PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI VIA BRUHAT-TITS THEORY

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1 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI VIA BRUHAT-TITS THEORY STEPHEN DEBACKER ABSTRACT. Let denote a field with nontrivial discrete valuation. We assume that is complete with perfect residue field. Let denote the group of -rational points of a reductive, linear algebraic group defined over. A -torus in is said to be unramified if it splits over an unramified extension of. We show that, up to equivalence, the set of pairs ̵, where is a facet in the Bruhat-Tits building of and Ì is a maximal -minisotropic torus in (the connected reductive -group associated to ), parameterizes the set of -conjugacy classes of maximal unramified tori in. When is quasi-finite and is connected and splits over an unramified extension, we use the above parameterization to explicitly describe the set of -conjugacy classes in the stable conjugacy class of an unramified strongly regular semisimple element of. Finally, we generalize the above parameterization to include -conjugacy classes of pairs À ܵ where À is a maximal rank unramified subgroup of and Ü is a hyperspecial vertex in the reduced building of À. 0. INTRODUCTION The main result of this paper is a uniform parameterization of the set of conjugacy classes of maximal unramified tori in a reductive Ô-adic group. This classification matches conjugacy classes of maximal unramified tori with certain equivalence classes that arise naturally from Bruhat-Tits theory. The motivation for this result comes from harmonic analysis; specifically, from J.-L. Waldspurger s papers [16, 17]. Using the parameterization scheme discussed in this paper, D. Kazhdan and I [6] have been able to generalize some of the results of [16] in a uniform manner. The main result. Let denote a field with nontrivial discrete valuation. We assume that is complete with perfect residue field. Let denote a fixed algebraic closure of. Let à denote the maximal unramified extension of in. Let denote the group of -rational points of a reductive, linear algebraic group defined over. Let Æ denote the group of -rational points of the identity component Æ of. Let µ denote the Bruhat-Tits building of Æ. Date: May 21, Mathematics Subject Classification. Primary 22G25; Secondary 17B45, 20G15. Key words and phrases. Bruhat-Tits building, maximal unramified tori, reductive group. This material is based upon work supported by the National Science Foundation under Postdoctoral Fellowship and Grant No This work was first announced at a Mathematical Sciences Research Institute conference at Banff in 2001 and significantly generalized while visiting the Institute for Mathematical Sciences (IMS) at the National University of Singapore (NUS) in 2002, visits supported by IMS and NUS. PREPRINT 1

2 2 STEPHEN DEBACKER A torus in is called unramified if it is the group of -rational points of a torus in Æ which splits over an unramified extension of. In this paper we classify -conjugacy classes of maximal unramified tori in in terms of equivalence classes of pairs ̵. Here is a facet in the building, is the connected reductive -group associated to, and Ì is an -minisotropic maximal torus in. In more detail: Let Á Ø denote the set of pairs ̵ where is a facet in µ and Ì is a maximal -torus in. In Ü3.2 we define on Á Ø an equivalence relation, denoted. In Ü3.3 we associate to each element ̵ ¾ Á Ø a -conjugacy class ̵ of maximal unramified tori in. The set Á Ø is too large, so we restrict our attention to the subset Á Ñ of minisotropic pairs in Á Ø. A pair ̵ ¾ Á Ø is said to be minisotropic if the maximal -split torus in Ì coincides with the maximal -split torus in the center of. We now state Theorem 3.4.1, the main result of this paper. Let Ì denote the set of - conjugacy classes of maximal unramified tori in. Theorem. There is a bijective correspondence between Á Ñ and Ì given by the map which sends ̵ to ̵. If is Ô-adic and is connected and -split, then this result can be derived from some work of Paul Gérardin [8]. If is Ô-adic and is unramified, then Waldspurger [16] stated a variant of this result as a hypothesis. We remark that if is algebraically closed, then Ì and Á Ñ both have one element. In this case, the element of Ì is the -conjugacy class of maximal -split tori in, and Á Ñ consists of those pairs ̵ where is an alcove in µ and Ì is a maximal torus in. From Lemma a maximal unramified torus in is the group of -rational points of a maximal Ã-split torus in which is defined over. From a theorem of Steinberg (see, for example, [13, Chap. II Ü3.3 and Chap. III Ü2.3]) Æ is quasi-split over Ã. Thus, the centralizer in Æ of a maximal unramified torus of is the group of -rational points of a maximal -torus in. Since this correspondence is one-to-one, our theorem also provides a classification of the -conjugacy classes of maximal tori of which arise in this way. Additional results. In Ü4 we use the above result to give an explicit description of the set of - conjugacy classes in certain stable conjugacy classes. More precisely: Suppose is quasi-finite and is connected and Ã-split. If ̵ ¾ Á Ñ and Ì ¾ ̵, then for ¾ Ì for which µ Ì we describe the set of -conjugacy classes in µ Finally, in Ü5 we present a generalization of the main result. Let denote the set of - conjugacy classes of pairs À ܵ where À is a maximal rank unramified subgroup in and Ü is a hyperspecial point in red Àµ, the reduced Bruhat-Tits building of À. We classify the elements of in terms of equivalence classes of pairs Àµ. Here is a facet in µ, and À is an -cuspidal maximal rank connected reductive subgroup in.

3 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI 3 Additional comments. The main result (Theorem 3.4.1) was circulated as a preprint in I later realized that the main result could be generalized to include Theorem The next version of the paper (not circulated) gave a proof of Theorem and presented Theorem as a corollary; unfortunately, the transparency of the original argument was lost. Consequently, in this version I have chosen clarity over brevity: the two proofs are separated (and another result, Theorem 4.5.1, has been added). I thank Gopal Prasad for his many helpful comments on the first version of this paper. I thank Mark Reeder for allowing me to use his proof of Lemma This paper has benefited from discussions with Jeff Adler, Roman Bezrukavnikov, David Kazhdan, Robert Kottwitz, Amritanshu Prasad, Gopal Prasad, Mark Reeder, Paul J. Sally, Jr., and Jiu-Kang Yu. It is a pleasure to thank all of these people. 1. NOTATION In addition to the notation discussed in the introduction, we will require the following Basic notation. Let denote the residue field of Ã. Note that is an algebraic closure of. Let Рõ which we also identify with Ð µ. We denote by Æ the group of -rational points of the derived group Æ of Æ. When we talk about a torus in, we mean the group of -rational points of a -torus in Æ. In order to avoid a proliferation of superscripts, we adopt the following convention. We shall call a subgroup of a parabolic subgroup of provided that it is a parabolic subgroup of Æ. We adopt a similar convention with respect to tori and Levi subgroups. If ¾, then ½. If Ë, then Ë ¾ Ë. If a group Ä acts on a set Ë, then Ë Ä denotes the set of Ä-fixed points of Ë Apartments, buildings, and associated notation. Let µ µ denote the (enlarged) Bruhat-Tits building of Æ. We identify µ with the -fixed points of õ, the Bruhat-Tits building of Æ Ãµ. Let red µ red µ Æ µ denote the reduced Bruhat-Tits building of Æ. According to [12], we have a decomposition µ red µ µ where denotes the center of. For a Levi -subgroup Å of a parabolic -subgroup of, we identify Å µ in µ. There is not a canonical way to do this, but every natural embedding of Å µ in µ has the same image. For Å µ, we let Ø Åµ denote the stabilizer of Å in and Ü Åµ denote the point-wise stabilizer of Å. Given a maximal -split torus Ë of which is defined over we have the torus Ë Ë µ in and the corresponding apartment ˵ Ë µ in µ. Let Ì be a maximal Ã-split -torus of containing Ë [3, Corollaire ]. We identify Ë µ with Ì Ãµ. For Š˵, we let ˵ Å denote the smallest affine subspace of ˵ containing Å. If is an affine root of with respect to, Ë, and, then denotes the root of with respect to and Ë which is the gradient of. We let Í denote the corresponding root subgroup of.

4 4 STEPHEN DEBACKER Suppose Ü ¾ µ. We will denote the parahoric subgroup of Æ attached to Ü by Ü, and we denote its pro-unipotent radical by Ü. Note that both Ü and Ü depend only on the facet of µ to which Ü belongs. If is a facet in µ and Ü ¾, then we define Ü and Ü. Recall that Ü is a subgroup of Ø Æ Üµ. For a facet in µ the quotient is the group of -rational points of a connected reductive group defined over. We denote the parahoric subgroup of Æ Ãµ corresponding to Ü ¾ õ by õ Ü. We denote the pro-unipotent radical of õ Ü by õ Ü. The subgroups õ Ü and õ Ü depend only on the facet of õ to which Ü belongs. If is a facet in õ and Ü ¾, then we define õ õ Ü and õ õ Ü. For a facet in õ, the quotient õ õ is the group of -rational points of a connected, reductive -group. Suppose is a -invariant facet in õ. In this case, ¼ is a facet in µ. Moreover, we have ¼ õ, ¼ õ, and ¼ (in particular, is defined over ). Sometimes, we will abuse notation and denote by (resp.,, resp., õ ¼, resp., õ ) the group ¼ ¼ (resp.,, resp., ¼ õ, resp., õ ) Unramified groups. The group is said to be Ã-split if it contains a Ã-split maximal torus. A vertex Ü ¾ red µ is said to be hyperspecial provided that is Ã-split and Ü is a -invariant special vertex in red õ. The group (and, similarly, the group ) is called unramified provided that is connected and there exists a hyperspecial vertex in red µ. If the dimension of is less than or equal to one (for example, if is finite), then this definition is equivalent to the usual one: The group is unramified provided that is connected, -quasisplit, and Ã-split. 2. TORI OVER AND In this section we show how to move between tori over and tori over Maximal unramified tori. We recall that a torus Ì of is unramified if Ì is the group of -rational points of a -torus Ì which splits over an unramified extension of. We shall call a torus Ì as above an unramified torus of. The following result will be used throughout the remainder of the paper. Lemma Suppose Ì is a torus in. The following statements are equivalent. (1) Ì µ is a maximal unramified torus of. (2) Ì is a maximal Ã-split -torus of. (3) Ì is a maximal Ã-split torus of and Ì is defined over. Proof. By definition, we have (1) and (2) are equivalent. Moreover, (3) implies (2). We now show (2) implies (3). Suppose Ì is a maximal Ã-split -torus of. Let Å Æ Ìµ. Then Å is a Levi -subgroup of a parabolic Ã-subgroup of. Let Ë ¼ be a maximal -split torus in Å. From [3, Corollaire ], there exists a maximal Ã-split torus Ë of Å such that Ë ¼ Ë and Ë is defined over. Note that Ë is also a maximal Ã-split torus in. Since

5 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI 5 Ë Å, we have Ì Ë. Since Ë and Ì are Ã-split -tori and Ì is maximal in with respect to this property, we must have Ì Ë From maximal unramified tori over to tori over. Suppose Ì is a maximal unramified torus in. Let Ì denote the maximal Ã-split -torus in such that Ì Ì µ. Define Ì Ãµ Ø ¾ Ì Ãµ ص ¼ for all ¾ ̵ From [15, Ü3.6.1] there is a natural embedding of Ì µ in µ; namely, Ì µ Ì Ãµ Ì Ãµ õ Ì Ãµ µ õ Ì Ãµ Ó We shall always think of Ì µ as being embedded in µ in this way. We now collect some facts about Ì µ. Lemma Suppose Ì is a maximal Ã-split torus of which is defined over. Let Ì denote the group of -rational points of Ì. (1) Ì µ is a nonempty, closed, convex subset of µ. Moreover, Ì µ is the union of the facets in µ which meet it. (2) There is a maximal -split torus Ë in such that Ì µ is an affine subspace of Ë µ. (3) For all facets in Ì µ, there exists ̵ ¾ Á Ø such that the image of Ì Ãµ õ in µ is Ì µ. Moreover, if is a maximal facet in Ì µ, then ̵ ¾ Á Ñ. (4) If ½ and ¾ are maximal facets in Ì µ, then for all apartments in µ containing ½ and ¾ we have ½ µ ¾ µ. Proof. (1) : Since Ì µ is the Bruhat-Tits building of Ì, the first half of the statement follows from the work of Bruhat and Tits [2, 3]. For any -invariant facet of õ, we have µ is a facet of µ. Consequently, for any -invariant facet of Ì Ãµ õ, we have that is a facet of µ which is contained in Ì µ. (2) : From [1, Proposition 8.15] we can write Ì Ì Ì where Ì the maximal -split torus in Ì and Ì is the maximal -anisotropic subtorus of Ì. Let Å Æ Ì µ. Then Ì Å and Å is a Levi -subgroup of a parabolic -subgroup. Let Å denote the group of -rational points of Å. We have that the image of Ì µ in red Å µ is a point, call it Ü Ì. Let Ë be a maximal -split torus in Å, and hence in, such that the apartment of Ë in red Å µ contains Ü Ì. Since Ì Ë, we have Ì µ Ë µ. (3) : Suppose is a facet in Ì µ. Let Ì be the maximal -torus in corresponding to the image of Ì Ãµ õ in µ. We have ̵ ¾ Á Ø. Now suppose that is a maximal facet in Ì µ. Choose Ì as in the previous paragraph. Let Ì be the maximal -split torus in Ì. Let Ì denote the -split torus in corresponding to the image of Ì Ãµ õ in µ. We have that Ì is the maximal -split torus in Ì. If we embed Ì Ãµ in Ì Ãµ õ in the natural way, then we have Ì µµ Ì µ.

6 6 STEPHEN DEBACKER As in the proof of part (2) we may choose a maximal -split torus Ë of such that Ì µ Ë µ and Ì Ë. Since is a maximal facet in Ì µ, we have that an affine root of with respect to Ë,, and is zero on Ì µµ Ì µ if and only if it is zero on. It follows that Ì is the maximal -split torus in the center of. Thus ̵ ¾ Á Ñ. (4) : Let be an apartment in µ containing ½ and ¾. Since ½ is maximal in Ì µ and Ì µ is convex, we conclude that ¾ ½ µ. Similarly, we have ½ ¾ µ. Thus ½ µ ¾ µ. The previous lemma gives us a way to associate to a maximal unramified torus in a pair ̵ ¾ Á Ñ. We now examine how unique this association is. Lemma Suppose Ì ½ and Ì ¾ are maximal Ã-split tori of which are defined over. If is a -invariant facet in Ì ½ õ Ì ¾ õ and the images of Ì ½ õ õ and Ì ¾ õ õ in µ coincide, then Ì ½ and Ì ¾ are -conjugate. Proof. Let Ì denote the maximal torus in whose group of -rational points is the image of Ì ½ õ õ in µ. Note that Ì is defined over. Let ½ denote the centralizer of Ì ½ in Æ. The group ½ is a Levi -subgroup of a parabolic Ã-subgroup (it is also a maximal torus) of. Note that ½ õ Ì ½ õ and so for all facets in Ì ½ õ we have ½ õ ½ õ õ and ½ õ ½ õ õ. There exists an ¾ õ such that Ì ½ Ì ¾. Let denote the image of in µ. By hypothesis, Ì Ì. Thus, ¾ Æ Ì µ. Consequently, there exist an Ò ¾ Æ ÆÌ ½ õ õ and a ¾ õ such that Ò. We have Ì ¾ Ì ½ Ì ½. For ¾, let µ ½ µ; is a one-cocycle. We will show that µ ¾ ½ õ for all ¾. Fix ¾. Since is -stable and ¾ õ, we have µ ¾ õ. Since µ Ì ½ Ì ½, we have µ ¾ Æ Æ Ì ½ µ õ. Thus Ì ½ õ is µ-stable. If is an alcove in Ì ½ õ such that, then µ fixes point-wise and therefore µ fixes Ì ½ õ. Thus, we conclude that µ ¾ ½ õ. Since À ½ ½ õ µ is trivial, there exists Þ ¾ ½ õ such that Þ is fixed by. We have Þ Ì ½ Ì ¾ and Þ ¾ õ From tori over to tori over. Suppose ̵ ¾ Á Ø. Let ¼ be the facet in õ whose set of -fixed points is. In the final paragraph of the proof of [3, Proposition ] Bruhat and Tits use [7, Exp. XI, Cor. 4.2] to show that there exists a maximal Ã-split torus Ì of such that Ì is defined over, the apartment Ì Ãµ contains, and the image of Ì Ãµ õ ¼ in µ ¼ µ is Ì µ. We record this result in the following lemma. Lemma If ̵ ¾ Á Ø, then there exists a maximal Ã-split torus Ì of such that Ì is defined over, the apartment Ì Ãµ contains, and the image of Ì Ãµ õ in µ is Ì µ. 3. A PARAMETERIZATION OF CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI In this section, we present a parameterization of Ì via Bruhat-Tits theory.

7 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI Strong associativity. Following [10, 11], in [5, Ü2.3] the concept of strong associativity is developed. We recall the definition and some of its consequences. Definition Two facets ½ and ¾ of µ are strongly associated if for all apartments containing ½ and ¾, we have ½ µ ¾ µ Remark Two facets ½ ¾ of µ are strongly associated if and only if there exists an apartment containing ½ and ¾ such that ½ µ ¾ µ. See [5, Lemma 2.3.3]. Remark Suppose ½ and ¾ are strongly associated facets in µ. There is an identification of ½ with ¾. Namely, the natural -equivariant map õ ½ õ ¾ µ is surjective with kernel õ ½ õ ¾ õ ½ õ ¾ õ ½ õ ¾. See, for example, [5, Lemma 2.5.1]. Definition If ½ and ¾ are strongly associated facets in µ, then we denote the natural identification of ½ and ¾ introduced above by ½ ¾ An equivalence relation on Á Ø. We first consider the action of on Á Ø. Suppose ¾ and ̵ ¾ Á Ø. From Lemma there exists a maximal Ã-split torus Ì of such that Ì is defined over, the apartment Ì Ãµ contains, and the image of Ì Ãµ õ in µ is Ì µ. Define ̵ ̵ where Ì is the maximal -torus in whose group of -rational points coincides with the image of Ì Ãµ õ in µ. From Lemma 2.2.2, this definition is independent of the torus Ì we choose to represent Ì. We are now prepared to introduce a relation on Á Ø. Definition Suppose ½ Ì ½ µ and ¾ Ì ¾ µ are two elements of Á Ø. We will write ½ Ì ½ µ ¾ Ì ¾ µ provided that there exist an apartment in µ and ¾ such that (1) ½ µ ¾ µ and (2) Ì ½ Ì ¾ in ½ ¾. Lemma The relation on Á Ø is an equivalence relation. Proof. We will verify that the relation is transitive. The proofs that the relation is reflexive and symmetric are easier and left to the reader. Suppose Ì µ ¾ Á Ø for ½ ¾. Suppose ½ Ì ½ µ ¾ Ì ¾ µ and ¾ Ì ¾ µ Ì µ. We want to show ½ Ì ½ µ Ì µ. There exist ¾ ¾ and apartments ½¾ and ¾ in µ such that and (1) ½¾ ½ µ ½¾ ¾ ¾ µ (2) ¾ ¾ µ ¾ µ

8 8 STEPHEN DEBACKER (1) Ì ½ ¾ Ì ¾ in ½ (2) Ì ¾ Ì in ¾ ¾ ¾ Since ¾ ¾ ½¾ ¾ ¾, there exists an element ¾ ¾ ¾ such that ¾ ¾ ½¾. We have ½¾ ½ µ ½¾ ¾ ¾ µ ¾ ¾ ¾ ¾ µ ¾ ¾ ¾ µ ¾ ¾ µ ½¾ ¾ µ Moreover, we have that ½ ¾ ¾ ¾ surjects, under the natural map, onto ½ µ (resp., ¾ ¾ µ, resp., ¾ µ). Thus, there exists ¼ ¾ ½ ¾ ¾ ¾ such that Ì ½ ¾ Ì ¾ ¼ ¾ Ì ¾ ¼ ¾ Ì in ½ ¾ ¾ ¾ ¾ ¾ 3.3. A map from Á Ø to Ì. From Lemmas and 2.3.1, the following definition makes sense. Definition Suppose ̵ ¾ Á Ø. Let Ì be any maximal Ã-split torus in such that Ì is defined over, the apartment Ì Ãµ contains, and the image of Ì Ãµ õ in µ is Ì µ. Define ̵ ¾ Ì by setting ̵ equal to the -conjugacy class of Ì µ. Remark If ¾ and ̵ ¾ Á Ø, then ̵ ̵. Lemma The map from Á Ø to Ì which sends ̵ ¾ Á Ø to ̵ induces a well-defined map from Á Ø to Ì. Proof. Suppose ½ Ì ½ µ and ¾ Ì ¾ µ are two elements of Á Ø. We need to show that if ½ Ì ½ µ ¾ Ì ¾ µ, then ½ Ì ½ µ ¾ Ì ¾ µ. Since ½ Ì ½ µ ¾ Ì ¾ µ, there exist ¾ and an apartment in µ such that and ½ µ ¾ µ Ì ½ Ì ¾ in ½ ¾ From Remark 3.3.2, we can assume that ½. From Lemma there exists a maximal Ã-split -torus Ì ¾ of such that ¾ Ì ¾ õ and the image of Ì ¾ õ õ ¾ in ¾ µ coincides with Ì ¾ µ. Note that ¾ Ì ¾ µ is the -conjugacy class of Ì ¾ µ. It follows from Lemma (2) that we can choose ¾ ¾ such that Ì ¾ µ. Since ½ µ ¾ µ Ì ¾ µ, we conclude that ½ Ì ¾ µ. Let Ì ¼ denote the maximal -torus in ½ such that the image of Ì ¾ õ õ ½ in ½ µ coincides with Ì ¼ µ. We have Ì ¼ Ì ¾ in ½ ¾ and Ì ½ Ì ¾ in ½ ¾

9 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI 9 Thus, there exists ¼ ¾ ½ ¾ such that ¼ Ì ½ ¼ Ì ¾ Ì ¾ Ì ¼ in ½ ¾ ¾ ½ In other words, ¼ Ì ½ Ì ¼ in ½. We conclude from Lemma that ½ Ì ½ µ is the - conjugacy class of ¼ µ ½ Ì ¾ µ, i.e., ½ Ì ½ µ ¾ Ì ¾ µ A bijective correspondence. We now prove the main result of this paper. Theorem There is a bijective correspondence between Á Ñ and Ì given by the map sending ̵ to ̵. Proof. From Lemma 3.3.3, this map is well defined. From Lemma (3) and Lemma the map is surjective. It remains to show that the map is injective. Suppose ½ Ì ½ µ and ¾ Ì ¾ µ are pairs in Á Ñ such that ½ Ì ½ µ ¾ Ì ¾ µ. We need to show that ½ Ì ½ µ ¾ Ì ¾ µ. For ½ ¾, from Lemma we can choose a maximal Ã-split -torus Ì of such that the -conjugacy class of Ì µ is Ì µ, the apartment Ì Ãµ contains, and the image of Ì Ãµ õ in µ is Ì µ. Since ½ Ì ½ µ ¾ Ì ¾ µ, there exists a ¾ such that Ì ¾ Ì ½. Let Ì Ì ¾ Ì ½ and let Ì Ì µ. Note that both ½ and ¾ lie in Ì Ãµ Ì µ. Since ½ Ì ½ µ is an minisotropic pair, ½ is a maximal facet in Ì µ. Similarly, ¾ is a maximal facet in Ì µ. From Lemma (4), the facets ½ and ¾ are strongly associated. Since the image of Ì Ãµ õ ½ õ ¾ in ½ µ (resp., ¾ µ) is Ì ½ µ (resp., Ì ¾ µ), we have Ì ½ Ì ¾ in ½ ¾ 4. CONJUGACY CLASSES IN STABLE CONJUGACY CLASSES OF UNRAMIFIED STRONGLY REGULAR ELEMENTS In this section we assume that is quasi-finite and is Ã-split and connected. We recall that is quasi-finite provided that is perfect and is isomorphic to. Suppose that is a strongly regular semisimple element of (that is, a regular semisimple element of whose centralizer is connected) such that µ is a maximal unramified torus of. In this section, we provide an explicit description of the set of -conjugacy classes in the stable conjugacy class of. When is finite, this description is useful for harmonic analysis Some fixed notation for Section 4. Fix a maximal -split torus Ë of. Let be a maximal Ã-split -torus of containing Ë [3, Corollaire ]. Let Ï denote the Weyl group Æ µ õ õ. Choose a topological generator ¾ for. Since preserves, it acts on Ï. Two elements Û ½ Û ¾ ¾ Ï are -conjugate provided that there exists Û ¼ ¾ Ï such that Û ¼ ½ Û ½ Û ¼ µ Û ¾. This defines an equivalence relation on Ï, and the partitions associated to this equivalence relation are called -conjugacy classes.

10 10 STEPHEN DEBACKER 4.2. Conjugacy classes of maximal tori over quasi-finite fields. For finite fields, Carter [4, Ü3.3] establishes a natural bijection between the set of conjugacy classes of maximal tori in a finite group of Lie type and the set of Frobenius conjugacy classes in its absolute Weyl group. For the field صµ of Laurent series over the complex numbers in an indeterminate Ø, Kazhdan and Lusztig [9, Ü1] show that the analogue of Carter s result holds for صµ-split groups. Since finite fields and Laurent series over an algebraically closed field of characteristic zero are what Serre [14, XIII.Ü2] describes as the only non-pathological examples of quasi-finite fields, it is natural to ask if there is a uniform proof of this result. In this section, we answer this question in the affirmative. Let µ denote the group of -rational points of a connected reductive -group. From [14, XIII. Propositions 3 and 5] we have that the dimension of is less than or equal to one. Thus, from Steinberg s theorem (see, for example, [13, III.Ü2.3]), we have that À ½ µ is trivial and is -quasi-split. Fix a maximal -split torus Ì of. Since is -quasi-split, it follows that, the centralizer of Ì in, is a maximal -torus. We let Æ denote the normalizer of Ì in and identify its absolute Weyl group with Ï Æ. As above, we partition Ï into -conjugacy classes. Lemma There is a natural bijective correspondence between the set of µ-conjugacy classes of maximal -tori in and the set of -conjugacy classes in Ï. Proof (Mark Reeder). Let denote the -variety consisting of all maximal tori in. Since each maximal torus in is µ-conjugate to, we have the exact sequence ¼ Æ ¼ where the second to last map sends to. From [13, I. Proposition 36] we have the exact sequence (as pointed sets) ¼ Æ µ À ½ Ƶ À ½ µ Since À ½ µ is trivial, we conclude that the set of µ-conjugacy classes of maximal -tori in is parameterized by À ½ Ƶ. From [13, III.Ü2.3, Corollary 3] we have that the canonical map from À ½ Ƶ to À ½ ϵ is bijective. Since Ï is finite and is a topological generator for, for each Û ¾ Ï, the map Ñ Û Ûµ Ñ ½µ Ûµ defines a one-cocycle Û. Moreover, for Û Û ¼ in Ï we have that Û is cohomologous to Û ¼ if and only if Û and Û ¼ lie in the same -conjugacy class. We conclude that the map Û Û from the set of -conjugacy classes in Ï to À ½ ϵ is bijective. Remark Suppose ½ ¾ ¾ µ such that ½ and ¾ are maximal -tori. The above argument shows us that the maximal -tori ½ and ¾ are µ-conjugate if and only if the projections of ½ ½ ½ µ ¾ Æ and ½ ¾ ¾ µ ¾ Æ into Ï lie in the same -conjugacy class. Moreover, for each -conjugacy class in Ï, there is a ¾ µ for which the image in Ï of ½ µ ¾ Æ lies in the class and is a maximal -torus.

11 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI A parameterization of stable conjugacy classes of maximal unramified tori. We say that two maximal -tori Ì ½ and Ì ¾ of are stably conjugate provided that there exists ¾ µ such that Ì ½ µ Ì ¾ µµ. This is equivalent to saying that there exist ¾ µ and a strongly regular Ø ¾ Ì ¾ µ such that Ø ¾ Ì ½ µ. The statement of the following lemma was suggested to me by R. Kottwitz, and its proof proceeds very much like Carter s proof [4, Ü3.3] of the classification of the set of conjugacy classes of maximal tori in a finite group of Lie type. Lemma Suppose is quasi-finite and is Ã-split and connected. There is a natural injective map from the set of stable conjugacy classes of maximal unramified tori in to the set of -conjugacy classes in Ï. If is also -quasi-split, then this map is surjective. Proof. Suppose that Ì ( ½ ¾) is a maximal unramified torus in. From Hilbert s Theorem 90, if Ø ½ ¾ Ì ½ µ and Ø ¾ ¾ Ì ¾ µ are strongly regular elements which are conjugate by an element of µ, then Ø ½ and Ø ¾ are conjugate by an element of õ. Consequently, two maximal unramified tori Ì ½ and Ì ¾ of are stably conjugate if and only if there exists a ¾ õ such that Ì ½ µ Ì ¾ µµ. Since there is a single õ-conjugacy class of maximal Ã-split tori in, all maximal unramified tori of are õ-conjugate to. If ¾ õ and is a maximal unramified torus in, then µ µ. Consequently, ½ µ ¾ Æ µ õ. Note that if ¼ ¾ õ such that ¼ and is defined over, then there exists an Ò ¾ Æ µ õ such that ¼ Ò and so ½ µ Ò ½ ¼ ½ ¼ µ Òµ. In this way, we get a well-defined map from maximal unramified tori in to the set of -conjugacy classes in Ï : µ is the -conjugacy class of ½ µ õ. Suppose ¼ ¾ õ. We first show that if and ¼ are two stably conjugate maximal unramified tori in, then µ ¼ µ. Since and ¼ are stably conjugate, there exist an ¾ õ and strongly regular Ø ¾ õ such that Ø ¾ µ µ and Ø ¾ ¼ µ µ. This implies that ½ µ ¾ µ õ µ µ õ. Since µ ½ µ õ ½ ½ µµ µ õ ½ µµ µ ½ ½ µµ µ õ ½ µ õ we conclude that µ µ. Since ¼, from the previous paragraph we have µ ¼ µ. Consequently, µ ¼ µ. Therefore, we have a map from the set of stable conjugacy classes of maximal unramified tori of to the set of -conjugacy classes in Ï. We now show that this map is injective. Suppose ¼ ¾ õ such that and ¼ are maximal unramified tori in and µ ¼ µ. By replacing ¼ with ¼ Ò for some Ò ¾ Æ µ õ, we may assume that (1) ½ µ ¾ ¼ ½ ¼ µ õ

12 12 STEPHEN DEBACKER Fix a strongly regular element Ø ¾ µ µ. It will be enough to show that ¼ ½ Ø ¾ ¼ µ µ. Thus, it is enough to show (2) ¼ ½ Ø ¼ µ µ ½ Ø However, Equation (2) is valid if and only if ½ Ø ¼ ½ ¼ µµ µ ½ Ø Since µ ½ Ø ¾ õ, it follows from Equation (1) that the map is injective. Finally, we must show that when is unramified, the map is surjective. Let Ü be a hyperspecial point in the apartment of Ë in red µ. Let be the maximal -split -torus in Ü corresponding to. Note that Ï is naturally isomorphic to Æ Ü µ µ µ. From Remark for every -conjugacy class Ç in Ï, there exists a ¾ Ü µ such that is a maximal -torus in Ü and the image of ½ µ in Ï lies in Ç. As in Lemma we can lift to a maximal unramified torus Ì ¼ in with Ü ¾ Ì ¼ µ Ì ¼ õ. Since Ü ¾ Ì ¼ õ õ, there exists a ¾ õ Ü such that Ì ¼. Let denote the image of in Ü µ. Since, from Remark the image of ½ µ, and hence that of ½ µ, lies in Ç. For Ì µ ¾ Á Ñ for ½ ¾, the proof of Lemma yields a simple criterion for determining if ½ Ì ½ µ and ¾ Ì ¾ µ lie in the same stable conjugacy class. Without loss of generality, we assume that ½ ¾ Ë µ. Let denote the maximal -torus in corresponding to. Let Ï Æ µ µ µ. Let Ç Ì µ be the -conjugacy class in Ï parameterizing the µ-conjugacy class of Ì µ. Let Ç Ì µ be the -conjugacy class in Ï obtained by lifting Ç Ì µ into Æ µ õ and then projecting onto Ï. Note that we have a natural embedding of Ï in Ï and of Ç Ì µ in Ç Ì µ. Corollary Suppose is quasi-finite and is Ã-split and connected. In the notation introduced above, we have ½ Ì ½ µ and ¾ Ì ¾ µ lie in the same stable conjugacy class if and only if Ç Ì ½ µ Ç Ì ¾ µ 4.4. Stable conjugacy classes in an unramified maximal torus. Suppose ¾ is a strongly regular semisimple element such that Ì µ is a maximal unramified torus of. Set Ë Ì µ Ì This is a finite set. For ¼ ¾ Ë Ì, we write ¼ provided that there is a ¾ such that ¼ ; this defines an equivalence relation on Ë Ì. In this subsection we give an explicit description of Ë Ì Definition Suppose is a facet in Ë µ. Let Ï µ denote the image in Ï of the stabilizer (not the fixator) in Æ µ õ of õ µ. For Û ¾ Ï we define the subgroups Ï ÛÆ Û ¼ ¾ Ï Û Û ¼ µµ Û ¼ and Ï µ ÛÆ Ï µ Ï ÛÆ

13 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI 13 of Ï. Lemma Suppose is quasi-finite and is Ã-split and connected. Choose ̵ ¾ Á Ñ such that Ë µ. Fix Ì ¾ ̵. For a strongly regular element ¾ Ì, we have that Ë Ì is in (natural) bijective correspondence with the coset space Ï Û ÌÆÏ µ Û ÌÆ Here Û Ì is any element of Ç Ìµ Ç Ìµ Ï (notation introduced prior to Corollary 4.3.2). Remark If the image of in red µ is a hyperspecial vertex, then Ï µ Ï and so µ Ì Ì Proof. If we replace Û Ì ¾ Ç Ìµ with a -conjugate, say Û ¼ Ì Û¼ ½ Û Ì Û ¼ µ for some Û ¼ ¾ Ï Ï, then Û ¼ Ï Û ¼ Ì Æ Ï Û ÌÆ and similarly Û ¼ Ï µ Û ¼ Ì Æ Ï µ Û ÌÆ Thus, it is enough to show that the bijection works for a single element of Ç Ìµ. Let Ì denote the maximal unramified torus of for which Ì Ì µ. From Hilbert s Theorem 90 we have µ Ì Ãµ Ì Therefore, the map Ò Ò ½ from Æ Ìµ õ to Ì Ãµ induces a bijective correspondence between the coset space Æ Ìµ ÃµÌ Ãµ Æ Ì µì and Ë Ì. (Here we think of Æ Ì µì as a subgroup of Æ Ìµ ÃµÌ Ãµ via the injection induced from the natural embedding of Æ Ì µ in Æ Ìµ õ.) Let Å be the Levi subgroup of generated by µ and the root groups Í for affine roots of with respect to Ë,, and which are constant on. Let Å denote the Levi - subgroup of a parabolic -subgroup for which Å Å µ. We have Šõ Šõ õ õ. Consequently, the pair ̵ occurs in the analogue of Á Ñ for Å and we may assume that Ì Å. From Lemma (2) and (3) we may also assume that Ì µ Ë µ µ Ë µ õ õ. Since Ì µ Ì Ãµ Ì Ãµ õ there is an Ñ Ì ¾ Šõ such that Ì Ñ Ì. Set Ò Ì ÑÌ ½ Ñ Ìµµ ¾ Æ µ õ and let Û Ì denote the image of Ò Ì in Ï. In the notation introduced prior to Corollary we have Û Ì ¾ Ç Ìµ. Note that both Ñ Ì and Ò Ì lie in õ Ý for all Ý ¾ Ì µ õ µ. To finish, we show that the group isomorphism Ò Ñ Ì ½ Ò from Æ Ìµ õ to Æ µ õ induces group isomorphisms Æ Ìµ ÃµÌ Ãµ Ï Û ÌÆ

14 14 STEPHEN DEBACKER and Æ Ìµ ÃµÌ Ãµ Æ Ì µì Ï µ Û ÌÆ We first show that is isomorphic to Ï Û ÌÆ. Choose Û ¾ Ï and let Ò ¾ Æ µ õ be a representative for Û. It is enough to show that (3) Û Û Ì Ûµ if and only if ÑÌ Òµ ½ ÑÌ Òµ ¾ Ì Ãµ Note that Û Û Ì Ûµ if and only if Ûµ ½ Û Ûµ ½. This last equality is equivalent to Òµ ½ Ò Ì ½ Òµ ¾ õ which happens if and only if ÑÌÒµ ½ ÑÌÒµ ¾ Ì Ãµ. We now show that Æ Ì µì is isomorphic to Ï µ Û ÌÆ. Since the map induced from ÑÌ ½ - conjugation carries Æ Ì µì into Ï µ Û ÌÆ, it is enough to show that for each Û ¾ Ï µ Û ÌÆ ½ there is an Ò ¼ ¾ Æ Ì µ such that Ñ Ì Ò ¼ µ õ Û. Fix Û ¾ Ï µ Û ÌÆ. Since Û ¾ Ï µ, there is an Ò ¾ Æ µ õ such that the image of Ò in Ï is Û and Ò, and hence Ñ ÌÒ ¾ Æ Ìµ õ, stabilizes Ì µ. Fix Ý ¾ Ì µ. We have Ì ½ ÑÌ Òµ Ý ÑÌ Òµ ݵ ÑÌ Òµ Ý Hence ÑÌÒµ ½ ÑÌÒµ ¾ Ü Ãµ ݵ. Since Û ¾ Ï Û ÌÆ, from Equation (3) we have ÑÌÒµ ½ ÑÌÒµ ¾ Ì Ãµ. Therefore ÑÌ Òµ ½ ÑÌ Òµ ¾ Ü Ãµ ݵ Ì Ãµ Ì Ãµ Ý Since À ½ Ì Ãµ Ý µ and À½ ̵ are both trivial, we conclude that À ½ Ì Ãµ Ý µ is trivial. Consequently, there exists a Ø ¾ Ì Ãµ Ý such that ÑÌÒµØ ¾ Æ Ì µ. Let Ò ¼ ÑÌÒµØ The result. The proof of the following theorem is immediate from the results of the previous two subsections. Theorem Suppose is quasi-finite and is Ã-split and connected. Choose ¼ Ì ¼ µ ¾ Á Ñ such that ¼ Ë µ. Fix Ì ¼ ¾ ¼ Ì ¼ µ. For a strongly regular element ¾ Ì ¼, the set of -conjugacy classes in µ is parameterized by the set of pairs Û Here ¾ Á Ñ is represented by some ̵ ¾ Á Ñ with Ë µ and Ç Ìµ Ç Ì ¼ µ, and if we fix Û Ì ¾ Ç Ìµ, then Û runs over the cosets in Ï Û ÌÆÏ µ Û ÌÆ. 5. A PARAMETERIZATION FOR MAXIMAL RANK UNRAMIFIED SUBGROUPS OF We return to our original assumptions on and. This section presents a generalization of the material in ÜÜ2 3. As a by-product of the way in which this paper was written, the material presented below borrows heavily from the presentation there.

15 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI Definitions. We recall that a subgroup À of is called unramified if À is the group of -rational points of a connected reductive Ã-split -subgroup À of and red À µ contains a hyperspecial point. We shall also call such an À an unramified subgroup of. Let Á denote the set of pairs Àµ where is a facet in µ and À is a maximal rank connected reductive -subgroup in. Note that Á Ñ Á Ø Á. We also consider the subset Á Á of cuspidal pairs in Á. A pair Àµ ¾ Á is said to be cuspidal if the maximal -split torus in À coincides with the maximal -split torus in the center of. (Equivalently, À is cuspidal in if and only if À lies in no proper parabolic -subgroup of.) Note that Á Ñ Á Á Ø Maximal rank subgroups over and. In this subsection we show how to move between maximal rank unramified -subgroups of and maximal rank connected reductive subgroups over. The following Lemma follows immediately from Lemma Lemma Suppose À is a maximal rank connected reductive Ã-split -subgroup of. Every maximal Ã-split -torus in À is a maximal Ã-split torus in From maximal rank unramified subgroups of to connected reductive -groups. Suppose that À is a maximal rank unramified subgroup of. We identify À õ with its image in õ. (As usual, there does not exist a canonical embedding of À õ in õ, but the image of any natural embedding is independent of the embedding.) We therefore have We now collect some facts about Àµ. Àµ À õ õ µ Lemma Suppose À is a maximal rank unramified subgroup of. Let À denote the group of -rational points of À. (1) Àµ is a nonempty, closed, convex subset of µ. Moreover, Àµ is the union of the facets in µ which meet it. (2) For all facets in Àµ, there exists Àµ ¾ Á such that the image of À õ õ in µ is À µ. Moreover, if is a maximal facet in the preimage in Àµ of a facet in red Àµ, then Àµ ¾ Á. (3) If ½ and ¾ are maximal facets in the preimage in Àµ of a facet in red Àµ, then for all apartments in µ containing ½ and ¾ we have ½ µ ¾ µ. Proof. (1) : The proof here is identical to that of Lemma (1). (2) : Suppose is a facet in Àµ. Let À be the maximal rank connected reductive - subgroup in corresponding to the image of À õ õ in µ. We have Àµ ¾ Á. Now suppose that is a maximal facet in the preimage in Àµ of a facet in red Àµ. Choose a subgroup À in as in the previous paragraph. Let Ë ¼ be a maximal -split torus in À so that Ë ¼ µ À µ. Let Ë be a maximal -split torus in such that Ë ¼ Ë. Let À denote the maximal -split torus in the center of À and let À denote the -split torus whose group of -rational points coincides with the image of À õ õ in µ. We have À Ë ¼ Ë

16 16 STEPHEN DEBACKER and we may, in the natural way, identify À µ Ë ¼ µ Ë µ Ë µ Since is a maximal facet in the preimage in Àµ red Àµ À µ of a facet in red Àµ, for all affine roots of with respect to Ë,, and, if is constant on, then is constant on À µ. Therefore, À is contained in the center of and so À cannot lie in a proper parabolic -subgroup of. (3) : This is clear. Remark In the notation used in the proof of (2) above we have that the torus À is the maximal -split torus in the center of exactly when is a maximal facet in the preimage in Àµ of a vertex in red Àµ. The previous lemma gives us a way to associate to a pair À ܵ, with À a maximal rank unramified subgroup in and Ü ¾ red Àµ a hyperspecial vertex, an element of Á From subgroups over to unramified subgroups over. Lemma Suppose À ½ and À ¾ are maximal rank unramified subgroups of. If is a - invariant facet in À ½ õ À ¾ õ which projects to a -fixed special vertex in red À ½ õ (resp. red À ¾ õ) and the images of À ½ õ õ and À ¾ õ õ in µ coincide, then À ½ and À ¾ are -conjugate. Proof. Let À denote the maximal rank reductive subgroup in whose group of -rational points is the image of À ½ õ õ in µ. Note that À is defined over. Let Ì be a maximal -torus in À which contains a maximal -split torus of À. Let À ̵ denote the -root system of À with respect to Ì. As in Lemma 2.3.1, we choose a maximal Ã-split -torus Ì in À lifting Ì. Note that, since the image of in red À µ is hyperspecial, À is completely determined by À ̵ and Ì. From Lemma 2.2.2, we conclude that À ½ and À ¾ are -conjugate. Lemma Suppose is a facet in µ and À is a maximal rank connected reductive -subgroup of. There exists a maximal rank unramified subgroup À in such that: (1) The facet belongs to À µ. (2) The image of À õ õ in µ is the group of -rational points of À. (3) The image of in red À µ is a hyperspecial vertex, Ü. Proof. Let Ì be a maximal -torus in À (and hence in ) which contains a maximal -split torus of À. Let À ̵ denote the -root system of À with respect to Ì. As in Lemma 2.3.1, let Ì be a lift of Ì to a maximal Ã-split -torus Ì in. We think of À ̵ as a subset of ̵, the root system of with respect to Ì. Let À be the Ã-split full rank subgroup of whose group of Ã-rational points is generated by Ì Ãµ and the root groups in õ corresponding to elements of À ̵. Note that À is defined over, and, by construction, the image of À õ in red À õ is a -fixed special vertex. The lemma follows. Remark If, in the statement of Lemma 5.2.5, À is also assumed to be -cuspidal, then is a maximal facet in the preimage in À µ of Ü.

17 PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI A parameterization of maximal rank unramified subgroups. Recall that denotes the set of -conjugacy classes of pairs À ܵ where À is a maximal rank unramified subgroup in and Ü is a hyperspecial point in red Àµ. In this subsection, we present a parameterization of via Bruhat-Tits theory An equivalence relation on I. Suppose ¾ and Àµ ¾ Á. From Lemma there exists a maximal rank unramified subgroup À of such that the building of Àµ contains, the image of À õ õ in µ is À µ, and the image of in red Àµ is a hyperspecial vertex. Define Àµ Àµ where À is the maximal rank connected reductive -group in whose group of -rational points coincides with the image of À õ õ in µ. From Lemma 5.2.4, this definition is independent of the unramified subgroup À of we choose to represent À. Definition Suppose ½ À ½ µ and ¾ À ¾ µ are two elements of Á. We will write ½ À ½ µ ¾ À ¾ µ provided that there exist an apartment in µ and ¾ such that (1) ½ µ ¾ µ and (2) À ½ À ¾ in ½ ¾. The proof of the following lemma is nearly identical to the proof of Lemma Lemma The relation on Á is an equivalence relation A map from Á to. From Lemmas and 5.2.5, the following definition makes sense. Definition Suppose Àµ ¾ Á. Let À be any maximal unramified subgroup of such that the building À õ contains, the image of À õ õ in µ is À µ, and the image of in red À µ is a hyperspecial vertex Ü. Define Àµ ¾ by setting Àµ equal to the -conjugacy class of the pair À µ Ü µ. Remark If ¾ and Àµ ¾ Á, then Àµ Àµ. Lemma The map from Á to which sends Àµ ¾ Á to Àµ induces a well-defined map from Á to. Proof. Suppose ½ À ½ µ and ¾ À ¾ µ are two elements of Á. We need to show that if ½ À ½ µ ¾ À ¾ µ, then ½ À ½ µ ¾ À ¾ µ. Since ½ À ½ µ ¾ À ¾ µ, there exist ¾ and an apartment in µ such that and ½ µ ¾ µ À ½ À ¾ in ½ ¾ From Remark 5.3.4, we can assume that ½.

18 18 STEPHEN DEBACKER From Lemma there exists a maximal rank unramified subgroup À ¾ of such that ¾ À ¾ õ, the image of À ¾ õ õ ¾ in ¾ µ coincides with À ¾ µ, and the image of ¾ in red À µ is hyperspecial. Note that ¾ À ¾ µ is the -conjugacy class of À ¾ µ Ü ¾ µ where Ü ¾ is the image of ¾ in red Àµ. Let Ì ¾ be a maximal -torus in À ¾. From Lemma we can choose a maximal Ã-split -torus Ì ¾ in À ¾ lifting Ì ¾ such that ¾ Ì ¾ µ. It follows from Lemma (2) that we can choose ¾ ¾ such that Ì ¾ µ. Since ½ µ ¾ µ Ì ¾ µ, we conclude that ½ Ì ¾ µ À ¾ µ. Let À ¼ denote the maximal rank connected reductive -subgroup in ½ such that the image of À ¾ õ õ ½ in ½ µ coincides with À ¼ µ. We have À ¼ À ¾ in ½ ¾ and À ½ À ¾ in ½ ¾ Thus, there exists ¼ ¾ ½ ¾ such that ¼ À ½ ¼ À ¾ À ¾ À ¼ in ½ ¾ ¾ ½ In other words, ¼ À ½ À ¼ in ½. We conclude from Lemma that ½ À ½ µ is the - conjugacy class of ¼ µ ½ À ¾ µ, i.e., ½ À ½ µ ¾ À ¾ µ A bijective correspondence. We now prove the final result of this paper. Theorem There is a bijective correspondence between Á and given by the map sending Àµ to Àµ. Proof. From Lemma 5.3.5, this map is well defined. From Lemma (2) the map is surjective. It remains to show that the map is injective. Suppose ½ À ½ µ and ¾ À ¾ µ are pairs in Á such that ½ À ½ µ ¾ À ¾ µ. We need to show that ½ À ½ µ ¾ À ¾ µ. For ½ ¾, from Lemma we can choose a maximal unramified subgroup À in such that the building À õ contains, the image of À õ õ in µ is À µ, the image of in red Àµ is a hyperspecial vertex Ü, and the -conjugacy class of the pair À µ Ü µ is À µ. Since ½ À ½ µ ¾ À ¾ µ, there exists a ¾ such that À ¾ À ½ and Ü ¾ Ü ½ in red À ½ µ. Let À À ¾ À ½ and let À À µ. Note that both ½ and ¾ lie in Àµ. Moreover, both ½ and ¾ lie in the preimage in Àµ of Ü ½ ¾ red Àµ. Since ½ À ½ µ is an -cuspidal pair, from Remark we have that ½ is a maximal facet in the preimage in Àµ of Ü ½ ¾ red Àµ. Similarly, ¾ is a maximal facet in the preimage in Àµ of Ü ½ ¾ red Àµ. From Lemma (3), the facets ½ and ¾ are strongly associated. Since the image of À õ õ ½ õ ¾ in ½ µ (resp., ¾ µ) is À ½ µ (resp., À ¾ µ), we have À ½ À ¾ in ½ ¾

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