# Freeness of spherical Hecke modules of unramified in characteristic

###### Abstract

Let be a non-archimedean local field of odd residue characteristic . Let be the unramified unitary group in three variables, and be a maximal compact open subgroup of . For an irreducible smooth representation of over , we prove that the compactly induced representation is free of infinite rank over the spherical Hecke algebra .

## 1 Introduction

In the last two decades, the area of -modular representations of -adic reductive groups has already had a vast development. In their pioneering work ([BL94], [BL95]), Barthel–Livn gave a classification of irreducible smooth -representations of the group over a local field of residue characteristic , leaving the supersingular representations as a mystery. Almost ten years later, C. Breuil ([Bre03]) classified the supersingular representations of , which was one of the starting points of the mod- and -adic local Langlands program, initiated and developed by Breuil and many other mathematicians (see [Bre10] for an overview). Recently, a classification of irreducible admissible mod- representations of -adic reductive groups has been obtained by Abe–Henniart–Herzig–Vignras ([AHHV17]), as a generalization of many previous works due to these authors. However, our knowledge of the mysterious supersingular representations is still very limited, and from the work of Breuil and Paknas ([BP12]) it seems that a ‘classification’ of that, even for the group with , is out of reach.

Smooth representations induced from maximal compact open subgroups, and their associated spherical Hecke algebras, are central objects in the study of -modular representation theory of -adic reductive groups, see [BL94], [Her11a], [Abe13], [AHHV17]. As such an induced representation is naturally a left module over its spherical Hecke algebra, a general question is to ask what we can say about the nature of this module ? Certainly, we might not expect much without any restrictions on the group under consideration. The only general result in this direction, as far as we know, is due to Groe-Klönne ([GK14]), see Remark 4.2 for a precise description of his result. In this note, we investigate this question for the unitary group in three variables.

We start to describe our result in detail. Let be an unramified quadratic extension of non-archimedean local fields with odd residue characteristic . Let be the unitary group defined over , and be a maximal compact open subgroup of . For an irreducible smooth representation of over , the compactly induced representation is naturally a left module over the spherical Hecke algebra . Our main result is as follows:

###### Theorem 1.1.

(Theorem 4.1)

The compactly induced representation is a free module of infinite rank over .

Theorem 1.1 is an analogue to a theorem of Barthel–Livn ([BL94, Theorem 19]) for the group , and we follow their approach in general. The underlying idea is naïve and depends on an analysis of the Bruhat–Tits tree of the group . However, there is an essential difference when we prove a key ingredient (Lemma 4.4) in our case, where we find a more conceptual approach and reduce it to some simple computations on the tree of .

Our freeness result has some natural applications, and we record some of them in this note.

The first one is that every non-trivial spherical universal module of is infinite dimensional (Corollary 4.6), which at least implies the existence of supersingular representations of containing a given irreducible smooth representation of . The existence of supersingular representations, to our knowledge, was only proved very recently for most simple adjoint -adic group ([Vig17]).

Next, following Groe-Klönne ([GK14, section 9]), we apply Theorem 1.1 to investigate integral structures in certain -adic locally algebraic representations of , and we formulate a conditional result for irreducible tamely ramified principal series (Theorem 5.1).

This note is organized as follows. In section 2, we set up the general notations and review some necessary background on the group and its Bruhat–Tits tree. In section 3, we study the Hecke operator in detail, and describe the image of certain invariant subspace of under . In section 4, we prove our main result. In section 5, we apply our main result to investigate -invariant norms in certain local algebraic representations of . In the final appendix 6, we provide a detail proof of the recursion relations in the spherical Hecke algebra of .

## 2 Notations and Preliminaries

Let be a non-archimedean local field of odd residue characteristic , with ring of integers and maximal ideal , and let be its residue field of cardinality . Fix a separable closure of . Let be the unramified quadratic extension of in . We use similar notations , , for analogous objects of . Let be a uniformizer of , lying in . Given a 3-dimensional vector space over , we identify it with (the usual column space in three variables), by fixing a basis of . Equip with the non-degenerate Hermitian form h:

, .

Here, denotes the non-trivial Galois conjugation on , inherited by , and is the matrix

The unitary group is the subgroup of whose elements fix the Hermitian form h:

Let (resp, ) be the subgroup of upper (resp, lower) triangular matrices of , where (resp, ) is the unipotent radical of (resp, ) and is the diagonal subgroup of . Denote an element of the following form in and by and respectively:

, ,

where satisfies . Denote by (resp, ), for any , the subgroup of (resp, ) consisting of (resp, ) with . For , denote by an element in of the following form:

We record the following useful identity in : for ,

(1) |

Up to conjugacy, the group has two maximal compact open subgroups and ([Hij63],[Tit79]), which are given by:

The maximal normal pro- subgroups of and are respectively:

Let be the following diagonal matrix in :

and put . Note that and .

Let be one of the two maximal compact open subgroups of above, and be the maximal normal pro- subgroup of . We identify the finite group with the -points of an algebraic group defined over , denoted also by : when is , is , and when is , is . Let (resp, ) be the upper (resp, lower) triangular subgroup of , and (resp, ) be its unipotent radical. The Iwahori subgroup (resp, ) and pro- Iwahori subgroup (resp, ) in are the preimages of (resp, ) and (resp, ) in . We have the following Bruhat decomposition for :

,

where denotes the unique element in , is either or .

We end this part by recalling some facts on the Bruhat–Tits tree of . Denote by the set of vertices of , which consists of all -lattices in , such that

,

where is the dual lattice of under the Hermitian form , i.e., .

Let be two vertices in represented by and . The vertices and are adjacent, if:

or .

When and are adjacent, we have the edge on the tree.

Let be the standard basis of . We consider the following two lattices in :

,

Denote respectively by the vertices represented by and , which are then adjacent. The group acts on in a natural way with two orbits, i.e.,

.

For , the stabilizer of in is exactly the maximal open compact subgroup , and the stabilizer of the edge is the intersection .

For a vertex , the number of vertices adjacent to is equal to , where is either or , depending on or . For a maximal compact open subgroup , we will write for , if is the unique vertex on the tree stabilized by .

Unless otherwise stated, all the representations of and its subgroups considered in this note are smooth over .

## 3 The spherical Hecke operator

### 3.1 The spherical Hecke algebra

Let be a maximal compact open subgroup of , and be an irreducible smooth representation of . As is pro-, factors through the finite group , i.e., is the inflation of an irreducible representation of .

It is well-known that and are both one-dimensional, and that the natural composition map is non-zero, i.e., an isomorphism of vector spaces ([CE04, Theorem 6.12]). Denote by the inverse of the composition map just mentioned. For , we have , where is the image of in . When viewed as a map in , the factors through , i.e., it vanishes on .

###### Remark 3.1.

There is a unique constant , such that , for . The value of is known: it is zero unless is a character ([HV12, Proposition 3.16]), due to the fact that . When is a character, is just the scalar .

###### Remark 3.2.

There are unique integers and such that and .

Let be the compactly induced smooth representation, i.e., the representation of with underlying space

and acting by right translation. In this note, we will sometimes call a maximal compact induction.

As usual ([BL94, section 2.3]), denote by the function in , supported on and having value at . An element acts on the function by , and we have for .

The spherical Hecke algebra is defined as , and by [BL94, Proposition 5] it is isomorphic to the convolution algebra of all compactly support and locally constant functions from to , satisfying for any and . Let be the function in , supported on , and satisfying . Denote by the Hecke operator in , which corresponds to the function , via the isomorphism between and .

When is hyperspecial, the following proposition is a special case of a theorem of Herzig ([Her11b]).

###### Proposition 3.3.

The algebra is isomorphic to .

###### Proof.

Here, we give a straightforward proof by explicit computations, and the recursion relations in the algebra will be used later.

It suffices to consider the algebra . Recall the Cartan decomposition of :

.

Let be a function in , supported on the double coset . Then, for any , satisfying , we are given . When , commutes with all . As is irreducible, by Schur’s Lemma is a scalar.

For , let . As , . Now , and we have . We see factors through . Similarly, for , , and we get , that is to say . In other words, only differs from by a scalar.

For , let be the function in , supported on , determined by its value on : .

###### Proposition 3.4.

consists of a basis of , and they satisfy the following convolution relations: for ,

(2) |

where is some constant depending on .

###### Proof.

The convolution formulae in the proposition give that , which will matter to us later. In particular, it follows that the algebra is commutative. We leave the proof to the appendix 6. ∎

Denote by the operator in which corresponds to . We then have similar composition of relations among , namely

,

and the assertion in the proposition follows. ∎

### 3.2 The formula

Let be a vector in , and by [BL94, (8)] we have

(3) |

As is supported on the double coset , we decompose into right cosets of :

,

and we need to identify with some simpler set. Note firstly that contains , and that

,

where, as mentioned in Remark 3.2, is the unique integer such that .

Secondly, we note the coset decomposition of with respect to :

.

In all, we may identify with the following set

,

and hence with the following set:

.

Using the previous identification, the above equation (3) becomes:

(4) |

where we note that .

Recall we have a Cartan–Iwahori decomposition:

(5) |

Based on (5), we may describe the -invariant subspace of . By Frobenius reciprocity and an argument like that of [BL94, Proposition 5], we have . Let be a function in , supported in , so is determined by its value at . For any such that , should satisfy .

For , and , we get , that is to say is fixed by . Similarly, for a negative , and , we have , which implies that is fixed by . Note that and , as and , where acts trivially on .

Recall that the subspace of -invariants in is one-dimensional, and from now on we fix a non-zero throughout this note.

Let be the function in , supported on , such that

By the remarks above, we have:

###### Lemma 3.5.

The set of functions consists of a basis of the -invariants of the maximal compact induction .

It is useful to rewrite the function in terms of a canonical -transition of .

, for ;

, for .

We now record the following formula , i.e., , and we will do such thing for all the other in the next subsection.

###### Proposition 3.6.

.

### 3.3 The formula for

The purpose of this part is to push Proposition 3.6 further.

For , denote by (resp, ) the subspace of functions in which are supported in the coset (resp, ). Both spaces are -stable. In our former notations, we indeed have , and , for . Then we may rewrite and as follows:

###### Lemma 3.7.

For ,

For , .

###### Remark 3.8.

Note that for , and for . From Lemma 3.5 and its argument, both and are one dimensional, hence they are generated by and respectively.

Naturally we are interested in how the above -subspaces are changed under the Hecke operator , and the following is the first observation.

###### Proposition 3.9.

. .

. .

. .

Here, we have put .

###### Proof.

This Proposition can be roughly seen from the tree of , but we want to make the inclusions in the statement more precisely, using the formula (4).

For , by the formula (4), we only need to note for .

For , as , we note firstly that for . It remains to check the following, which completes the argument of :

For , let be a non-negative integer. At first, we see

,

for . Next, we check the following, which finishes the proof of :

∎

###### Remark 3.10.

The argument tells us more: for , we can indeed detect the parts of which lie in and .

###### Corollary 3.11.

For , we have

,

for some constant , and is given by

###### Proof.

Suppose is a negative integer, and we will prove that

for some .

By (2) of Proposition 3.9, . As is -invariant, and preserves -invariants, Lemma 3.5 and Remark 3.8 imply that

,

for some .

We need to evaluate the function at and . Recall that , and by Proposition 3.6 we get:

.

We need to estimate in which Cartan–Iwahori double cosets the elements and might belong, for .

We have firstly that:

Here, in the second inclusion above, is some integer smaller than , depending on . To see that, let be the largest integer such that , and the assumption means that . We now apply the equality (1):

where , . The last expression of above identity gives us that

,

and the assertion on follows.

Now we may determine the value of . The list above immediately gives that , for any . As is supported on , the above list reduces us to look at the sum

,

which is clearly zero by splitting it as a double sum, observing that . In all we have shown .

In a similar way, we have

Here, in the second inclusion above, is some integer smaller than , depending on , which is seen by applying (1) again.

Therefore, we have , for any , and the following