On Lusztig’s map for spherical unipotent conjugacy classes in disconnected groups

Let 𝐺 be a simple algebraic group over an algebraically closed field and let 𝜃 be a graph-automorphism of 𝐺 . We classify the spherical unipotent conjugacy classes in the coset 𝐺𝜃 . As a by-product, we show that J.-H. Lu’s characterization in characteristic zero of spherical conjugacy classes in 𝐺𝜃 by the dimension formula also holds for spherical unipotent conjugacy classes in 𝐺𝜃 in positive characteristic. If 𝜃 has order 2, we provide an alternative description of the restriction to spherical unipotent conjugacy classes in 𝐺𝜃 , of Lusztig’s map Ψ from the set of unipotent conjugacy classes in 𝐺𝜃 to the set of twisted conjugacy classes of the Weyl group of 𝐺 . We also show that a twisted conjugacy class in the Weyl group has a unique maximal length element if and only if it has maximum in the Bruhat order (a result previously proved by X. He).


INTRODUCTION
Let˜be a connected reductive algebraic group over an algebraically closed field of characteristic ⩾ 0 and let be its Weyl group. In [24,25], Lusztig introduced a surjective map Φ from the set of conjugacy classes of to the set of unipotent classes of˜, and a right inverse Ψ, which conjecturally coincides with the map previously defined by Kazhdan and Lusztig over the complex numbers in [20] (this conjecture has been proved by Yun in [33]). In [7] a direct combinatorial description of the restriction to spherical unipotent conjugacy classes of the map Ψ was given. In [26] Lusztig extended his construction to the case where˜is not necessarily connected, by defining the map Φ from twisted conjugacy classes (in a certain sense) in to unipotent conjugacy classes in a fixed connected component of˜. If is a twisted conjugacy class in , then Φ( ) is the minimal unipotent conjugacy class in , with respect to Zariski closure, having non-empty intersection with the Bruhat double coset in corresponding to a minimal length element in . It is a non-trivial result that this construction works. The general case is reduced to the connected case and to four other cases, namely, where the connected component of˜is simple and admitting a graph automorphism: Moreover˜= ⋊ ⟨ ⟩, where is a graph-automorphism of order 2 in (a), (b), (d), and of order 3 in (c), = . The twisted action of on itself is defined by 1 ⋅ = 1 ( 1 ) −1 for 1 , ∈ . A right inverse Ψ of Φ is defined by taking, for a given unipotent class in , the unique twisted class in the fiber Φ −1 ( ) for which a certain invariant ( ) reaches its minimum [26,Theorem 1.16]. Also in this case, the fact that this procedure actually works is a deep result.
In this note we focus on the previous four cases. We first classify the spherical unipotent conjugacy classes in , that is, the unipotent classes for which there is a dense orbit of a Borel subgroup of . We prove that in the cases (a), (b), and (d) a unipotent conjugacy class in is spherical if and only if it consists of involutions. In case (c) there is only one spherical unipotent conjugacy class in , the -orbit of .
It has been shown in [4,5,12] that spherical conjugacy classes in may be characterized (in any characteristic) by means of a dimension formula involving the maximal Weyl group element for which meets a class. More precisely, let us define, for a conjugacy class  in , the element  ∈ as the unique element in for which  ∩  is Zariski dense in . Then  is spherical if and only if dim  = (  ) + rk(1 −  ), where is the length function on and rk is the rank in the geometric representation of . This result was extended to the non-connected case in [23] in characteristic zero, and in [6] in good odd characteristic for automorphisms of order 2. As a consequence of the classification of spherical unipotent classes in we obtain the dimension formula for these cases: let be a unipotent conjugacy class in = as above, ∈ be the unique element in for which ∩ is dense in . Then is spherical if and only if dim = ( ) + rk (1 − ). In the second part of our work we obtain an analog of the results of [7] for the non-connected cases (a), (b), and (d). In this case the automorphism has order 2, and we denote it by . The -twisted class of in is denoted by ⋅ . The main result of the second part is the following, dealing with cases (a), (b), and (d).
Theorem. Let be a simple algebraic group over an algebraically closed field of characteristic 2, a graph-automorphism of of order 2. If is a spherical unipotent conjugacy class in , then In case (c), the situation is different. If = ⋅ is the unique spherical unipotent conjugacy class in , the fiber Φ −1 ( ) consists only of the -twisted class of 1 in , denoted bỹ2 in [26, p. 465], hence Ψ( ) = ⋅ 1, while = 0 2 and 0 2 does not lie in ⋅ 1.
We also give some results on the map defined by ( ) = for any conjugacy class in . It was proved in [9,Corollary 2.15] that lies in the set , = { ∈ | is the unique maximal length element in its -twisted conjugacy class in }.
We analyze the image of the restriction of to the set , ℎ of spherical unipotent conjugacy classes in .
Proposition. If is of type , , 6 and has order 2 (hence = 2) we have ( , ℎ ) = , . For of type 4 and of order 3 This allows us to prove the following.

Theorem. The set
, coincides with the set of elements in such that is the maximum element its -twisted conjugacy class, with respect to the Bruhat order.
In [26, § 5.11] Lusztig extends his results to the set of almost unipotent bilinear forms in characteristic not 2, that is, elements g ∈ for of type , ⩾ 2, with Jordan decomposition g = , such that ∈ , 2 = 1, unipotent. In this paper we shall also deal with this case.

NOTATION AND PRELIMINARIES
Let be a semisimple algebraic group over an algebraically closed field . We fix a maximal torus of and a Borel subgroup containing , with unipotent radical . Then Φ is the set of roots relative to and determines the set of positive roots Φ + and the simple roots Δ = { 1 , … , }. For ∈ Φ we put = { ( ) | ∈ }, the root subgroup corresponding to . Let be the Euclidean space spanned by Φ: is the reflection with respect to ∈ Φ, the simple reflection associated to for = 1, … , . We identify the Weyl group of with ∕ , where is the normalizer of ; 0 is the longest element of . If is an automorphism of , then will be the subset of points in fixed by and if is an eigenvalue of , we shall write ( ) for the eigenspace of relative to . A -variety is called spherical if has a dense open orbit in . It is well known (see [2,32] in characteristic 0, [16,21] in positive characteristic) that is spherical if and only if the set of -orbits in is finite. For algebraic groups we use the notation in [8,19]. In particular, let = {1, … , }: for ⊆ , Δ = { | ∈ }, Φ is the corresponding root system, = Span ℝ Δ , the Weyl group, the standard parabolic subgroup of , the standard Levi subgroup of , with derived subgroup ′ . For ∈ we put = ∩ ( 0 ) −1 0 . Then the unipotent radical of is 0 , where is the longest element of . Moreover ∩ = is a maximal unipotent subgroup of and = 0 . We put = ′ ∩ , a maximal torus of ′ : then = is a Borel subgroup of ′ . Let be a Dynkin diagram automorphism. We shall still denote by the permutation of such that ( ) = if ( ) = , the isometry of , the graph-automorphism of , and the automorphism of induced by . In particular, ( ) = ( ) for all in Φ. If is an involution (that is, if it has order 2), we shall write instead of .
Let be a group, an automorphism of . Consider the -twisted conjugation action (ℎ, ) ↦ ℎ ⋅ ∶= ℎ (ℎ −1 ) of on itself. The orbits ⋅ for ∈ are called -twisted conjugacy classes. An element ℎ ∈ is called a -twisted involution if (ℎ) = ℎ −1 . Twisted conjugacy probably occurs first in Gantmakher's paper [14] on automorphisms for = ℂ. Also, [31] contains relevant results. In the literature (such as in [15, Chapter 3, § 3.8] and [27]) twisted conjugacy occurs in another form, being the identity component of a non-connected algebraic groupc ontaining . Let us consider the semidirect product˜= ⋊ ⟨ ⟩, with the product defined by ℎ −1 = (ℎ) for any ℎ in . Then the study of the -conjugation action on the coset is equivalent to the study of -twisted conjugacy in : for , ℎ in one has ℎ (ℎ −1 ) = ℎ ℎ −1 −1 , hence ( ⋅ ) = ⋅ , where ⋅ is the -conjugacy class of (in ). In particular ⋅ is a spherical -twisted class in if and only if ⋅ is a spherical conjugacy class in . In this paper we shall consider interchangeably -twisted classes in and conjugacy classes in .
We shall be interested in the cases where = and = for = . We put˜= ⋊ ⟨ ⟩, = ⋊ ⟨ ⟩, a group of isometries of . We will denote the set of unipotent conjugacy classes in by . By we denote the set of -twisted classes in . From the Bruhat decomposition = ∪ ∈ , we get the decomposition = ∪ ∈ , and we may consider as the double coset relative to ∈ . We have = ∪ ⩽ . Let be a conjugacy class in : there exists a unique ∈ such that ∩ = . We put Proof. It is clear that dim is independent of ∈ , since is a -conjugacy class in . If is elliptic, the result follows by the above observation. Therefore we suppose that is not elliptic. Let be a proper -stable subset of such that ∩ is an elliptic -twisted conjugacy class in and let ∈ ∩ . Consider the basis { | ∈ ⧵ } of the orthogonal ⟂ of in , where are the fundamental weights. We observe that both and ⟂ are -invariant and -invariant. Moreover acts as a permutation on both bases Δ and { | ∈ ⧵ }, and acts trivially on ⟂ . Since ∩ is elliptic in , we have ker( − Id) ∩ = {0}, so the kernel of − Id on is the kernel of − Id on ⟂ . If is a -orbit in ⧵ , and = ∑ ∈ , then { | a -orbit in ⧵ } is a basis of ker( − Id). Hence ( ) = dim . □ Remark 2.3. Proposition 2.2 gives an alternative proof of the fact that the definition of ( ) is independent of . Remark 2.4. As already observed in [23, Remark 1.2; 6, p. 617], if is an automorphism of , then = Int(g) • ′ for a certain g ∈ and a (possibly trivial) graph-automorphism ′ . Then, right translation by g induces a -equivariant isomorphism between the -twisted conjugacy class of an element and the ′ -twisted conjugacy class of g. Therefore the study of -twisted conjugacy classes in is reduced to the study of ′ -twisted conjugacy classes in .

CLASSIFICATION OF SPHERICAL UNIPOTENT CLASSES IN
In this section we classify the spherical unipotent orbits in . The set of unipotent -orbits in is partially ordered by inclusion of closures. By [28,II.2.21. Corollaire], it has a unique minimal element: the -orbit of , which is the unique quasi-semisimple unipotent orbit in (an element is quasi-semisimple if it fixes a Borel subgroup and a maximal torus of it).
We shall make use of the description of unipotent conjugacy classes, of their dimensions, and of the partial order on given in [28]. In the following, denotes a simple algebraic group over and denotes the characteristic of . We recall that, by [11,Theorem 3.4], if is a non-trivial unipotent conjugacy class of a simple algebraic group in characteristic 2, then is spherical if and only if it consists of involutions. Moreover, if is an involution in , then the conjugacy class of is spherical [11, § 4].
We also recall that, by [

even
The unique minimal unipotent orbit in is the orbit 1 +1 of . Since dim 1 +1 = dim , this is the only possible spherical unipotent orbit in : it is made up of involutions and is spherical by [11, § 4.1]. Note that in [11, § 4.1] it is stated that ( ) is isomorphic to ( + 1), but in fact it is isomorphic to ( ) (there is an isogeny from ( + 1) to ( ) and these groups are isomorphic as abstract groups).

odd
The unique minimal unipotent orbit in is the orbit 1 +1

3.2
Type , ⩾ , = , of order 2 To deal with of type we shall consider = (2 ). Then the outer involutions of are obtained by conjugation with involutions of (2 ) ⧵ (2 ). Note that if = 4, and is adjoint or simply-connected, there are other outer involutions in Aut : however, they are conjugate in Aut .
. Therefore = and we are done. □ Our aim is to show that a unipotent conjugacy class in is spherical if and only if is an involution. By [11, § 4.3], we are left to show that if the unipotent class in does not consist of involutions, then  is not spherical.
Assume that is a unipotent element of order greater than 4, and let be an element of order 4 in the subgroup generated by . Then lies in and, by [11,Proposition 3.12], ⋅ is not spherical. Since ( ) ⩽ ( ), it follows that also ⋅ is not spherical. We are therefore left to consider the subset of of conjugacy classes of elements of order 4. By [21,Theorem 2.2], it is enough to show that the minimal elements in are not spherical.

Type , = , of order 2
The unique minimal unipotent orbit in is the orbit ∅ of . Moreover the orbit 1 is the unique minimal element in ⧵ { . } (that is, ⧵ { . } has minimum 1 ) and dim 1 = dim , [28, p. 160, 250]. Hence ∅ and 1 are the only possible spherical unipotent orbits in : these are made up of involutions and are spherical by [11, § 4.4].
This concludes the classification of spherical unipotent classes in .

Theorem 3.3. Let be a simple algebraic group, a graph-automorphism of , and a unipotent conjugacy class in : (i) if = 2 and has order 2, then is spherical if and only if it consists of involutions;
(ii) if = 4 , = 3 and has order 3, then only the class of is spherical.
We shall make use of the following result.  If we restrict to graph-automorphisms of order 2, we get the following theorem.

Theorem 3.6. Let be a simple algebraic group over an algebraically closed field of characteristic 2, and a graph-automorphism of of order 2. Let be a spherical unipotent conjugacy class in , and let ∩ be non-empty. Then, is an involution, that is, is a -twisted involution.
Proof. Let be an element of ∩ . From the classification of spherical unipotent conjugacy classes in it follows that is an involution. Thus, = −1 ∈ ( ) −1 = −1 = ( −1 ) , forcing = −1 . Hence = 1. □ Let be the map defined by ( ) = for any conjugacy class in . It was proved in [9, Corollary 2.15] that lies in the set , = { ∈ | is the unique maximal length element in its -twisted conjugacy class in }.

THE RESTRICTION OF TO SPHERICAL UNIPOTENT ORBITS
We recall the construction of the surjective map Φ ∶ → and of its section Ψ defined in [26]. In [26, § 5.11] Lusztig also considers the set of almost unipotent bilinear forms in characteristic not 2, that is, elements g ∈ for of type , ⩾ 2, of order 2, with Jordan decomposition g = , such that 2 = 1, unipotent. The set denotes the set of conjugacy classes in , Φ ′ ∶ → . We shall deal both with spherical classes in assuming char = 2 and of order 2 or char = 3 and of order 3, and in assuming char ≠ 2 and of type for ⩾ 2. For in we define Γ = { ∈ (resp. ) | ∩ ≠ ∅}. Let ∈ , and let min be the set of all ∈ of minimal length. By [26, Theorem 1.3 (a), § 5.11], given ∈ min , Γ has minimum (that is, ⊆ ′ for all ′ ∈ Γ ): then Φ( ) ∶= (Φ ′ ( ) ∶= , respectively). This definition is independent of the chosen ∈ min . Note that is the unique class of minimal dimension in Γ .
The right inverse Ψ of Φ (Ψ ′ of Φ ′ , respectively) constructed in [26] is defined as follows. Let be in (in , respectively). Then by [26,Theorem 1.16,§ 5.11] there exists a unique element We recall the following result. In the remainder of this section, unless otherwise stated, has order 2 and, as usual, we denote it by . We shall deal with classes in , assuming therefore = 2, and in , assuming ≠ 2. We denote by ℎ ( ℎ ) the spherical conjugacy classes in (in , respectively). We are going to prove that for these classes Proposition 4.1 holds for every in the -twisted conjugacy class of .

Theorem 4.2.
Let be a spherical conjugacy class in , assume in addition that is unipotent if char( ) = 2. If ∩ ≠ ∅, then is an involution, that is, is a -twisted involution.
Proof. If char( ) ≠ 2, this is [6, Theorem 6] (note that in the paper the base field is of zero or good odd characteristic, but the arguments used in Section 2 only use char( ) ≠ 2). If char( ) = 2, this is Theorem 3.6. □ Lemma 4.3. Let be a spherical conjugacy class in , assume in addition that is unipotent if char( ) = 2, and let be a -twisted class in . If ∩ ≠ ∅ for some ∈ , then ∩ ≠ ∅ for every ∈ .
Proof. Let ∈ . Let be of minimal length, say, such that = ( ) −1 (that is, We will show by induction on that ∩ ≠ ∅ for = 0, … , . If = 0, the assertion follows by hypothesis. Assume that ∩ ≠ ∅ for a given and let ∈ ∩ . Then, iḟis a representative of in ( ) for = 1, … , , We recall that if is a conjugacy class in , then is the unique element in such that ∩ = and is the unique maximal length element in its -twisted conjugacy class, that is, lies in , . Hence has the form recalled in Section 3: = 0 with the following properties: is both an involution and a -twisted involution.
We have the following result involving the maps Φ and Φ ′ . Proof. Let = .
We conclude this section by underlying a property of , . This had been previously proved by X. He in [17,Corollary 4.5] (note that He's statement refers to minimal length elements, but multiplying by 0 one may obtain the result for maximal length elements, by a different twist).

Theorem 4.9. The set
, coincides with the set of elements in such that is the maximum element its -twisted conjugacy class with respect to the Bruhat order.
Proof. Let ∈ , = ⋅ . If has maximum with respect to the Bruhat order, then clearly is the unique maximal length element in , that is, lies in , . Conversely, assume that is in , and let ∈ . If has order 2, by Proposition 3.7 we have = for some spherical unipotent conjugacy class in . By Lemma 4.3, ∩ ≠ ∅, hence ⩽ . If is of order 3 (in type 4 ), then either = 0 , or = 0 2 . If = 0 , then ⩽ . For = 0 2 , we checked by direct calculation that ⩽ 0 2 . □  In the connected case the fiber of Φ over the minimal unipotent class {1} in always has a single element, namely, the conjugacy class of 1 in . This is not the case in general in the disconnected case. Let be of type , ⩾ 2, even. Then there is a unique conjugacy class of involutions in , the class of , and this is the unique minimal unipotent class in . In there is a unique -twisted conjugacy class with a unique maximal length element, the class of 0 , so that 0 intersects every class in . In particular we have ∩ 0 ≠ ∅ (see also [11, § 4
Remark 5.2. In [29] Spaltenstein introduces a partial order ≤ on the set of conjugacy classes in by means of the loop group of , where is over ℂ. A priori this partial order depends on the Lie algebra of . It is shown in [33,Corollary 4.6] that in fact ≤ is independent of . Moreover, over the complex numbers, in [33, Corollary 11.1] it is proved that Φ ∶ ( , ⩽) → ( , ⩽) is order-preserving; Ψ ∶ ( , ⩽) → (ImΨ, ⩽) is an isomorphisms of posets.
In [17, § 4.7], He introduces a partial order ⩽ on the set of elliptic conjugacy classes of as follows: ⩽ ′ if and only if there exist minimal length elements ∈ and ′ ∈ ′ such that ⩽ ′ under the Bruhat order. In any characteristic, the map Φ restricted to is injective: let be its image (the subscript bas stands for "basic," a terminology introduced by Lusztig in [24, § 4.1] and used in [33]).
In [1, Theorem 1.1] the authors prove that Φ ∶ ( , ⩽ ) → ( , ⩽) is order-reversing. Therefore the partial orders ⩽ and ≤ on are opposite to each other. On the other hand, in [7,Remark 2] it is observed that we may endow the set (the set of conjugacy classes in with a unique maximal length element), as follows: ⩽ ′ if and only if ⩽ ′ in the Bruhat order, where ( ′ , respectively) is the maximal length element in (in ′ ). In any characteristic , the map ∶ ℎ → , ↦ ⋅ is a poset isomorphism onto its image, and there exists for which is also surjective. Therefore the partial orders ⩽ and ≤ on are the same. To our knowledge no partial order on , on the lines of [29], has been defined. If such an order exists, at least when has order 2, it may coincide with the opposite order on , of elliptic -twisted classes in , as defined in [17, § 4.7] and with the corresponding order on , of -twisted classes in with a unique maximal length element.

A C K N O W L E D G E M E N T S
It is a pleasure to thank Giovanna Carnovale for proposing the problem and for helpful suggestions. This research was produced during the Ph.D. studies of the first author and it was partially supported by DOR2120551/21 "Representations of Quantum groups, finite -algebras, and Weyl groups," BIRD203834 "Grassmannians, flag varieties and their generalizations" of Francesco Esposito and by MUR-Italy via PRIN "Group theory and applications." Finally, we thank an anonymous referee for a careful reading and useful comments and suggestions.
Open Access Funding provided by Universita degli Studi di Padova within the CRUI-CARE Agreement.

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