is abelian and dim CB/M (xM ) = r. Proof. The action of B on U/M factors through B/M and if x ∈ U is regular, then B·(xM ) = (B/M ) · (xM ) is dense in U/M . Therefore, using (1.1) we see that CB/M (xM ) has dimension r. By Proposition 3.2, CB/M (xM ) has an abelian subgroup of dimension r. It follows that CB/M (xM )0 is abelian.  6

Next we show that if G is of adjoint type, then CB/M (xM ) is connected when x ∈ U is regular unipotent and M is an NT-subgroup of U . Our starting point is the following result of Springer from [27]. Proposition 3.4. Assume G is adjoint. Let x ∈ U be regular unipotent. Then CG (x) = CU (x) is connected. We also want the following two lemmas. Lemma 3.5. Let N be a normal subgroup of an algebraic group H. Let V be a H/N variety and v ∈ V . Then CH (v) is connected if and only if CH/N (v) is connected. Proof. Let πN : H → H/N be the natural map. It is clear that πN (CH (v)) = CH/N (v). Therefore, πN induces a bijection between the subgroups of CH (v) containing N and the subgroups of CH/N (v). It follows that |CH (v) : CH (v)0 | = |CH/N (v) : CH/N (v)0 | and the result follows.



Lemma 3.6. Let x ∈ U be regular unipotent. Then dim CU (l) (x) = |{γ ∈ Ψ+ : ht(γ) = l + 1}|. Proof. Using Corollary 2.2 we note that it suffices to show that dim CU (l) (X) = |{γ ∈ Ψ+ : ht(γ) = l + 1}| for X ∈ u regular nilpotent. We note that U (l) · X ⊆ {Y ∈ u : Y − X ∈ u(l+1) }. The variety on the right hand side of the above expression has dimension |{γ ∈ Ψ+ : ht(γ) ≥ l + 2}| and dim U (l) = |{γ ∈ Ψ+ : ht(γ) ≥ l + 1}|. Therefore, using (1.1) we see that dim CU (l) (X) ≥ |{γ ∈ Ψ+ : ht(γ) = l + 1}|. It follows from [4, Prop. 6.7] that dim cu(l) (X) ≥ dim CU (l) (X). Therefore, it suffices to show that dim cu(l) (X) = |{γ ∈ Ψ+ : ht(γ) = l + 1}|. This is the main result in [27, §2].  Proposition 3.7. Assume G is adjoint and let M be an NT-subgroup of U . (i) Let x ∈ U be regular unipotent. Then CB (xM ) is connected. (ii) Let X ∈ u be regular nilpotent. Then CB (X + m) is connected. Proof. We only prove (i) the proof of (ii)Qis similar. By [30, III, 1.13] we may assume x = α∈Π uα (1). Let U = M0 ⊇ · · · ⊇ MN = {1} be an NT-sequence of subgroups of U and suppose that M = Mi . We work by (reverse) induction on i to show that CB (xMi ) is connected, the base case i = N being Proposition 3.4. So suppose 0 ≤ i < N and CB (xMi+1 ) is connected, let β = βi+1 and suppose ht(β) = l. First assume that l ≥ 2. By considering the Chevalley commutator relations ([29, Prop. 8.2.3]) we see that the action of B on U/Mi factors through U (l−1) . Then by Lemma 3.5 we have that CB/U (l−1) (xMi+1 ) is connected. 7

The Chevalley commutator relations imply that   Y U (l−2) · x ⊆ y ∈ U : y = uα (1)  α∈Π

Y

uγ (λγ ) , λγ ∈ k

{γ:ht(γ)≥l}

 

.



We denote the variety on the right hand side of the above expression by Al ; it is a closed, irreducible subset of U . Clearly we have dim Al = |{γ ∈ Ψ+ : ht(γ) ≥ l}| and we also have dim U (l−2) = |{γ ∈ Ψ+ : ht(γ) ≥ l − 1}|. Therefore, by Lemma 3.6 and (1.1) we see that dim U (l−2) · x = dim Al . Since U (l−2) is a unipotent group, U (l−2) · x is closed in Al , by [4, Prop. 4.10] (see Lemma 4.1 below). Thus, the irreducibility of Al implies that U (l−2) · x = Al . Therefore, we may find v=

Y

uγ (µγ ) ∈ U (l−2)

{γ:ht(γ)=l−1}

such that v · xU (l) = xuβ (1)U (l) . For t ∈ k define v(t) =

Y

uγ (µγ t).

{γ:ht(γ)=l−1}

Since U

(l−2)

/U

(l−1)

is abelian, V = {v(t)U (l−1) : t ∈ k}

is a subgroup of U (l−2) /U (l−1) isomorphic to the additive group k. We note that the action of U (l−2) on U/U (l) factors through U (l−2) /U (l−1) . By considering the Chevalley commutator relations we see that V · xU (l) = {yU (l) : y = xuβ (λ), λ ∈ k}. Let yU (l−1) ∈ CB/U (l−1) (xMi ). Then y · xMi+1 = xuβ (µ)Mi+1 for some µ ∈ k. There exists w ∈ V such that w · xuβ (µ)Mi+1 = xMi+1 . So that wyU (l−1) ∈ CB/U (l−1) (xMi+1 ). Therefore, CB/U (l−1) (xMi ) ⊆ V CB/U (l−1) (xMi+1 ). Hence, as V, CB/U (l−1) (xMi+1 ) ⊆ CB/U (l−1) (xMi ) we see that CB/U (l−1) (xMi ) = V CB/U (l−1) (xMi+1 ). 8

is connected. Thus, by Lemma 3.5 CB (xMi ) is connected. In case ht(β) = 1 we can give an analogous argument where V is replaced by the onedimensional torus {yU ∈ B/U : y · uα (1)U = uα (1)U for all α ∈ Π \ {β}}.  The above results allow us to prove the next proposition which is key for the proof of Theorem 3.9. For the statement of Proposition 3.8 not to be vacuous we require that, for x ∈ U regular unipotent, there exists X ∈ Lie CU (x) which is regular nilpotent. One way to see this is to check that there exists X ∈ u regular nilpotent such that x · X = X (see [30, 3.11]), then use the separability of the orbit map U → U · x (which follows from the proof of Lemma 3.6). The argument in the proof of Proposition 3.8 is that of [30, III, 3.11]. Proposition 3.8. Assume G is adjoint. Suppose M is an NT-subgroup of U . Let x ∈ U be regular and let X ∈ Lie CB (x) be regular nilpotent. Then CB (xM ) = CB (X + m). Proof. Since CB/M (xM ) is abelian, by Corollary 3.3, Lemma 3.5 and Proposition 3.7, the adjoint action of CB (xM ) on its Lie algebra is trivial. Since X ∈ Lie CB (x), which implies X + m ∈ Lie CB/M (xM ), we get CB/M (xM ) ⊆ CB/M (X + m). So since CB (X + m) is connected by Proposition 3.7(ii) and dim CB/M (X + m) = dim CB/M (xM ) = r using (1.1) and that (B/M ) · (X + m) is dense in u/m, and Corollary 3.3, we have that CB/M (X + m) = CB/M (xM ). Hence, we get CB (X + m) = CB (xM ).



Using the fact that the map B · x → B · X (for x and X as in Proposition 3.8) extends to a B-equivariant isomorphism U → u, we can prove Theorem 3.9 where we remove the assumption that G is adjoint. Theorem 3.9. Suppose M is an NT-subgroup of U . Then there exists a B-equivariant isomorphism φ˜ : U/M → u/m. Proof. We begin by assuming that G is adjoint. Let x ∈ U be regular and let X ∈ Lie CB (x) be regular. The proof of the construction of Springer isomorphisms shows that CB (x) = CB (X) and implies that the isomorphism φ : B · x → B · X extends to an isomorphism φ : U → u . By Proposition 3.8, CB (xM ) = CB (X + m). So we have an isomorphism φ˜ : B · (xM ) → B · (X + m). We write Λ = k[U ] = k[Tβ : β ∈ Ψ+ ] for the coordinate ring of U . By an abuse of notation we also write Λ = k[u] for the coordinate ring of u. Then we have Λ0 = k[B · x] = Λ[Tα−1 : α ∈ Π] and by another abuse of notation we write Λ0 = k[B · X]. The isomorphism φ : B · x → B · X induces an isomorphism of k-algebras φ∗ : Λ0 → Λ0 . The fact that φ extends to φ : U → u means that φ∗ sends Λ onto Λ. We also have ΛM = k[U/M ] = k[Tβ : β ∈ Ψ+ \ Ψ(m)] 9

and ΛM = k[u/m]. Then Λ0M = k[B · (xM )] = ΛM [Tα−1 : α ∈ Π \ Ψ(m)] and Λ0M = k[B · (X + m)]. The isomorphism φ˜ : B · (xM ) → B · (X + m) induces an isomorphism of k-algebras φ˜∗ : Λ0M → Λ0M . We get the commutative diagram φ∗

Λ0 −−−→ x  i

Λ0 x  i

φ˜∗

Λ0M −−−→ Λ0M where i denotes the natural inclusion. Therefore, as φ∗ sends Λ onto Λ, it follows that φ˜∗ sends ΛM onto ΛM and therefore induces an isomorphism ΛM → ΛM . Hence, we see that φ˜ extends to an isomorphism φ˜ : U/M → u/m. We are left to show φ˜ is B-equivariant. This is easy, since for any b ∈ B the morphisms ˜ ˜ · yM ) are equal on B · xM . Thus, as B · xM is dense in yM 7→ b · φ(yM ) and yM 7→ φ(b U/M , the above morphisms are equal on U/M for any b ∈ B. ˆ be an isogeny where G ˆ is adjoint. Now suppose that G is not adjoint. Let π : G → G Then we see that the restriction of π to U induces isomorphisms from U onto Uˆ = π(U ) and ˆ . Similarly the derivative dπ : g → gˆ of π induces isomorphisms from u onto from M onto M ˆ →u ˆ = dπ(u) and from m onto m. ˆ Therefore, an isomorphism from Uˆ /M ˆ/m ˆ induces an u isomorphism U/M → u/m. One can check that this is B-equivariant.  We now define a relative Springer isomorphism to be a B-equivariant isomorphism as constructed in Theorem 3.9. We do not claim here that relative Springer isomorphisms always exist, see Remark 3.12 below. Definition 3.10. Let φ : U → u be a Springer isomorphism and let M be a unipotent normal subgroup of B. Then an isomorphism φ˜ : U/M → u/m such that the diagram φ

U −−−→  πM y

u  πm y

φ˜

U/M −−−→ u/m commutes is called a relative Springer isomorphism. In Section 2 we noted that the bijection between the orbits of G in U and N induced by a Springer isomorphism φ : U → N is independent of the choice of φ. In the appendix we consider the induced bijection between the B-orbits in U/M and u/m induced by a relative Springer isomorphism φ˜ : U/M → u/m. We show that the induced bijection between the ˜ sheets of B on U/M and u/m is independent of φ. Remark 3.11. We note that there are B-equivariant isomorphisms from U/M to u/m which are not relative Springer isomorphisms. For example, consider G = SL3 (k), B the upper triangular matrices in G and M = {(xij ) ∈ U : x12 = x23 = 0}. Then one can check that 10



   1 x12 ∗ 0 ax12 ∗ the map  0 1 x23  7→  0 0 bx23  , where a 6= b are non-zero elements of 0 0 1 0 0 0 k, is a B-equivariant isomorphism from U/M to u/m which is not induced by a Springer isomorphism. Remark 3.12. The author has recently proved that relative Springer isomorphisms exist for any unipotent normal subgroup M of B. The proof of this fact is rather technical so we choose not to include it here; a proof will appear in [12]. 4. Orbit maps and centralizers In this section we consider orbits and centralizers in the action of U on a quotient u/m of u, where M is an NT-subgroup of U . First we recall that the orbits in such an action are closed. Then we prove that the orbit maps are separable and the centralizers are connected. We begin by recalling [4, Prop. 4.10]. Lemma 4.1. Let R be a unipotent algebraic group acting morphically on a quasi-affine variety V . Then all orbits of R in V are closed. In Corollary 4.3 below we show that the orbit map U → U · (X + m) is separable for an NT-subgroup M of U and any X ∈ u. We require Proposition 4.2 for its proof. In the proofs of these two results we frequently use the equivalent conditions for an orbit map to be separable given by [4, Prop. 6.7]; we do not make this reference in the proofs. Proposition 4.2. Let M be a NT-subgroup of U and let x ∈ U . The orbit map U/M → (U/M ) · xM is separable. Proof. Let φ : U → u be a Springer isomorphism and let φ˜ : U/M → u/m be the corresponding relative Springer isomorphism. We begin by showing that φ˜−1 (cu/m (xM )) = CU/M (xM ). Let Y +m ∈ cu/m (xM ) then xM ∈ CU/M (Y +m) = CU/M (φ˜−1 (Y +m)) by U/M -equivariance ˜ Therefore, φ˜−1 (Y + m) ∈ CU/M (xM ), which implies the inclusion φ˜−1 (cu/m (xM )) ⊆ of φ. CU/M (xM ). A similar argument gives the reverse inclusion. In particular, dim cu/m (xM ) = dim CU/M (xM ) so that the orbit map U/M → (U/M ) · xM is separable.  Corollary 4.3. Let M be an NT-subgroup of U and let X ∈ u. The orbit map U → U · (X + m) is separable. Proof. Let φ : U → u be a Springer isomorphism and let φ˜ : U/M → u/m be the corresponding relative Springer isomorphism. The isomorphism φ˜ transforms the orbit map U/M → (U/M ) · φ˜−1 (X + m) to the orbit map U/M → (U/M ) · (X + m). Since the former map is separable so is the latter. Therefore, dim CU/M (X + m) = dim cu/m (X + m). It is clear that dim CU (X + m) = dim CU/M (X +m)+dim M and dim cu (X +m) = dim cu/m (X +m)+dim M . Thus dim CU (X + m) = dim cu (X + m) and the orbit map U → U · (X + m) is separable, as required.  Remark 4.4. Let φ : U → u be a Springer isomorphism, M an NT-subgroup of U and φ˜ : U/M → u/m the corresponding relative Springer isomorphism. For X ∈ u, arguments ˜ U/M (X + m)) and like those used in Proposition 4.2 show that we have cu/m (X + m) = φ(C therefore that cu (X+m) = φ(CU (X+m)). This would give an alternative proof of Proposition 4.5 below. 11

We now show that centralizers in the action of U on u/m are connected. The proof of the following proposition in similar to the proof of [30, III, 3.15]. Proposition 4.5. Let M be an NT-subgroup of U . (i) For y ∈ U , the centralizer CU (yM ) is connected. (ii) For Y ∈ u, the centralizer CU (Y + m) is connected. Proof. Let φ˜ : U/M → u/m be a relative Springer isomorphism given by Theorem 3.9. For zM ∈ U/M and t ∈ k we define ˜ (zM )t˜ = φ˜−1 (tφ(zM )). φ

If zM ∈ CU/M (yM ), then the B-equivariance (and therefore U/M -equivariance) of φ˜ implies that (zM )tφ˜ ∈ CU/M (yM ) for any t ∈ k. For t = 0 we have (zM )0φ˜ = M and for t = 1 we have (zM )1φ˜ = zM . So M, zM ∈ {(zM )tφ˜ : t ∈ k}, also {(zM )tφ˜ : t ∈ k} is isomorphic to k as an algebraic variety. Therefore, as k is connected, we see that zM ∈ CU/M (yM )0 and hence that CU/M (yM ) is connected. Thus by Lemma 3.5 we see that CU (yM ) is connected and we have proved (i). The second part of the proposition now follows from part (i) applied to φ˜−1 (Y +m), noting ˜ that CU (Y + m) = CU (φ˜−1 (Y + m)) by B-equivariance of φ.  5. The adjoint orbits of U in u In this section we consider the action of U on u. We note that some of the definitions and results in this section generalize definitions and results of A. V´era-Lopez and J.M. Arregi from [34]; we include all proofs for completeness. We fix an NT-sequence of B-submodules of u: u = m0 ⊇ · · · ⊇ mN = {0}. Then we consider the action of U on successive quotients ui = u/mi . Suppose X ∈ u and consider the set X + keβi + mi = {X + λeβi + mi : λ ∈ k} ⊆ ui . We consider which elements of X + keβi + mi are U -conjugate in ui . We have the following lemma (which generalizes [34, Lem. 3.1]), the existence of relative Springer isomorphisms and therefore Proposition 4.5 are crucial to its proof. Lemma 5.1. Let X + mi−1 ∈ ui−1 . Either (i) all elements of X + keβi + mi are U -conjugate or (ii) no two elements of X + keβi + mi are U -conjugate. Proof. Let λ ∈ k, and consider the orbit map ψλ : CU (X + mi−1 ) → CU (X + mi−1 ) · (X + λeβi + mi ) ⊆ X + keβi + mi . By Proposition 4.5, CU (X + mi−1 ) is connected, thus the image of ψλ is connected. Since CU (X + mi−1 ) ⊆ U is unipotent, the image of ψλ is closed by Lemma 4.1. Therefore, since X + keβi + mi is isomorphic to k as an algebraic variety, we have that im ψλ is equal to either {X + λeβi } or X + keβi + mi .  The definition below generalizes the definition after [34, Lem. 3.4]. 12

Definition 5.2. Let X ∈ u. (i) If Lemma 5.1(i) holds, then we say i is an inert point of X. (ii) If Lemma 5.1(ii) holds, then we say i is a ramification point of X. We now define a partial order ≤i (i = 1, . . . , N ) on ui , cf. [34, §2]. P P Definition 5.3. Let X, Y ∈ u and let X+mi = ij=1 aj eβj +mi and Y +mi = ij=1 bj eβj +mi . Then X + mi 0 i and U · (Z + mi−1 ) contains a unique ≤i−1 -minimal element. Let X = j=1 aj eβj and Pi Y = j=1 bj eβj be such that X + mi and Y + mi are ≤i -minimal elements of U · (Z + mi ). Then X + mi−1 and Y + mi−1 are ≤i−1 -minimal elements of U · (Z + mi−1 ). Therefore, by induction X + mi−1 = Y + mi−1 is the unique ≤i−1 -minimal representative of U · (Z + mi−1 ) and so aj = bj for j ≤ i − 1. Now if ai 6= bi , then i is an inert point of X so that Pi−1 0 0 X 0 + mi = j=1 aj eβj + mi ∈ U · (Z + mi ). Now X ≤i X and X ≤i Y . Therefore, ≤i -minimality of X + mi and Y + mi forces X + mi = X 0 + mi = Y + mi .  We note that the minimal representatives of U -orbits do depend on the chosen order of Ψ . Also we note that, if X + mi is the minimal representative of its U -orbit in ui , then X + mi−1 is the minimal representative of its U -orbit in ui−1 . We now describe when an element of ui is a ≤i -minimal element of its U -orbit. P Lemma 5.5. Let X ∈ u. Then X + mi = ij=1 aj eβj + mi is the unique ≤i -minimal element of its U -orbit if and only aj = 0 whenever j is an inert point of X. P Proof. We work by induction on i to show that, if X + mi = ij=1 aj eβj + mi is the unique ≤i -minimal element of its U -orbit, then aj = 0 whenever j is an inert point of X. The case i = 0 is trivial. P Let i > 0 and let X + mi = ij=1 aj eβj + mi ∈ ui be the unique ≤i -minimal element of its U -orbit in ui . Then X + mi−1 is the ≤i−1 -minimal representative of its U -orbit, so by induction, aj = 0 whenever j ≤ i − 1 is an inert point of X. If i is a ramification point of X, then trivially aj = 0 whenever j ≤ i is an inert point of X. If i is an inert point of X, then +

13

P X + mi is in the same U -orbit as i−1 j=1 aj eβj + mi and thus ≤i -minimality of X + mi implies that ai = 0. P We now work by induction on i to show that, if X + mi = ij=1 aj eβj + mi ∈ ui satisfies aj = 0 whenever j ≤ i is an inert point of X, then X + mi is the ≤i -minimal representative of its U -orbit. The base case i = 0 isP trivial. Let i > 0 and suppose X + mi = ij=1 aj eβj + mi ∈ ui satisfies aj = 0 whenever j ≤ i is an inert point of X. Let Y + mi be the ≤i -minimal element of U · (X + mi ). Then Y + mi−1 is ≤i−1 -minimal in U · (X + mi−1 ), so by induction Y + mi−1 = X + mi−1 . If ai 6= bi , then i must be an inert point of X and therefore ai = 0 by assumption. Thus ≤i -minimality of Y implies bi = 0 so X + mi = Y + mi is ≤i -minimal.  In Lemma 5.7 we give the dimension of a U -orbit in ui ; in its statement we use the notation introduced in the following definition. Definition 5.6. Let X ∈ u. The number of inert points of X less than or equal to i is denoted by ini (X). Lemma 5.7. Let X ∈ u. Then, dim CU (X + mi ) = N − ini (X) and so dim U · (X + mi ) = ini (X). Proof. We work by induction on i. If i = 0, then the result is trivial. Let i > 0. First suppose i is a ramification point of X. Then CU (X + mi ) = CU (X + mi−1 ) and ini (X) = ini−1 (X). So dim CU (X + mi ) = N − ini (X). Second suppose i is an inert point of X. Consider the orbit map ψ : CU (X + mi−1 ) → CU (X + mi−1 ) · (X + mi ). The image of ψ is isomorphic to k as an algebraic variety and the fibre ψ −1 (X + mi ) is CU (X + mi ). It follows from [4, Thm. AG.10.1] that dim CU (X + mi ) = dim CU (X + mi−1 ) − 1. Therefore, we have dim CU (X + mi ) = dim CU (X + mi−1 ) − 1 = N − ini−1 (X) − 1 = N − ini (X). The second equality in the statement of the lemma is due to (1.1).



Our results lead to an algorithm for determining the U -orbits in u, by finding all minimal representatives of the U -orbits in u. We outline this algorithm below. 0th step: There is one U -orbit in u0 , its ≤0 -minimal representative is 0 + m0 . ith step: Suppose we know the ≤i−1 -minimal representatives of all the U -orbits in ui−1 . We wish to determine the ≤i -minimal P representatives of all the U -orbits in ui . By Lemma 5.5, X + mi = ij=1 aj eβj + mi is the ≤i -minimal representative of its U -orbit, if X + mi−1 is the ≤i−1 -minimal representative of its U -orbit and ai = 0 if i is an inert point of X. Using Corollary 4.3 we see that we can determine whether i is an inert or ramification point of X by calculating dim cu (X + mi ) – this can be reduced to linear algebra, see [13, Rem. 4.4]. Therefore, we can determine all the ≤i -minimal representatives of U -orbits in ui by calculating dim cu (X + mi ) for each X such that X + mi−1 is a ≤i−1 minimal representative of its U -orbit in ui−1 . After the N th step we have calculated all the orbits of U in u. We illustrate this algorithm when G is of type B2 and A3 in the examples below. Example 5.8. We illustrate the calculation of the U -orbits in u when G is of type B2 . We note that this example was given in [7, p. 29]. Since the structure of the U -orbits is quite 14

∗ ∗ ∗ ∗

#PPP PP # PP # PP # PP # PP #

k× ∗ ∗ ∗

0 ∗ ∗ ∗

#bb # b b # b # b # b # b

0 ∗ ∗ 0

l  ll  l  l  l

0 k× ∗ 0

0 0 ∗ 0

A  A  A  A  A

0 0 0 0

0 0 k× 0

0 k× 0 0

0

T

T

T

T

T

0

∗ ∗ k×

k× ∗ ∗ 0

k×

∗ ∗ k×

0

0 ∗ k×

k× 0 ∗ 0

k×

0 ∗ k×

k× 0 0 0

k×

0 0 k×

T

T

T

T

T

0 0 k×

0

0 k× k×

Figure 1. The U -orbits in u for G = Sp4 (k) simple in this case, the algorithm of B¨ urgstein and Hesellink gives the same result as our algorithm, see Remark 5.10. We consider G = Sp4 (k) = {x ∈ GL4 (k) : xt Jx = J}, where   0 0 0 1  0 0 1 0  J =  0 −1 0 0  . −1 0 0 0 We take the upper triangular matrices in G to be the Borel subgroup B and T to be the diagonal matrices in G. Then u consists of the strictly upper triangular matrices in g, i.e. matrices of the form   0 a b c 0 0 d b    0 0 0 −a  . 0 0 0 0 Using the notation of [6, Planche II] the positive roots of G are β1 = 01, β2 = 10, β3 = 11, β4 = 12. We use the above enumeration of Ψ+ . Then the mi are as depicted below where we only show the (1, 2)th, (1, 3)th, (1, 4)th and the (2, 3)th entries. m0 =

k k k 0 k k 0 k k , m1 = , m2 = , k k 0 15

m3 =

0 0 k 0 0 0 , m4 = . 0 0

So for example, m2 consists of matrices of the form   0 0 b c 0 0 0 b    0 0 0 0 . 0 0 0 0 Figure 1 is a tree which illustrates the calculation of the U -orbits in u, using the algorithm described above. The ith row (i = 0, 1, 2, 3, 4) shows the minimal representatives of the U -orbits in ui . We only show the (1, 2)th, (1, 3)th, (1, 4)th and the (2, 3)th entries of these minimal representatives. The entry k × means that one can take any non-zero element of k and the entry ∗ means that the entry is factored out. An edge is drawn between minimal representatives of the form X + mi−1 and X + mi . So there are two edges coming from a minimal representative X + mi−1 if i is a ramification point of X and one edge if i is an inert point of X. The most interesting feature of the tree is that there are two edges coming from 0 0 ∗ , k× meaning that 4 is a ramification point of 0 0 0 λ for any λ ∈ k × . This is because “Uβ2 has been used to make the coefficient of eβ3 zero so it can not also be used to make the coefficient of eβ4 zero”. Example 5.9. We now give an example of part of the calculation of the U -orbits in u in case G is of type A3 . We take G = SL4 (k), U = U4 (k) and T to consist of the diagonal matrices in G. Then u consists of strictly upper triangular matrices in gl4 (k). Using the notation from [6, Planche I] the positive roots of G are β1 = 100, β2 = 010, β3 = 001, β4 = 110, β5 = 011, β6 = 111. We use the above enumeration of Ψ+ . The submodules mi are then as depicted below where we show only the entries above the diagonal: k k k 0 k k 0 k k 0 k k k k , m1 = k k , m2 = 0 k , m3 = 0 k , m0 = k k k 0 0 0 k 0 0 k 0 0 0 0 k , m5 = 0 0 , m6 = 0 0 . m4 = 0 0 0 Due to space restriction we only show in Figure 2 the branch of the tree illustrating the calculation of the U -orbits (as explained in the previous example) from the 2nd row of the 16

k×

k×

k×

k×

k×

∗ 0

∗ ∗ ∗

" " QQ " Q " Q " Q " Q " " Q ∗ 0

∗ ∗ 0

k×

∗ 0

∗ ∗ k×

0 0

∗ ∗ 0

k×

0 0

∗ ∗ k×

%S % S % S % S % S

%e % e % e % e % e

0 0

∗ 0 0

k×

0 0

∗ k× 0

k×

0 0

∗ 0 k×

k×

0 0

∗ k× k×

0 0

0 0 0

k×

0 0

0 k× 0

k×

0 0

0 0 k×

k×

0 0

0 k× k×

Figure 2. Some of the U -orbits in u for G = SL4 (k) tree beginning with k× ∗ ∗ 0 ∗ . ∗ The most interesting point in the tree is at k× 0 0

∗ ∗ , k×

where there are two edges, meaning that 5 is a ramification point of λ 0 ∗ 0 0 µ for any λ, µ ∈ k × . This is because “Uβ2 has been used to make the coefficient of eβ4 zero so it can not also be used to make the coefficient of eβ5 zero”. Remark 5.10. We note that the orbits of U on u are more complicated when the rank of G is large. There exist instances where there is a subset J ⊆ {1, . . . , i − 1} and aj , bj ∈ k × 17

(j ∈ J) such that i is an inert point of see [34, §3 Ex. 1].

P

j∈J

aj eβj and a ramification point of

P

j∈J

bj eβj ,

6. Relation to conjugacy classes of finite groups In this section we assume that char k = p > 0. Assume that G is defined and split over the field of p elements Fp . Let q be a power of p and denote by G(q) the finite group consisting of Fq -rational points of G. The groups constructed by Chevalley in [8], known as Chevalley groups are of this form; for this reason G(q) is sometimes called a Chevalley group. We write F for the Frobenius morphism such that G(q) = GF = {g ∈ G : F (g) = g}. We refer the reader to [9, §3] as a general reference for algebraic groups defined over finite fields and to [4, 18.6 and 18.7] for information about split groups. The Frobenius morphism induces a map on g, which by an abuse of notation we also denote by F . We recall that a subvariety S of G or g is F -stable if and only if it is defined over Fq , [9, Prop. 3.3]. If S is F -stable we write S F = {s ∈ S : F (s) = s} this is equal to the Fq -rational points of S, which we denote by S(q). By [9, 3.15] we may assume that B is F -stable so that U is F -stable and U (q) is a Sylow p-subgroup of G(q). Since G is split, we may choose the isomorphisms uβ : k → Uβ so that the action of F is given by F (uβ (t)) = uβ (tq ), for each β ∈ Ψ. Then F acts on gβ and this action is given by F (aeβ ) = aq eβ . We now recall some results which we use to relate the adjoint orbits of U in u to the conjugacy classes of U (q). First we need to introduce some notation. Let H be an F -stable subgroup of G and M an F -stable normal subgroup of H. Then F acts on both H/M and h/m in a natural way and we have (H/M )(q) ∼ = H(q)/M (q), (h/m)(q) ∼ = h(q)/m(q), see [9, Cor. 3.13]. Let X ∈ h. We recall that the set H 1 (F, CH (X+m)) is defined to be the set of equivalence classes of CH (X + m) under the relation ∼, where x ∼ y if there exists z ∈ CH (X + m) such that x = zyF (z)−1 . The following proposition combines parts of [30, I, 2.7 and 2.8]. Proposition 6.1. (i) The orbits of H(q) in (H · (X + m))F are in correspondence with the elements of the set H 1 (F, CH (X + m)). (ii) There is a bijection between H 1 (F, CH (X + m)) and H 1 (F, CH (X + m)/CH (X + m)0 ). In particular, if CH (X + m) is connected, then (H · (X + m))F is a single H F -orbit. Let u = m0 ⊇ · · · ⊇ mN = {0} be an NT-sequence of submodules of u and define ui = u/mi as in Section 5. It follows from [3, Cor. 9.3.4] that there exists a Springer isomorphism φ : U → u which commutes with F . For such φ, it is clear that the relative Springer isomorphisms φ˜ : U/Mi → u/mi also commute with F . We can now deduce: Proposition 6.2. The orbits of U (q) on (U/Mi )(q) are in correspondence with the F -stable orbits of U in u/mi . 18

In particular, the conjugacy classes of U (q) are in correspondence with the F -stable adjoint orbits of U in u. Proof. The discussion above, concerning Springer isomorphisms which commute with F , implies that the orbits of U (q) in (U/Mi )(q) correspond to the orbits of U (q) in ui (q). Then, by Propositions 6.1 and 4.5, we see that orbits of U (q) in (u/mi )(q) are in correspondence with the F -stable orbits of U in ui .  We now consider the F -stable adjoint orbits of U in u. Lemma 6.3. Let X + mi ∈ ui be the ≤i -minimal representative of its U -orbit. The orbit U · (X + mi ) is F -stable if and only if X + mi ∈ ui (q). Proof. It is clear that if X + mi ∈ ui (q), then U · (X + mi ) is F -stable. If X + mi is ≤i -minimal in U · (X + mi ), then F (X + mi ) is ≤i -minimal in F (U · (X + mi )). Therefore, if U · (X + mi ) is F -stable, then the uniqueness in Proposition 5.4 implies that X + mi ∈ ui (q).  By Lemma 6.3, the conjugacy P classes of U (q) correspond to the minimal representatives of the U -orbits in u of the form β∈Ψ+ aβ eβ with aβ ∈ Fq for all β ∈ Ψ+ . For instance, in Example 5.8 (with F (xij ) = (xqij )) we have 1 + (q − 1) + (q − 1) + (q − 1) + (q − 1)2 + (q − 1) + (q − 1)2 = 2q 2 − 1 U (q)-conjugacy classes. Our next proposition generalizes [34, Thm. 3.1]. Proposition 6.4. Let X + mi ∈ ui (q). Then |U (q) · (X + mi )| = q ini (X) . Proof. We work by induction on i, the base case i = 0 being trivial. Assume |U (q) · (X + mi−1 )| = q ini−1 (X) . Consider the natural map π : ui (q) → ui−1 (q). Let Y + mi−1 ∈ (U · (X + mi−1 ))F and consider its preimage π −1 (Y + mi−1 ) ⊆ (U · (X + mi ))F . One can see that |π −1 (Y + mi−1 )| = q if i is an inert point of X and |π −1 (Y + mi−1 )| = 1 if i is a ramification point of X. The result follows.  For X ∈ u(q) an application of the formula |CU (q) (X + mi )| = |U (q)|/|U (q) · (X + mi )| gives: Corollary 6.5. Let X + mi ∈ ui (q). Then |CU (q) (X + mi )| = q N −ini (X) . 7. The adjoint orbits of B in u In this section we discuss the adjoint B-orbits in u. In case char k > 0, we also consider the conjugacy classes of B(q) in U (q) (using the notation of the previous section). We define minimal representatives of B-orbits in u and show that they exist. In general minimal representatives are not unique, but form an orbit of a finite group. Determining the B(q)conjugacy classes in U (q) is not as straightforward as determining the conjugacy classes of U (q); because of the non-uniqueness of minimal representatives, and because for X ∈ u the centralizer CB (X) need not be connected. 19

Pick an enumeration β1 , . . . , βN of Ψ+ such that ht(βi ) ≤ ht(βi+1 ) and define mi =

N M

gβj .

j=i+1

Then u = m0 ⊇ · · · ⊇ mN = {0}, is an NT-sequence of submodules of u and we define ui = u/mi . Our definition below says that X + mi is the minimal representative of its B-orbit in ui “if it is ≤i -minimal in its U -orbit and as many coefficients as possible are normalized to 1”. The definition is inductive. P Definition 7.1. Let X + mi = ij=1 aj eβj + mi ∈ ui . Then X + mi is called a minimal representative of its B-orbit in ui if: (i) X + mi is the minimal representative of its U -orbit in ui ; (ii) X + mi−1 is a minimal representative of its B-orbit in ui−1 ; (iii) if ai 6= 0 and βi is linearly independent of {βj : j < i and aj 6= 0}, then ai = 1. We now discuss the existence of minimal representatives and show that in general they are not unique, but form an orbit of a finite group. We start with the following lemma. Lemma 7.2. Suppose X + mi ∈ ui is the ≤i -minimal representative of its U -orbit and let t ∈ T . Then t · (X + mi ) is the ≤i -minimal representative of its U -orbit. Proof. This follows from the fact that CU (Y + mj ) = tCU (Y + mi )t−1 (for each j = 1, . . . , i), and Lemmas 5.5 and 5.7.  Now we have to introduce some notation. Let X(T ) be the character group of T and Y (T ) the co-character group of T . We recall that there is a perfect pairing between X(T ) and Y (T ), defined by hχ, ψi = n, where χ(ψ(t)) = tn , χ ∈ X(T ) and ψ ∈ Y (T ). Further, given a Z-basis {ψ1 , . . . , ψr } of Y (T ), the elements of T are of the form t = ψ1 (t1 ) · · · ψr (tr ), where ti ∈ k × . Let {γ1 , . . . , γs } ⊆ Ψ+ be linearly independent. We can find a subset {λ1 , . . . , λs } of Y (T ) ⊗Z Q such that hγi , λj i = δij ; and therefore, a subset {µ1 , . . . , µs } of Y (T ) and positive integers c1 , . . . , cs such that hγi , µj i = ci δij . Then for t1 , . . . , ts ∈ k × , s Y j=1

µj (tj ) ·

s X

bi eγi =

i=1

s X

tci i bi eγi .

i=1

Along with Lemma 7.2, the above calculation implies that minimal representatives of Borbits in ui exist. Now let X + mi be Pa minimal representative of its B-orbit in ui . Let I ⊆ {1, . . . , i} be such that X + mi = j∈I aj eβj + mi with aj 6= 0, for all j ∈ I. Define J = {j ∈ I : βj is linearly independent of {βj 0 : j 0 ∈ I, j 0 < j}}. Then, since X + mi is a minimal representative of its B-orbit, part (iii) of Definition 7.1 implies that aj = 1 for all j ∈ J. Suppose Y + mi is another minimal representative of the B-orbit of X + mi and let b ∈ B be such that b·(X +mi ) = Y +mi . Then CU (Y +mj ) = bCU (X +mi )b−1 for each j = 1, . . . , i. 20

Thus j is an inert point of Y if and only if j is an inert point of X, this follows from Lemma 5.7. Therefore, using Lemma 5.5 and the fact that Y + Pmi is the ≤i -minimal representative of its U -orbit in ui , we see that Y + mi is of the form j∈I bj eβj + mi . Since B is the semidirect product of T by U , we can write b = tu with t ∈ T and u ∈ U . Suppose u · (X + mi ) 6= X + mi , and let l be minimal such that u · (X + ml ) 6= X + ml . Then we l must be an inert point of X, so l ∈ / I, by Lemma 5.5. But t−1 · (Y + mi ) = P see that −1 j∈I bj βj (t) eβj + mi , which cannot be equal to u · (X + mi ). This contradiction means we may assume u = 1. Now define Γ = {βj : j ∈ I} and write ZΓ for the sublattice of X(T ) generated by Γ. The theory of finitely generated abelian groups says that we can choose a basis χ1 ,. . . ,χr for X(T ) such that d1 χ1 , . . . , dl χl is a basis for ZΓ, where l ≤ r, and dj divides dj+1 for j = 1, . . . , l − 1. Q Let ψ1 , . . . , ψr be the basis of Y (T ) dual to χ1 ,. . . ,χr . Then t can be written in the form t = rj=1 ψj (tj ). Q We see that ( rj=l+1 ψj (tj )) · (X + mi ) = X + mi . Let S be the subtorus of T consisting Q of elements of the form s = lj=1 ψj (sj ), where s1 , . . . , sl ∈ k × . Then we may assume t ∈ S. d

Further, we see that s ∈ CS (X + mi ) if and only if sj j = 1 for j = 1, . . . , l. In particular, CS (X + mi ) is a P finite group. 0 Let X +mi = j∈J eβj +mi and ∆ = {βj : j ∈ J}. Then arguments similar to those above show that CS (X 0 + mi ) is a finite group that contains CS (X + mi ). In fact one can show that CS (X 0 + mi )/CS (X + mi ) is isomorphic to ZΓ/Z∆; one requires the assumption that p is good for G and [30, 4.3] to ensure ZΓ/Z∆ has order co-prime to p. Further, since X +mi and Y +mi are minimal representatives of their B-orbits, it is easy to see that t ∈ CS (X 0 +mi ). It follows that the set of minimal representatives of the B-orbit of X + mi is acted on faithfully and transitively by the finite group MS (X + mi ) = CS (X 0 + mi )/CS (X + mi ). In particular, X + mi is the unique minimal representative of its B-orbit if and only if ZΓ = Z∆. The discussion above proves the following proposition. Proposition 7.3. Let O be an orbit of B in ui . There exists a minimal representative X +mi of O. If Y + mi is another minimal representative of O, then there is a unique element of t¯ ∈ MS (X + mi ) such that t¯ · (X + mi ) = Y + mi , where MS (X + mi ) is as defined above. Remark 7.4. We note that in case G is of type Ar , we do have uniqueness of minimal representatives of B-orbits in ui . This is because a root system of type Ar has no proper maximal rank subsystems; this implies that we always have ZΓ = Z∆, in the above notation. In some cases not of type A we still get unique minimal representatives of B-orbits in u. For instance, in Example 5.8 we get the following (unique) minimal representatives of the 7 B-orbits; in fact the arguments above imply that if B acts on u with finitely many orbits, then the B-orbits have unique minimal representatives. We use the same notation as in the example. 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 In the following example, we show that for G of type G2 we do not have unique minimal representatives of all B-orbits in u. 21

Example 7.5. Suppose G is of type G2 . Using the notation for the roots in Ψ from [6, Planche IX], we may enumerate Ψ+ as β1 = 10, β2 = 01, β3 = 11, β4 = 21, β5 = 31, β6 = 32. Then one can check that for any a2 , a4 , a5 ∈ k × , a2 eβ2 + a4 eβ4 + a5 eβ5 is the minimal representative of its U -orbit in u. The corresponding minimal representatives of B-orbits are of the form X(a5 ) = eβ2 + eβ4 + a5 eβ5 , where a5 ∈ k × . We have that Z[β2 , β4 , β5 ] ⊆ X(T ) is equal to the root lattice and Z[β2 , β4 ] has index 2 in the root lattice. Therefore, MS (X + mi ) is of order 2. The nonidentity element of MS (X + mi ) maps X(a5 ) to X(−a5 ). We now use the notation given in the Section 6 to discuss the B(q)-conjugacy classes in U (q). The lack of uniqueness of B-orbit minimal representatives makes it difficult to determine the F -stable B-orbits in general. Let X +mi be a minimal representative of its B-orbit in ui . Then we have that B ·(X +mi ) is F -stable if and only if F (X + mi ) is in the MS (X + mi )-orbit of X + mi ; this can be proved similarly to Lemma 6.3. For instance, in Example 7.5 the B-orbit of X(a5 ) is F -stable if q × and only if a5 ∈ F× q or a5 ∈ Fq 2 satisfies a5 = −a5 . In general it is difficult to determine the B(q)-conjugacy classes in U (q) from the F stable B-orbits in u, because CB (X) need not be connected for an element X ∈ u, as we demonstrate in the following example. Example 7.6. Let G = SL2 (k) and assume char k 6= 2. Let F be defined by F (xij ) = (xqij ), let B be the subgroup of upper triangular matrices in G and let T be the subgroup of diagonal matrices. Let   0 1 X= . 0 0 The B-orbit of X is {λX : λ ∈ k × } and the centralizer of X in B is    a b 2 CB (X) = ∈ B : a, b ∈ k, a = 1 . 0 a We also have 0

CB (X) =



1 b 0 1



 ∈B:b∈k .

So that CB (X) is disconnected. One can see that (B · X)F splits into two B(q)-orbits, namely B(q) · X = {λX : λ = µ2 for some µ ∈ F× q } and B(q) · νX = {λνX : λ = µ2 for some µ ∈ F× q } where ν ∈ Fq is not a square. Let X ∈ u(q) be the minimal representative of its U -orbit. Our next proposition implies that to determine how (B · X)F splits into orbits of B(q), one only needs to consider CT (X). Proposition 7.7. Let X + mi ∈ ui be the ≤i -minimal representative of its U -orbit in ui . Then we have the factorization CB (X + mi ) = CU (X + mi )CT (X + mi ). 22

P Proof. Write X + mi = j∈J aj eβj + mi where J ⊆ {1, . . . , i} and aj 6= 0 for all j ∈ J. Let bP∈ CB (X + mi ). We may write b = ut where u ∈ U and t ∈ T . We have t · (X + mi ) = / CT (X + mi ) and let j be minimal such that βj (t) 6= 1. j∈J βj (t)aj eβj + mi . Suppose t ∈ Since aj 6= 0, j is a ramification point of X by Lemma 5.5. But u · (X + (βj (t) − 1)aj eβj + mj ) = X + mj which implies that j is an inert point of X. This contradiction means that t ∈ CT (X + mi ). Therefore, we also have u ∈ CU (X + mi ) and hence b ∈ CU (X + mi )CT (X + mi ).  Using Propositions 7.7 and 4.5, we see that CB (X + mi )0 = CU (X + mi )CT (X + mi )0 , if X + mi is the minimal representative of its U -orbit in ui . Therefore, we have a natural isomorphism CB (X + mi )/CB (X + mi )0 ∼ = CT (X + mi )/CT (X + mi )0 . We have a bijection between H 1 (F, CB (X + mi )) and H 1 (F, CB (X + mi )/CB (X + mi )0 ), by [30, I, 2.7]. Thus we get: Corollary 7.8. Let X + mi ∈ ui (q) be the ≤i -minimal representative of its U -orbit in ui . Then the B(q)-orbits in (B · X + mi )F are in correspondence with the elements of H 1 (F, CT (X + mi )/CT (X + mi )0 ). Appendix A. Correspondence of B-orbits in U and u We mentioned in Section 2 that J.-P. Serre has shown that different Springer isomorphisms φ : U → N afford the same bijection on the G-orbits (see [26]). We now discuss the situation for Springer isomorphisms φ : U → u and relative Springer isomorphisms φ˜ : U/M → u/m. We imitate Serre’s proof that the bijection between the G-orbits is independent of the choice of Springer isomorphism to show (in Theorem A.5 below) that the bijection between the ˜ Proposition A.1 (see below) is key for sheets of B on U/M and u/m is independent of φ. showing the bijection of G-orbits is independent of the choice of Springer isomorphism, it was proved by Serre in [26]. We need to introduce some notation before we can state it. For a subset S of U we denote the set of regular unipotent elements in S by S r . Let x ∈ U be regular unipotent and let X ∈ Lie CU (x) be regular nilpotent. For y ∈ CG (x)r we know there exists a Springer isomorphism φy,X which sends y to X. Further, one can see that any Springer isomorphism must be of the form φy,X for some y ∈ CG (x)r . Proposition A.1. There exists a morphism Φ : CG (x)r × U → N of algebraic varieties such that Φ(y, z) = φy,X (z) for every y ∈ CG (x)r and z ∈ U. We may easily deduce: Corollary A.2. There exists an algebraic morphism Φ : CU (x)r ×U → u such that Φ(y, z) = φy,X (z) for every y ∈ CU (x)r and z ∈ U . Given a Springer isomorphism φy,X and an NT-subgroup M of U we write φ˜y,X for the corresponding relative Springer isomorphism from U/M to u/m. Using arguments similar to those in Theorem 3.9 we can prove ˜ : CU (x)r × U/M → u/m Proposition A.3. There exists a morphism of algebraic varieties Φ ˜ zM ) = φ˜y,X (zM ) for every y ∈ CU (x)r and z ∈ U . such that Φ(y, 23

Proof. The proof that Φ exists in Proposition A.1 in [26] shows that the map (y, z) 7→ φy,X (z) from CG (x)r × G · x to G · X extends to Φ : CG (x)r × U → N . This means that the map (y, z) 7→ φy,X (z) from CU (x)r × B · x to B · X extends to Φ : CU (x)r × U → u. Now we may apply arguments as in the proof of Theorem 3.9 to show that the map (y, zM ) 7→ φ˜y,X (zM ) ˜ : CU (x)r × U/M → u/m. from CU (x)r × B · xM to B · X + m extends to Φ  To prove Theorem A.5 we need the following proposition which is proved similarly to the lemma in [26]. Proposition A.4. Let R be a connected algebraic group and let V and W be varieties on which R acts morphically. Let X be an irreducible algebraic variety and suppose Φ : X ×V → W is a morphism such that the map v 7→ Φ(x, v) is an isomorphism of R-varieties, for every x ∈ X. Then for every sheet S of V the image Φ(x, S) is independent of x ∈ X. Proof. We recall from Section 1 that Vi = {v ∈ V : dim R · v = i} and the sheets of R on V are the irreducible components of the Vi for all i ∈ Z≥0 . It is clear that for every x ∈ X the isomorphism Φx : v → Φ(x, v) maps Vi to Wi . Therefore, we may assume that V = Vi and W = Wi . Moreover, the irreducibility of X implies that the Φx s map a given irreducible component of V into a fixed irreducible component of W . Therefore, we may assume that V and W are irreducible. The result now follows.  We now easily deduce Theorem A.5 from Propositions A.3 and A.4 Theorem A.5. The bijection between the sheets of B on U/M and u/m given by a relative ˜ Springer isomorphism φ˜ is independent of the choice of φ. In particular, if B acts on U/M with finitely many orbits, then the bijection between the ˜ orbits of B on U/M and u/m induced by φ˜ is independent of φ. Remark A.6. In Remark 3.11 we noted that there exist B-equivariant isomorphisms U/M → u/m which are not relative Springer isomorphisms. We note here that Theorem A.5 does not consider such isomorphisms. Remark A.7. We do not have an example of two relative Springer isomorphism which induce different bijections on the B-orbits in U/M and u/m. Some calculations for small rank type A cases (where B acts with infinitely many orbits) show that the bijection of B-orbits is independent in these cases. References [1] C.A.M. Andr´e, Basic characters of the unitriangular group (for arbitrary primes), Proc. Amer. Math. Soc. 130 (2002), no. 7, 1943–1954. [2] H. Azad, M. Barry and G. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), no. 2, 551–562. ´ [3] P. Bardsley and R.W. Richardson, Etale slices for algebraic transformation groups in characteristic p, Proc. London Math. Soc. (3) 51 (1985), no. 2, 295–317. [4] A. Borel, Linear algebraic groups, Graduate Texts in Mathematics 126, Springer-Verlag 1991. ¨ [5] W. Borho, Uber Schichten halbeinfacher Lie-Algebren, Invent. Math. 65 (1981/82), no. 2, 283–317. [6] N. Bourbaki, Groupes et alg`ebres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1975. 24

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School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT, United Kingdom. E-mail address: [email protected] URL: http://web.mat.bham.ac.uk/S.M.Goodwin

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