On the orbits of a Borel subgroup in abelian ideals
Let be a Borel subgroup of a semisimple algebraic group , and let be an abelian ideal of . The ideal is determined by certain subset of positive roots, and using we give an explicit classification of the -orbits in and . Our description visibly demonstrates that there are finitely many -orbits in both cases. Then we describe the Pyasetskii correspondence between the -orbits in and and the invariant algebras and , where . As an application, the number of -orbits in the abelian nilradicals is computed. We also discuss related results of A. Melnikov and others for classical groups and state a general conjecture on the closure and dimension of the -orbits in the abelian nilradicals, which exploits a relationship between between -orbits and involutions in the Weyl group.
Key words and phrases:Root system, ad-nilpotent ideal, strongly orthogonal roots, the Pyasetskii duality
2010 Mathematics Subject Classification:14L30, 17B08, 20F55, 05E10
April 26, 2014
- 1 Preliminaries on linear actions with finitely many orbits and abelian ideals
- 2 Classification of -orbits in
- 3 Classification of -orbits in and the Pyasetskii duality
- 4 Algebras of -invariants and a dimension estimate
- 5 Counting -orbits in the abelian nilradicals
- 6 Closures of -orbits and involutions in the Weyl group
Let be a connected semisimple algebraic group with Lie algebra . Fix a Borel subgroup and a maximal torus . Let be an abelian ideal of , i.e., , , and . It is easily seen that and hence is a sum of certain root spaces. Therefore is the closure of a nilpotent -orbit in . By [18, Theorem 2.3], is the closure of a spherical -orbit. That result is based on the characterisation of the spherical nilpotent -orbits obtained in [14, (3.1)]. Consequently, the -module has finitely many orbits (that is, the set is finite). By a general result of Pyasetskii , this is equivalent to that, for the dual -module , the set is finite.
In this article, a direct approach to the study of -orbits in is provided. We prove that is finite, without using the sphericity of , and point out a representative for each -orbit in . Describing -orbits in abelian ideals immediately reduces to simple Lie algebras and, from now on, we assume that is simple. Let be the subset of positive roots corresponding to . We say that is strongly orthogonal, if each pair of roots in is strongly orthogonal in the usual sense, cf. Definition 1 below. We establish a natural bijection between and the set, , of all strongly orthogonal subsets of . Namely, let us fix nonzero root vectors and, for any , set . Then is a complete set of representatives of -orbits in (Theorem 2.2). Quite independently, without using Pyasetskii’s result , we obtain a similar set of representatives for the -orbits in , also parameterised by (Theorem 3.2). Both classifications rely on the following simple observation. Let be strongly orthogonal roots in . Set and . Then and (Lemma 1.2).
For , let (resp. ) denote the corresponding -orbit in (resp. ). We point out two sets that give rise to the dense -orbits in and , respectively. Furthermore, Pyasetskii’s theory yields a natural one-to-one correspondence (duality) between and (see 1.1), and we explicitly describe it. More precisely, given , let denote the Pyasetskii dual orbit in . Then for some , and we determine via , see Theorem 3.9.
Set . Since both and contain dense -orbits, it follows from [22, § 4, Prop. 5] that the algebras of -invariants and are polynomial. We show that their Krull dimensions equal and , respectively. This also implies that the number of -orbits of codimension in (resp. ) equals (resp. ). Moreover, the description of holds true upon replacing with an arbitrary -ideal , see Section 4.
Let a standard parabolic subgroup of with and the nilradical of . The abelian nilradicals (=ANR) yield the most interesting examples of abelian ideals of , and for any such we compute the total number of -orbits and also the number of orbits such that is a prescribed integer, see Section 5.
It is a fundamental problem to describe the closures of -orbits in and , i.e., the natural poset structure on and . In general, these two posets are rather unrelated and one has two different problems. (The Pyasetskii duality tends to behave as a poset anti-isomorphism, but only to some extent!) Although no general solution to either of the problems is known, we have a conjecture on the case in which is an ANR . Let denote the standard Levi subgroup of . Then and are dual -modules and the -orbits in coincide with the -orbits. This implies that the posets and are naturally isomorphic, and it is more convenient to state our conjecture for -orbits in . To any one associates the involution that is the product of reflections corresponding to all roots in . Let be the length function on . For any , we regard as an endomorphism of . It is well-known that is the minimal length for presentations of as a product of arbitrary reflections in , which is also called the absolute length of . For , we conjecture that (i) if and only if w.r.t the Bruhat order and (ii) , see Conjecture 6.2. But both assertions are false for arbitrary maximal abelian ideals, see Example 6.3.
We also give in Section 6 an account on related results for classical algebras that are due to Melnikov and others [1, 5, 8, 11, 12]. In fact, our approach provides a unified treatment for problems studied independently for different series of simple Lie algebras.
The ground field is algebraically closed and of characteristic zero.
is the set of roots of , is the set of positive roots corresponding to , and is the set of simple roots in ; is the Weyl group of and is the highest root in . For , is the root subgroup of and . Then .
If an algebraic group acts on an irreducible affine variety , then is the algebra of -invariant regular functions on and is the field of -invariant rational functions. If is finitely generated, then .
If , then is the stabiliser of in and .
Acknowledgements. Part of this work was done while I was able to use rich facilities of the Max-Planck Institut für Mathematik (Bonn).
1. Preliminaries on linear actions with finitely many orbits and abelian ideals
1.1. Representations with finitely many orbits
Let be a representation of a connected algebraic group such that . By , the dual representation also enjoys this property. More precisely, Pyasetskii provides a natural bijection between two sets of -orbits and thereby obtains that . It works as follows. Consider the moment map and its reduced zero-fibre . Under the assumption that , is a (-stable) variety of pure dimension , and the set of irreducible components of , , is in a one-to-one correspondence with the set of -orbits in , or with the set of -orbits in . Namely, let and be two projections. If , then , , contains a dense -orbit and this yields the bijection between and , which is called the Pyasetskii duality (correspondence). It is obtained as the composition of natural bijections:
For , the Pyasetskii dual -orbit in is denoted by . The passage is described directly as follows. For , let denote the annihilator of in . Then is irreducible and contains the dense -orbit, which is . This also shows that the component corresponding to (or ) is the closure of the conormal bundle of (or ). Below is a slight extension of the Pyasetskii result.
If and are -modules and is finite, then there is a natural one-to-one correspondence between and . In particular, .
The moment map associated with the -module can also be regarded as the moment map for the -module . ∎
1.2. Ad-nilpotent and abelian ideals of
Let be a -stable subspace of . Then is an ideal of that consists of ad-nilpotent elements, and we say that is an ad-nilpotent ideal of . Every ad-nilpotent ideal is a sum of root spaces, i.e., , where , and is called a combinatorial ideal in . Abusing the language, we will often omit the word ‘combinatorial’ and refer to as an ideal, too. If , then is an abelian ideal of (and is a combinatorial abelian ideal), and we use the letter ‘’ for such ideals. That is, is always an abelian ideal of . Although we are primarily interested in -orbits related to abelian ideals and their duals, we also obtain some results that hold for arbitrary ad-nilpotent ideals. The combinatorial ideals has the following characteristic property:
if , , and , then ,
and the abelian ideals have the additional characteristic property
if , then .
We equip with the usual partial ordering ‘’. This means that if is a non-negative integral linear combination of simple roots. For any , let (resp. ) denote the set of its minimal (resp. maximal) elements with respect to ‘’. A combinatorial ideal is fully determined by . If , then is the combinatorial ideal generated by , i.e., . Then and is the minimal -stable subspace containing all , . If for some abelian ideal , then is also abelian.
Two different roots are said to be strongly orthogonal, if neither of is a root. In this case, one has , where is a -invariant scalar product in . A subset is strongly orthogonal, if each pair of roots in is strongly orthogonal.
If is simply-laced, then ‘strongly orthogonal’ is the same as ‘orthogonal’. Therefore, we omit the word ‘strongly’ in our further examples related to the ADE-cases.
Our results on -orbits in and rely on the following simple observation.
Suppose that are strongly orthogonal.
If for some , then ;
If for some , then .
(i) Assume that as well. Excluding the case of , which is easy
to handle directly (see below), we then have and .
1. Suppose that one of these scalar products is negative, say . Then and hence , which contradicts the fact that is abelian.
2. If , then . Then , which contradicts the strong orthogonality.
(ii) If both and belong to , then these two roots are strongly orthogonal. Then applying part (i) to them yields a contradiction. ∎
For of type , the unique maximal abelian ideal is -dimensional. If , where is short, then . Here contains no pairs of orthogonal roots!
2. Classification of -orbits in
For every , we fix a nonzero root vector . Let be an abelian ideal of and the corresponding set of positive roots. For a nonempty , we set . If , then .
The linear span of , , equals .
It follows from the construction that is the smallest -stable subspace containing . ∎
For the future use, we record the following obvious fact:
• If , then either of and is a strongly orthogonal set.
Let denote the set of all strongly orthogonal subsets of .
There is a natural one-to-one correspondence
This correspondence takes to the orbit .
Our proof consists of two parts (assertions):
(a) For any , the orbit contains an element of the form for some .
(b) If , then .
Part (a). For , we set . We describe below a reduction procedure that gradually transforms into such that is strongly orthogonal. Consider the strongly orthogonal set and
By Lemma 1.2(i), we have if . (Note, however, that the sets , , are not necessarily disjoint.) This implies that using root subgroups with , one can consecutively get rid of all root summands of whose roots belong to
More precisely, one can write
where represents the sum related to the roots outside
Given , there are and such that . Then there exists a unique such that (i.e., we kill the summand with ). By Lemma 1.2(i), this transformation does not affect the -group of summands. It may change other summands in the -group and also change , but the important thing is that generates a smaller ideal than does. Continuing this way, we eventually kill all summands in the -group. In other words, there is such that
where is strongly orthogonal to . Then we set and play the same game with and the strongly orthogonal set . Again, in view of Lemma 1.2(i), making further reductions with , does not change the sum , and we can kill all the summands with weights in . Eventually, we obtain a vector , where the set is strongly orthogonal, , and all coefficients are nonzero. Finally, since the roots in are linearly independent, we can make all using a suitable element of .
Part (b). Assume that and .
Clearly, , and this is an abelian ideal inside . If , then in view of Lemma 2.1. Set , and consider the corresponding decompositions
Suppose that and with . Then
If , then has nonzero summands corresponding to some roots in , which cannot occur in the right-hand side. (For, if , , then .) Hence and . Therefore, and are -conjugate. Since and are strongly orthogonal subsets in a smaller combinatorial ideal, arguing by downward induction on , we conclude that . Thus, . ∎
Because the set is clearly finite, we obtain
The set of -orbits in , , is finite.
Along with the bijection , we produced a representative in every -orbit. We say that is the canonical representative in (it depends only on the normalisation of root vectors , ). As a by-product of Lemma 1.2 and our proof of Theorem 2.2, one obtains the following description of the tangent space of at .
For , the tangent space is -stable and the corresponding set of roots is , where . More precisely, is the set of roots of and is the set of roots of . In particular, .
Our next goal is to describe the strongly orthogonal set in corresponding to the dense -orbit in . We define the lower-canonical set inductively, as follows. We begin with and put . If and are already constructed, then we set and . Eventually, we get and define . By the construction, the difference of two roots in is not a root; and since we are inside an abelian ideal, the sum of two roots is never a root. Thus, is strongly orthogonal. Whenever we wish to stress that is determined by , we write for it.
The lower-canonical set gives rise to the dense -orbit in .
The above construction of as a union shows that any belongs to a unique . Therefore, there exists such that . By Lemma 1.2(i), all the roots are different since is strongly orthogonal. This implies that . Hence is dense in . ∎
It is not true that for each there exists a unique . We just pick one with the required property.
For , we take to be the algebra of traceless upper-triangular matrices. We stick to the usual matrix interpretation, hence is represented by the right-justified Young diagram . See below the diagram for :
Each box of the diagram represents a positive root, with usual -notation. For instance, the north-east box is the highest root . The ad-nilpotent ideals of correspond to the right-justified Young diagrams that fit inside the above diagram of . Then the maximal abelian ideals of are the nilradicals of maximal parabolic subalgebras, i.e., these are the rectangles , . The maximal abelian ideals for are depicted below:
(We do not draw the boxes outside the ideals!) An arbitrary abelian ideal
corresponds to a diagram that fits inside one of such rectangles. Then is the set of
south-west corners of the diagram. Furthermore, for any , the
set is the hook with south-west corner
For instance, consider the abelian ideal in , , with rows . That is, . Here and . The following diagram depicts , i.e., the union of hooks through : . (The roots in are denoted by bullets.) The only remaining box is , and this gives . Thus, is represented by the diagram: . It is not hard to compute that in this example.
3. Classification of -orbits in and the Pyasetskii duality
For any ad-nilpotent -ideal , we think of the -module as the quotient , where is the orthocomplement of in with respect to the Killing form. The set of -weights in is , and we fix a nonzero weight vector for any . (The -weight of is .) It is convenient to choose roots vectors , , and then define as the image of in . This yields a choice of weight vectors in all that is compatible with the surjections if .
Although the set of weights of is , we prefer to think of it in terms of . As this reverses the root order on the weights of , we will have to consider the maximal elements for subsets of in our constructions related to . Modulo such alterations, the classification of -orbits in is being obtained in a fairly similar way. For , we set . (Again, if , then .) Let be the largest combinatorial ideal in such that . Set
Obviously, and .
We have .
It is easily seen that is the smallest -stable subspace of containing . ∎
Note that , since is assumed to be simple. Therefore, any non-empty combinatorial ideal in contains . This means that if and only if if and only if .
There is a natural one-to-one correspondence
This correspondence takes to the orbit .
The argument is similar to that in Theorem 2.2. One should use Lemma 1.2(ii) in place of Lemma 1.2(i) and Lemma 3.1 in place of Lemma 2.1. For the reader convenience and future reference, we outline the argument.
Part (a). For , we consider and . Then we set
Accordingly, we write . For any , consider . By Lemma 1.2(ii), the union is disjoint. Therefore, using root subgroups with in this union, we may gradually kill the whole -group of summands for , without affecting the -group of summands. That is, there is such that
and each root in is strongly orthogonal to , and so on. Eventually we obtain a representative in whose support is strongly orthogonal.
Part (b) is similar to the respective part in the proof of Theorem 2.2. ∎
If are two abelian ideals and , then the -orbit is also a -orbit in . That is, the notation is unambiguous and can be used with any abelian ideal such that . But this is not the case for the -orbits in the dual spaces! The orbit depends on the ambient space . If we write temporarily , then the surjection takes to , and the corresponding orbit dimensions are usually different.
For , the tangent space is -stable and the corresponding set of weights (negative roots) is , where . More precisely, is the set of roots of and is the set of roots of . In particular, .
Warning. To describe a tangent space of a -orbit in , we use the set , which is the same as , because is an ideal. However, is usually a proper subset of .
Next, we describe the strongly orthogonal set corresponding to the dense -orbit in . The upper-canonical set is defined inductively, as follows. We begin with , which incidentally is just , and put . When and are already constructed, we define and . Eventually, we obtain and set . It is quite clear that is strongly orthogonal. Whenever we wish to stress that is determined by , we write for it.
The upper-canonical set gives rise to the dense -orbit in .
Our procedure of constructing the upper-canonical set in applies perfectly well to arbitrary subsets
of . But the resulting ‘canonical’ set may not be strongly orthogonal. (For
instance, because the sum of two roots in can be a root.)
However, for , the procedure does provide a strongly orthogonal set,
see [7, Sect. 2], .
We call it Kostant’s cascade (of strongly orthogonal roots) in and set
is Kostant’s cascade, then
is a representative of the dense -orbit in .
Furthermore, if is an ad-nilpotent ideal of , then our construction shows that . Hence is strongly orthogonal for any . Another good case, which we need below, is that of an arbitrary subset . Here the upper-canonical set in is always strongly orthogonal, since the sum of two roots in is never a root.
For , Kostant’s cascade is . It consists of the positive roots along the antidiagonal. We continue to consider the abelian ideal of shape in , , cf. Example 2.6. Here and it is depicted by the diagram . Comparing two canonical sets shows that it may happen that . Furthermore, different abelian ideals may have the same upper-canonical set, whereas this is not the case for the lower-canonical sets. Indeed, contains and any ideal is completely determined by its minimal elements.
Our next goal is to describe the Pyasetskii duality for and in terms of . There are the bijections
and the question is: what is in terms of ?
We already know the answer in the two extreme cases:
• If , then and is the dense -orbit in . Hence ;
• Likewise, for and the dense -orbit in , we get .
To discuss the situation for an arbitrary , we recall that the tangent space is -stable and the corresponding set of roots is , where