A characterization of groups of parahoric type
Let be a local henselian nonarchimedean field of residual field , and let be the group of -points of a connected reductive group defined over . It is well-known that the quotient of any parahoric subgroup of by its pro-unipotent radical is isomorphic to the group of -points of a reductive group defined over , and conversely. In this paper, we generalize this result by studying a class of linear algebraic groups named groups of parahoric type; we prove that, under certain conditions, any such group is -isomorphic to the quotient of a parahoric subgroup of some reductive group over by its -th congruence subgroup for a suitable .
Let be a perfect field, and let be a henselian local field of discrete valuation with residual field . Let be the group of -points of a connected reductive group defined over , and let be a parahoric subgroup of . It is well-known (see for example [23, section 2.5] or [24, section 3.2]) that the quotient of by its pro-unipotent radical is the group of -points of a connected reductive group over ; conversely, any such group is isomorphic to the quotient of a parahoric subgroup of a suitable reductive -group by its pro-unipotent radical. This is a very useful tool in the theory of representations of : if is a smooth representation of such that the subspace of elements of fixed by is nontrivial, the restriction of to factors into a representation of on , and the theory of representations of , which is usually easier, in particular when is finite (see for example ), gives us lots of important informations about our representation . With ifinite, that possibility has been, for example, widely used by the author in  to study invariant distributions in the case of level ; it is also an important ingredient in the theory of types, since the study of level representations of a -adic group can be reduced to the study of representations of finite groups of Lie type, which already contain all the relevant data (see for example  for general facts about types, and works of Morris (, ), or the section of , for the level case).
But of course smooth representations of do not always have nontrivial -fixed vectors. A natural question which arises then is: what do the quotients of by normal bounded open subgroups smaller than look like ? It is not hard to prove, directly in the equal characeristic case and using  in the mixed-characteristic case (see proposition 3.14) that these groups are the groups of -points of connected linear algebraic groups defined over , but such groups are not reductive in general; is it possible to generalize the definition of a reductive group to groups of that kind ? Unfortunately, finding a definition which is as simple and elegant as in the reductive case while encompassing all of our quotients and only them seems to be really hard; that explains our choice of giving a definition of groups of parahoric type (first introduced in ) largely inspired by the definition of Bruhat-Tits’ valued root data (see [7, Section I, 6.2]): that kind of definition does not really qualify as simple and elegant, but is useful enough for our purposes.
An important particular case of such normal bounded open subgroups of are the congruence subgroups introduced in  by Schneider and Stuhler (see subsection for the definition of these subgroups). They define a filtration of a parahoric subgroup of by subgroups , , being the pro-unipotent radical of , and then use them to define the Schneider-Stuhler coefficient systems over the Bruhat-Tits building of : for every smooth representation of and every nonnegative integer , the Schneider-Stuhler coefficient system of depth of is the collection of the subspaces , with being any parahoric subgroup of and being its -th congruence subgroup, of -fixed points of on which acts; these subspaces are nothing else than representations of the groups , which are finite when is a finite field. These coefficient systems have turned out to be a useful tool in the recently developed homological/cohomological theory of smooth representations of -adic groups; they allow us for example to construct projective resolutions of representations of -adic groups, which are then useful to compute their Ext groups ((see  and ). You can also csee for example the works of Broussous (, ) about the coefficient systems themselves,, and of Opdam-Solleveld (,) about extension functors. In  and , Meyer and Solleveld also use them to prove in the modular case some results of Harich Chandra (see  or ) about the trace formula. Such an interest in coefficient systems shos that a more detailed study of the representations they involve can be of some interest.
Note also that in the particular case where is a special maximal parahoric subgroup of , the quotient is -isomorphic to the group , where is the unit ring of and its maximal ideal, and that the representations of these groups have already been studied in particular by Lusztig (see ) and Stasinski (see , , , ); unfortunately, that special case is not enough to be able to deal with the Schneider-Stuhler coefficient systems, which happen to involve all possible quotients of parahoric subgroups of our -adic group by their congruence subgroups of some given depth. In that context, a more general study of these quotients is definitely of some interest.
It is easy to check that, apart from a few degenerate cases, all quotients of parahorics by these congruence subgroups match our definition (since it was tailor-made for that). But what about the converse ? In other words, given a group of parahoric type of given depth defined over a perfect field , we want to find a henselian local field of discrete valuation with residual field , a reductive group defined over and, if is the maximal unramified extension of , a parahoric group of the group of -points of , stable by the Galois group , such that is -isomorphic to , where is the -th congruence subgroup of .
In this paper we restrict ourselves to the case where is quasisplit over (for the definition of a quasisplit group of parahoric type, see the beginning of section : that definition is simply a generalization of the definition of a quasisplit reductive group.) It turns out that at least in this case, when the characteristic of is either or not too small, as soon as we have a suitable field , it is possible, at least when the center of satisfies a structural condition (see section ), to find suitable and as well. More precisely, for every element of the absolute root system of , we can put a ring structure on the root subgroup , and that ring is the quotient of the ring of integers of some henselian local field by its ideal of elements of valuation at least either or , depending whether lies inside or outside the reductive part of . Our main result (theorem 6.1) states that as soon as all the , with lying inside (resp. outside) the reductive part of are isomorphic to each other, then is isomorphic to a quotient of a parahoric subgroup. Unfortunately, there are groups of parahoric type which do not satisfy that condition, but it holds at least for large classes of groups, as we will see in section of this paper. We conjecture that our result extends to non-quasisplit groups of parahoric type satisfying the required condition. By slightly generalizing the definition of a group of parahoric type, we also expect to be able to prove a similar result with the Schneider-Stuhler filtrations being replaced with the full Moy-Prasad filtrations (see  for their definition), which allow for a finer study of smooth representations oof a -adic group.
This paper is organized as follows. In section , we recall the definition of a group of parahoric type and prove a few results which will be needed in the sequel. We also observe that, apart from the aforementioned degenrate cases, the quotients of parahoric subgroups of reductive groups defined over a local field by their -th congruence subgroup actually match our definition.
In section , we introduce the notion of a truncated valuation ring. We also attach a ring (resp. ) to every root subgroup of a group of parahoric type (resp. to the group of its -points), and prove that in most cases, either the rings attached to two different roots or suitable quotients of them are isomorphic.
In section , we prove our main result (with some additional hypotheses) in the particular case of groups with a root system of rank .
In section , we make a detailed study of the commutator relations, in order to prove that the structural constants involved in them are similar to the Chevalley constants of reductive groups (see ). An additional difficulty compared to the case of reductive groups is that we want our constants to lie in (quotients of) a truncated valuation ring of residual field , and since the Lie algebra of the -group is not a -module, we are forced to work with directly, which makes things a bit messier. We also study the action of a Cartan subgroup on the root subgroups.
In section , we prove our main result in the general case.
In section , we give a few examples of cases in which the conditions of the theorem are fulfilled, as well as an explicit example of a case in which they are not.
2 Definition and first properties
2.1 Definition of a group of parahoric type
Let be a connected algebraic group defined over some perfect field , and let be a positive integer. Let be a maximal torus of , and let be the centralizer of in ; is a Cartan subgroup of . Let (resp. )) be the group of characters (resp. cocharacters) of .
Since we are working with groups over very general fields, we will not define any topology on the groups over other than the Zariski topology. In this paper, topology-related terms (closed, open, dense subsets of for example), when applied to algebraic groups over , always refer to the Zariski topology.
For every algebraic group , we denote by the unipotent radical of .
Let be the set of weights of with respect to , and let be the root system of the reductive group , also with respect to , identified to a maximal torus of ; is then a root system included in .
Let be the Lie algebra of . For every , let be the unique closed connected subgroup of whose Lie algebra is the eigenspace of for the action of . When and is nontrivial, it will be called a root subgroup of . Since is then a nilpotent subalgebra of , is unipotent. Note also that .
The group is said to be of parahoric type of depth if it satisfies the following conditions:
(PT1) is a reduced root system, and if is the corresponding set of coroots in , the quadruplet is a root datum;
(PT2) for every , the intersection of with the root subgroup of with respect to is of dimension ;
(PT3) the group is abelian, of dimension , where is the rank of , and for every , the intersection of its unipotent radical with the subgroup of generated by and is of dimension . Moreover, for every such that , , and the product of the when runs over is of dimension , where is the rank of ;
(PT4) there exists a concave function from to and, for every and every integer , a connected subgroup of satisfying the following conditions:
(PT4b) for every , , and if is nontrivial, ;
(PT4c) the commutator relations: for every such that , and , we have ;
(PT4d) for every such that and , is of dimension and we have .
As for reductive groups, the group is defined over if is defined over .
Note that the above definition of a group of parahoric type is slightly different from the definition used in .
Since all maximal tori of are conjugates, these properties do not depend on the choice of .
Remember that a (-valued) concave function on is a function from to satisfying the following properties:
for every , ;
for every such that , .
The function in the above definition is not uniquely determined by , but we can easily check that it is entirely determined by its values on the elements of any arbitrarily chosen set of simple roots of , and that those values can also be chosen arbitrarily. In particular we have:
Let be any arbitrarily chosen element of ; it is possible to choose in such a way that .
Since is reduced by (PT1), we deduce from [4, §1, proposition 11] that there exists a set of simple roots of which contains ; the result then follows from the above remark.
(Note that it is not always possible to have -valued and for every at the same time: for example, when is of type and is the subsystem of long roots of , if we set for every long , then we must have for every short . This is a consequence of propositions 2.9 and 2.10 (see below) and of the fact that the Weyl group of acts transitively on both and in that particular case.)
Let be a closed root subsystem of , and let be the subgroup of generated by and the , . Then is a group of parahoric type, and is defined over as soon as is.
The proof is straightforward. As the concave function from to associated to in [PT4), we can simply choose the restriction to of .
Note that it is also often possible to choose as a -valued concave function which is not the restriction to of any -valued concave function on . For example, in the case we were considering above, setting , we can simply choose .
Set and . We say that is of semisimple parahoric type if the following equivalent conditions are satisfied:
the rank of is equal to the rank of ;
the rank of is equal to the rank of ;
generates as a -vector space;
generates as a -vector space.
That notion is a generalization of the notion of semisimple groups in the same way as the notion of group of parahoric type is a generalization of the notion of reductive groups. Of course a group which is both reductive and of semisimple parahoric type is semisimple.
For every , is an abelian group.
The group acts on the subspace of via the weight . Since is reduced, it cannot contain , hence that subspace must be trivial.
Assume , and . Then the subalgebta is the nilpotent radical of the intersection of with the subalgebra of generated by .
The subalgebra is a nilpotent subalgebra of of dimension , hence must be .
Let be a positive integer; for every , the subalgebra of , with , and , depends only on .
We will proceed by induction on . When , the result is simply lemma 2.4. Assume now and write, for every , for suitable such that ; by induction hypothesis depends only on . Let be such that , and ; we now prove that we have:
and the result follows immediately.
By (PT4d), we have:
On the other hand, since is abelian, we have:
The result then follows immediately from the Jacobi identity.
For every and every (resp. every ), we call valuation of in (resp. valuation of on ) and denote by (resp. ) the largest integer such that (resp ). By convention the valuation of the identity element is infinite.
2.2 Parahoric groups and groups of parahoric type
In this section, we give the definition of the -th congruence subgroup (in the sense of Schneider-Stuhler) of a parahoric group of some algebraic group over a nonarchimedean local field with discrete valuation. These filtrations of parahoric subgroups were first introduced in  for , then in  for quasi-simple and simply-connected, and finally generalized to any reductive group in . We now check that the quotients of parahoric subgroups by these subgroups are actually groups of parahoric type.
Let be a connected reductive algebraic group defined over a henselian local field with discrete valuation and perfect residual field , and split over the unramified closure of , and let be the group of -points of . In the sequel, the topology we will be using on groups over is the usual analytic topology.
Let be the Bruhat-Tits building of . Let be a -split maximal torus of , and let be the apartment of associated to ; is isomorphic as an affine space to , where is the group of -points of the neutral component of the center of , and that isomorphism is canonical up to translation. We can thus identify with by setting the origin at some arbitrarily chosen point of ; moreover, if we take as a special vertex of ; the walls of are precisely the hyperplanes defined by equations of the form for some and some . Let be the subgroup of generated by ; for every and every , we can set , where is the usual pairing between and , whose restriction to factors through , extended to .
For every , let be the root subgroup of associated to , and for every , we denote by the subgroup of fixing every element of such that . We have for every , and the family is a fundamental system of open neighborhoods of unity in for the -adic topology.
Let be the derived subgroup of , and let be the maximal bounded subgroup of . The group is an open normal subgroup of , and we deduce from [17, corollaries 2.19 and 2.21] that is the semidirect product of by a discrete group (the group denoted by in ). For every part of , the connected fixator of in is the subgroup of elements of which fix pointwise; when is bounded, contains open subgroups of and the , hence is itself an open subgroup of , hence also of .
Let now be any facet of , and for every , set:
By [7, I. 6.4.3], is a concave function; moreover, since every wall of either contains or does not meet it, we have for every . Let be the connected fixator of in ; is a parahoric subgroup of . Let now be the function defined by:
We will check that is also concave. For every , we have:
since . Now let be elements of such that ; we have:
we are using here the fact that by concavity of . Moreover, since , ; hence and is concave.
Let be the subgroup of defined by:
with being the pro-unipotent radical of ; is the pro-unipotent radical of . Similarly, for every , set:
where is the group of cocharacters of , and:
by [29, I.2], is an open normal subgroup of , which we call the -th congruence subgroup of , and the form a fundamental system of open neighborhoods of unity in .
Assume the residual characteristic of and the root system of satisfy one of the following conditions:
and has no irreducible component of type ;
and every irreducible component of is of type for some .
Let be the parahoric subgroup of associated to the facet ; assume is stable by the action of over . For every integer , let be the -th congruence subgroup of . Then is an algebraic group of parahoric type of depth defined over the residual field of .
The fact that satisfies (PT1) to (PT3) is an easy consequence of the definitions. Moreover, the concave function of (PT4) is the function defined above, and for every and every , we obviously have , hence (PT4a) and (PT4b) hold. Finally, when satisfies the required conditions, the commutator relations of (PT4c) come from [10, Theorem 1], and the relations of (PT4d) are easy to check directly. Hence (PT4) holds as well.
Note that when and do not satisfy the condition of the above proposition, we can also deduce from [10, Theorem 1] that (PT4c) does not hold. Hence in most of these cases, cannot be a group of parahoric type although it comes from a quotient of a parahoric subgroup by some congruence subgroup. The author has chosen to keep these degenerate cases out of the definition, in order not to make it even more complicated than it already is.
2.3 More properties of groups of parahoric type
The groups and, for every , , are connected.
Since the torus acts on , that group is the product of the weight subgroups for that actions, which are and the intersections of with the , . Since is connected, these intersections must then be connected too.
Assume . Then is simply a reductive group.
By lemma 2.7, the intersections of with and the , , are connected. On the other hand, for every , is of dimension by (PT2), hence trivial, and since , must be trivial too. The proposition follows.
Assume the characteristic of is not . Let be an element of ; then , and .
For every subspace of , set , and for every , define inductively by and when . By (PT4d) and an obvious induction, we must have for every , hence if , for large enough. On the other hand, since , has a nontrivial image in , and it is easy to check by induction that the image of is equal to the image of for every , which leads to a contradiction if . Hence must be zero.
Now we prove that . Since these groups are both connected (by respectively lemma 2.7 and (PT4)) and have the same dimension it is enough to prove that the latter is contained in the former; we will in fact prove the inclusion for their respective Lie algebras, which is equivalent since they are connected. Assume some element of has a nontrivial image in ; a simple computation in the Lie algebra shows that there exists then such that also has a nontrivial image in . On the other hand, we have by (PT4d), and by an easy induction (consider and so on), applying (PT4d) again at each step, we find an element of the trivial space with a nontrivial image in which is of course impossible. Hence the result.
Assume is an element of which does not belong to . Then .
If , then is reductive, hence and there is nothing to prove; assume then . Let be any element of ; the dimension of , which is contained in the Lie algebra of , is equal to by (PT3); on the other hand, it is also equal to by (PT4d). Hence we must have and the result follows immediately.
Let be two elements of such that either or lies in . Assume is also a root. Then .
Assume for example . We have and . On the other hand, since , we have , hence:
Hence the first two inequalities are equalities and the proposition is proved.
Let be two elements of such that . Assume in addition that none of the three lies in .
Assume . Then we also have and .
Assume . Then we also have and .
First we prove the first assertion. We will only prove the first equality, the proof of the second one being similar. We have:
The second assertion is an immediate consequence of the first one.
Let be two elements of such that . Then .
since is concave and -valued, we must then have and the proposition is proved.
Now we look at the commutator relations. We can easily check in a similar way as in proposition 8.2.3 of  that for every such that , the set of commutators is contained in the product of the , with and being positive integers. Moreover, for every and every , if we write , with , does not depend on the choice of the order in which the product is taken. Hence the map from to is a morphism of algebraic groups. We can then rewrite as follows:
(PT4c’) for every such that , and , the projection of on is surjective.
Now we consider the subsets of the form of , the product being taken in some given order; we first remark that they are of the same dimension as the subgroup of they generate, hence open and dense in . Since they are obviously normalized by , there exists an open dense subset of such that if belongs to , we can write , with for every ; moreover, since normalizes both and , depends only on the relative position of and for the order we have taken, and for every