Nilpotent orbits of orthogonal groups

Nilpotent orbits of orthogonal groups over -adic fields, and the DeBacker parametrization


For local non-archimedean fields of characteristic or sufficiently large, with odd residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups, and then separately give an algorithmic construction for representatives of each orbit. We then, in the general setting of groups , or classical groups, give a new characterisation of the “building set” (defined by DeBacker) related to an -triple in terms of the building of its centralizer. Using this, we prove our construction realizes DeBacker’s parametrization of rational nilpotent orbits via elements of the Bruhat-Tits building.

Key words and phrases:
p-adic groups; nilpotent orbits; DeBacker classification; quadratic forms; Bruhat-Tits buildings
1991 Mathematics Subject Classification:
20G25 (17B08, 17B45)
This research is supported by a Discovery Grant from NSERC Canada.

1. Introduction

Rational, or arithmetic, nilpotent adjoint orbits of algebraic groups over a local field arise in representation theory in several contexts. For example, the Harish-Chandra–Howe character formula locally expresses a character of a representation as a linear combination of (Fourier transforms of) nilpotent orbital integrals. As another example, the orbit method would parametrize representations by admissible coadjoint orbits, with the admissible nilpotent orbits corresponding to core singular cases.

Algebraic, or geometric, nilpotent adjoint orbits can be thought of as those under the algebraic group over the algebraic closure of the local field. These orbits can be parameterized in multiple ways, including the Bala-Carter classification (extended to low characteristic by McNinch and others), weighted Dynkin diagrams, and partition-type classifications (for classical groups).

The rational points of an algebraic orbit form zero or more rational orbits, and these can in principle be counted using Galois cohomology; yet it remained an open combinatorial problem to count these orbits for orthogonal groups. Solving this is the first goal of this paper, in Section 4.

Our second goal is to present an algorithm for generating representatives for all rational nilpotent orbits of orthogonal and special orthogonal groups over , in the spirit of the one presented by Collingwood and McGovern in [9] over ; our solution is presented in Section 5.

Our third and most important goal is to offer insight into a new geometric parametrization of rational nilpotent orbits proposed by DeBacker in [10]. To this end we prove, in the more general setting of , and classical groups, that DeBacker’s “building set” attached to a Lie triple can be identified with the building of the centralizer of that Lie triple. This kind of “functoriality result” gives a coherent interpretation of the geometry of the DeBacker parametrization, and is presented in Section 6.

Finally, combining these results, we attach a representative of each nilpotent orbit to a facet of the building in our standard apartment, and prove that this gives an explicit realization of the DeBacker correspondence, in Section 7.

Let us now summarize the results of this paper in more detail.

For symplectic, orthogonal or special orthogonal groups the algebraic nilpotent orbits can be parametrized by partitions. Let be a partition of in which even parts occur with even multiplicity and let denote the corresponding nilpotent adjoint orbit of the algebraic group . Let be the number of parts in of multiplicity , the number of odd parts with multiplicity , and the number of odd parts occuring in with multiplicity at least . Assume the characteristic of is either zero or sufficiently large (see Section 3) and that . Then it is known that the rational orbits occurring are parametrized by certain tuples of quadratic forms (Theorem 3.1). Our first result is to compute their number.

Theorem (Theorem 4.3).

Let be a nondegenerate -dimensional quadratic space of anisotropic dimension . Then the number of -rational orbits under in is

where unless either and is even, or and is odd, in which cases . The number of -rational orbits in under is the same unless , when there are two.

In contrast, the number of -orbits in one algebraic nilpotent orbit of a special linear group is where is the gcd of the parts of the corresponding partition. For a symplective group, the number of rational orbits in the algebraic orbit corresponding to partition is simply where is the number of even parts of multiplicity , is the number of even parts of multiplicity , and is the number of even parts of multiplicity greater than or equal to ; this is easily deduced from [19].

Next, for each pair parametrizing a rational nilpotent orbit of , we choose, following Proposition 5.1, a partition of the set , where denotes the multiplicity of in . We construct an explicit orbit representative and associated Jacobson-Morozov triple by aligning subspaces of with the parts of in Sections 5.3 to 5.9. This kind of explicit parametrization has many applications to representation theory, including: computing Fourier coefficients of automorphic forms in [1] and [14]; geometrizing invariant distributions coming from nilpotent orbits [8]; and proving the motivic nature of Shalika germs in [13], building on work of [11]. Note that although determining a complete set of representatives in the case of special linear and symplectic groups is a direct generalization of the real case (see [19]), orthogonal and special orthogonal groups present a special challenge.

Using a “generalized Bala-Carter” philosophy, DeBacker parametrized the rational nilpotent orbits of groups over, among others, local non-archimedean fields (with restrictions on residual characteristic) using the Bruhat-Tits building of the corresponding group. The key construction is of a “building set” of a Lie triple, denoted . Namely, the DeBacker parametrization attaches to each rational nilpotent orbit one or more degenerate pairs, which for our purposes we may take to be pairs , where for a Lie triple extending (see Section 7). When is maximal in , the pair is called distinguished; associativity classes of distinguished pairs are in bijection with rational nilpotent orbits [10]. The challenge in this description is that it does not suffice to work within a single apartment: a facet may be maximal in without being distinguished.

We prove the following general result in Section 6.

Theorem (Theorem 6.1).

Suppose is , or a classical group, and suppose is a Lie triple in , with corresponding homomorphism . Let be the centralizer of in . Assume the nilpotency degree of is less than . Then there is a natural identification as -sets

An immediate consequence is a formula for the dimension of all the maximal facets in , whence it suffices to produce a pair attached to of the correct dimension in order to deduce that it is distinguished. We apply this approach for orthgonal and special orthogonal groups in Proposition 7.1 and Theorem 7.2.

DeBacker’s parametrization has only been explored in a handful of cases, including [19] for the special linear and symplectic groups, where the dimensions of the maximal facets were established via a combinatorial arguments.

The current paper arose initially from an NSERC USRA project of the first author on counting the number of rational nilpotent orbits for orthogonal groups. The third author complemented this with a parametrization of these orbits and circulated a preprint, whereupon the second and fourth authors shared their preprint [22]. In it, J.W. Yap constructs distinguished representatives of all rational nilpotent orbits of split even orthogonal groups (correcting an error in the proof of [19, Theorem 4]), and proves Theorem 6.1 in that case. J.-J. Ma and J.W. Yap had also gone on to prove Theorem 6.1 as it appears here.

Several interesting questions remain open.

For one, although the proof of Theorem 6.1 currently relies on a realization of the Bruhat-Tits building, the second author has shown separately the existence of the map (6.2) for any connected semisimple of adjoint type over , and conjectures that there should be a natural inverse map .

For another, it is an open question to determine known invariants of rational nilpotent orbits in terms of the data of their DeBacker parametrization. Together with [19], we now have the complete parametrization for all split classical groups, which opens the possibilities for study. Part of the problem would be to give a combinatorial description of the associativity classes of facets in , and more particularly of the -associativity classes for each , which are greater in number and offer a finer parametrization.

Our counting results rely on Jacobson-Morozov theory to describe the nilpotent orbits, and thus entail a restriction on the characteristic of . It would be interesting to count rational orbits, and give explicit representatives, in these missing cases.

This paper is organized as follows. In Section 2 we establish our notation and some necessary results about quadratic forms. In Section 3 we present the orthogonal groups, nilpotent adjoint orbits and their partition-based parametrization. Section 4 is devoted to the proof of Theorem 4.3, counting the number of rational orbits. In Section 5, we present an algorithm for generating representatives of each orbit. To do so explicitly, we set the notation for root vectors in Section 5.1 and describe the overall strategy in Section 5.2, with details for each of the subcases in Sections 5.3 to 5.9. In Section 6 we revert to the case of general and briefly recall the DeBacker parametrization, before proving Theorem 6.1. In Section 7 we attach to each of our orbit representatives a distinguished pair, thus establishing a new dictionary from the partition-based to the building-based parametrizations of rational nilpotent orbits for orthogonal and special orthogonal groups.

2. Notation and the Witt group

Let be a local non-archimedean field of residual characteristic , with integer ring and maximal ideal generated by a uniformizer . Denote by the residue field of . Let be a fixed nonsquare in with image in .

The following theory is concisely presented in [2] and based on [15, Chapter 1]. A quadratic space over a field such that is a finite-dimensional vector space over equipped with a regular quadratic form ; when needed, its associated (nondegenerate) bilinear form is denoted , a matrix form is , and the dimension of is , the degree of . Denote by the quadratic hyperbolic plane.

If and are two quadratic spaces we write if they are isomorphic and if the isomorphism classes of the quadratic forms and differ by a sum of hyperbolic planes. Then defines an equivalence relation on the monoid of nondegenerate quadratic forms, and the resulting quotient is the Witt group of with trivial element denoted or . Write for the image of in , which we can identify up to isomorphism with the anisotropic kernel of . Then is the anisotropic dimension of .

Each quadratic space admits a basis relative to which is diagonalized; in this case we write for some but even up to permuting and scaling each coordinate by elements of this representation of is not necessarily unique.

If , then since we have , which has the structure of if (that is, if is congruent to mod ) and of otherwise. The identification of sets is thus not a homomorphism but it is easy to check that it satisfies the very useful property that for all ,


If , then the map induces a well-defined injection . In fact, the map which sends to the class of defines an isomorphism . (We may write when we want to specify the group structure.)

We list the distinct elements of in the second and third columns of Table 1, in terms of the favoured representatives for , and grouped by their anisotropic dimension (given in the first column). Write for the unique class of anisotropic dimension , which is the quaternionic class. We now collect some facts needed for Section 4.

Representative for Number of Common
Choices Representative
Table 1. Representatives of elements of (in two forms: simple ones dependent on the sign of in , and more complex ones which are independent thereof), together with the number of choices of distinct diagonal representatives of each up to .
Lemma 2.1.

Let be a local non-archimedean field of odd residual characteristic.

  1. The number of isometry classes of quadratic forms of degree is if , if , and if .

  2. The number of choices of distinct diagonal representations of each anisotropic form or hyperbolic plane, counting order but modulo , is an invariant of the anisotropic dimension and is independent of the class of mod .

  3. The map extends to a bijection such that for all ,


The first statement is well-known, but can also be inferred from Table 1 directly. We have recorded the number of choices of distinct diagonal representatives for each class of anisotropic form or hyperbolic plane, counting order but modulo scaling in each factor by , in the fourth column of Table 1; this establishes the second assertion. The map extends via the isomorphisms . Since for any , (2.2) follows from (2.1). For convenience, we have recorded the common representatives defining the map in the last column of Table 1. ∎

We say that two tuples of quadratic forms and are isometric if and for all , . Let denote the corresponding isometry class; then is a well-defined element of . We say that represents if .

Now let be a partition of . The multiplicity of in is number of times occurs in . Let denote the set of partitions of in which even parts occur with even multiplicity. For , let if has no odd parts; otherwise, let denote its distinct odd parts and let

be the set of isometry classes of -tuples of quadratic forms of the stated degrees.

Given a quadratic space of degree , we set

If is even, let be the subset of partitions of which have no odd parts; these are called very even partitions. If then for each very even partition we attach two distinct copies of , to give

3. Nilpotent adjoint orbits of the orthogonal group

Let be a semisimple algebraic group defined over and its Lie algebra. We assume that the characteristic of is either zero, or else is greater than , the maximal value of the Coxeter number of any irreducible component of the root system of . If , then we also assume that for each nilpotent element , we have where this denotes the -operation on the restricted Lie algebra .

A Lie triple is a nonzero set such that , and . Then Jacobson-Morozov theory [3, XIII,§11], [16] asserts a bijection between the nonzero nilpotent orbits of on (respectively, of on ) and conjugacy classes of Lie triples in under (respectively, in under ), given by associating the triple to the orbit of its nilpositive element . Moreover, by [16] there is a group homomorphism defined over for which


We often denote the Lie triple by . Under these hypotheses, we have that , the centralizer of in , coincides with , the stabilizer under the adjoint action of the Lie triple .

Now let be an -dimensional quadratic space over . The special orthogonal Lie algebra is

Observe that for any , so from Table 1 we infer there is a single isomorphism class of Lie algebra for each anisotropic dimension, except for . In this latter case, by the Kneser-Tits classification [20], there are two isomorphism classes, corresponding to Lie algebras splitting over a ramified or an unramified extension respectively. The orthogonal group is

and it contains as the index-two subgroup of elements of determinant equal to . These groups are compact if and only if is anisotropic. We think of them as the -points of the corresponding inner forms of the algebraic groups and , respectively.

Given a geometric nilpotent orbit under the algebraic group , then its set of rational points may be empty, or may decompose as a union of one or more rational nilpotent orbits. In the latter case, using the arguments of [18, Prop 4.1], one can deduce that the set of rational orbits is in bijection with the kernel of the map of pointed sets in Galois cohomology


where is an -triple for a base point of and is its centralizer.

The algebraic group is a product of symplectic and orthogonal groups (see below, as (3.4)). For a group preserving a nondegenerate -dimensional bilinear form, counts the number of -isometry classes of forms of degree ; thus it is trivial if the form is symplectic, and if the form is symmetric, it has order , or if is , or at least , respectively. The kernel of the map (3.2) can thus parametrized by tuples of quadratic forms whose sum is equivalent to the chosen form in the Witt group. This correspondence is made explicit in the proof of the following theorem, which is a known result; for example, for archimedean local fields see [9, Ch. 9] and for extensions of see [21, I.6].

Theorem 3.1.

Let be a nondegenerate -dimensional quadratic space over . The nilpotent orbits on are parametrized by the set . If , then the nilpotent orbits coincide with those under but otherwise, is even and the nilpotent -orbits are parametrized by the set .


Let be nilpotent and let be a corresponding Lie triple. Then is a subalgebra of isomorphic to which acts on , decomposing it into pairwise orthogonal isotypic components . Each is the sum of all irreducible submodules of degree equal to , so we may write


where denotes the unique irreducible -module of degree , whose multiplicity space is . If set and in (3.3).

Each carries an -invariant nondegenerate bilinear form , which is symplectic if is even and symmetric if is odd. When is odd, we fix a choice of form such that . For each such that , let denote the restriction of to . Let be the unique (up to isometry) bilinear form on such that . Then if is even, is symplectic so is symplectic as well; thus is a split quadratic space. If is odd, then and are both symmetric. Writing for the associated quadratic form we have , which implies in .

This defines the map sending to : setting for each , the decomposition (3.3) implies , whence this data defines a partition of . If is even, then is symplectic, implying is even and thus . The tuple lies in . Since , the pair lies in . In fact, in this way, each element of defines a decomposition (3.3) that is unique up to isometry.

Note that lies in if and only if it acts as an intertwining operator on viewed as an -module, so our map induces a well-defined map on orbits. Moreover, it follows that acts on each nonzero occuring in (3.3) by an element of , the corresponding isometry group, whence


Now suppose and are two Lie triples in and and are the corresponding decompositions of into isotypic components. Then, as above, these decompositions are isometric if and only if and are conjugate via an element of . As is in bijection with the set of isometry classes of such decompositions, the first statement of the theorem follows.

To understand the orbits, suppose that gives . Thus . From (3.4), and that the symplectic factors have determinant , we conclude that contains an element of determinant if and only if contains at least one odd part, in which case and , showing that the and orbits coincide.

If, however, is even for all , then no such exists. In this case, each is a split quadratic space and so is a sum of hyperbolic planes, whence . Since in this case, and has index two in , we deduce that each of the -orbits corresponding to decompose as a disjoint union of two -orbits. ∎

4. Counting rational nilpotent orbits

Let be a partition and let be the number of odd parts with multiplicity , the number of odd parts with multiplicity , and the number of odd parts with multiplicity or greater. Let be an element of the associated algebraic orbit and an associated Lie triple. From the form of in (3.4), and Lemma 2.1, we deduce that

whereas , depending on . Thus from the discussion preceding the statement of Theorem 3.1, if one expects about rational orbits in , with some variation depending on and the choice of rational form of the algebraic group .

On the other hand, Theorem 3.1 gives a direct means of counting the number of rational orbits in : they are parametrized by . That is to say, it suffices to count the number of isometry classes of tuples (of degrees prescribed by the multiplicities of the odd parts in ) that represent . This is a nontrivial counting problem, and the subject of this section.

We begin with the simple case that each odd part of has multiplicity equal to .

Lemma 4.1.

Let and set . Let have the same parity as . The number of isometry classes of -tuples of degree-one quadratic forms representing is


We prove this formula by induction on even and odd , respectively. When , we have if but if , so we can see that (4.1) holds. When , there are distinct isometry classes of pairs of quadratic forms. By Lemma 2.1, regardless of the sign of in , each of the six anisotropic quadratic forms with is represented by exactly two such pairs, accounting for pairs; the remaining four pairs represent the hyperbolic plane (which has ). In particular no pair can represent (which has ). This count agrees with (4.1) for and each . Thus the formula for holds for and all of the same parity as .

Suppose now that and that is as given, for all such that has the same parity as ; in particular, since the right side of (4.1) depends only on the anisotropic dimension, we may define . Let and suppose it is represented by a -tuple of degree-one quadratic forms where denotes an -tuple. Set ; then .

Set and ; then necessarily .

Suppose first that . Then the four pairs that yield give and thus whereas the twelve others give . Therefore by induction we have

Next suppose that . Then for each we have that has anisotropic dimension two, so is represented by exactly two choices of pairs . Thus the remaining choices of correspond to such that . This yields

as required. Finally, suppose . If , then we must have ; this accounts for pairs . For each of the three other elements of the same anisotropic dimension as , a quick calculation using Table 1 and Lemma 2.1 yields that has anisotropic dimension and hence is representable by exactly two choices of . This accounts for pairs. The remaining six choices of therefore yield such that , so necessarily . We thus infer

and the formula follows. ∎

Now consider the case that each odd part of has multiplicity exactly two.

Lemma 4.2.

Let such that , and let . Then the number of isometry classes of -tuples of degree-two quadratic forms representing is

where is the closest integer to .


We can write the formula as where

Notice that if then , for all .

When , then and for all so the formula holds. Assume and let us count the number of ways, up to isometry, to construct an -tuple of degree-two quadratic forms representing . There are choices for the form , of which are anisotropic. Set and let . By induction is an invariant of anisotropic dimension so we can set .

Suppose . If then and . Otherwise, has anisotropic dimension . Therefore by induction we have

On the other hand, if , then if but if . Each of the remaining five choices of gives such that . This yields the final relation

Since , the formula follows. ∎

Theorem 4.3.

Let be a vector space of dimension and suppose is decomposed as a direct sum of subspaces of dimension , subspaces of dimension and subspaces of dimension at least . Let and set . If is a nonnegative even integer, then the number of ways of assigning a nondegenerate quadratic form to each subspace such that the sum is equivalent to in the Witt group is

where unless either and is even, or and is odd, in which cases .


First suppose that , and let be one of the designated subspaces of of dimension at least . The number of choices of quadratic forms on the remaining spaces is . Given such a choice, let be Witt class of their sum. Then has anisotropic dimension of the same parity as . Since , each of the 8 possible choices of can be realized on . The formula follows.

Now suppose that , so that all designated subspaces have dimension or . If or then we apply Lemmas 4.1 and 4.2, respectively. Otherwise, letting denote the subgroup of the Witt group of all quadratic forms of even anisotropic dimension, we deduce that


where at this last step we have used that ranges over all Witt classes of quadratic forms of dimension of the same parity as , and thus all possible -tuples of degree-one quadratic forms.

When is even, is nonzero only when , in which case , so the final summand is

whereas when is odd, the only nonzero factor is and the term corresponding to has ; this yields

Thus the final summand in (4) is precisely Expanding as in Lemma 4.2, we obtain, for