Reductive groups over a local field.
(Groupes réductifs sur un corps local.)

*(French)*Zbl 0254.14017[Joint “Looking back” review with Zbl 0597.14041.]

The two articles under review contain the foundations of the theory of Bruhat-Tits buildings.

When Bruhat and Tits developed the theory of the buildings now bearing their names, the structure theory of reductive algebraic groups over an arbitrary field (as an algebraic avatar of the theory of Lie groups) was already quite well understood. The foundations of this theory were laid by Armand Borel, Claude Chevalley, Jacques Tits and many others.

Bruhat and Tits embarked on the project to understand reductive algebraic groups over a field \(K\) with a non-archimedean absolute value. Their goal was to define a new geometric object taking into account the valuation on the ground field, which might be seen as a \(p\)-adic avatar of the Riemannian symmetric space \(G/K\) associated to a semisimple real Lie group \(G\) and a maximal compact subgroup \(K\).

Note that a non-archimedean absolute value satisfies the triangle inequality in the following strong form

\[ |a+b| \leq \max\{|a|, |b|\}. \]

One example of a field with a non-archimedean absolute value is the field of formal Laurent series \(k((X))\) over an arbitrary ground field \(k\), which is endowed with the absolute value \(|\sum_{n \geq n_0} a_n X^n| = e^{-n_0}\) if \(a_{n_0} \neq 0\). The field \(K = k((X))\) is discretely valued, which means that the value group \(|K^\times|\) is a discrete subgroup of \(\mathbb{R}\).

Other prominent examples of fields with a non-archimedean absolute value are the completions \(\mathbb{Q}_p\) of \(\mathbb{Q}\) with respect to the \(p\)-adic absolute value. These fields are local, i.e., they are locally compact in the topology induced by the absolute value. The fields \(\mathbb{Q}_p\) and their extensions are important for local questions in number theory.

However, the topology induced by a non-archimedean absolute value has disadvantages. The fields \(k((X))\) and \(\mathbb{Q}_p\) are for example totally disconnected, i.e., the only connected subsets are the empty set and the one-point sets. These topological flaws create difficulties if one seeks for non-archimedean analogies of archimedean constructions. In the case of Bruhat-Tits buildings this explains why a new object had to be constructed: The quotient space of a semisimple group by a maximal compact subgroup, which works fine in the archimedean world, is not an interesting topological space in the non-archimedean world.

Now we consider a reductive algebraic group \(\mathcal{G}\) over a field \(K\) with a non-archimedean valuation. Recall that a linear algebraic group is a group variety over \(K\) which can be embedded in the \(K\)-variety given by a general linear group. It is called reductive if it does not contain any non-trivial connected unipotent normal subgroup. If it does not even contain any non-trivial connected solvable normal subgroup, then \(\mathcal{G}\) is called semisimple. The general linear group is reductive. The special linear group, the symplectic group or the special unitary group are examples of semisimple groups. Since for many interesting questions it is crucial to work with fields \(K\) which are not algebraically closed, one has to work with group schemes rather than groups. A group scheme over \(K\) can be thought of as an object encoding information on all groups \(\mathcal{G}(L)\), where \(L\) runs through the extension fields of \(K\).

Before Bruhat and Tits developed their general theory, O. Goldman and N. Iwahori had constructed a space of non-archimedean norms which afterwards turned out to be the Bruhat-Tits building associated to the general linear group [“The space of \(p\)-adic norms,” Acta Math. 109, 137–177 (1963; Zbl 0133.29402)]. Besides, N. Iwahori and H. Matsumoto [“On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups,” Publ. Math. IHES 25, 5–48 (1965; Zbl 0228.20015)] had investigated split groups before Bruhat and Tits embarked on the general theory.

The Bruhat-Tits building associated to a reductive group \(\mathcal{G}\) over a field with a non-archimedean absolute value is a metric space endowed with a continuous action by the \(K\)-rational points \(\mathcal{G}(K)\). If the valuation on \(K\) is discrete, then it carries a simplicial structure. Its existence depends on rather general hypotheses. For example, Bruhat-Tits buildings exist if the ground field \(K\) is discretely valued and henselian with a perfect residue field. Moreover, they always exist for split groups, i.e., for reductive groups \(\mathcal{G}\) such that the maximal split torus is defined over \(K\). Note that the Bruhat-Tits building of a reductive group coincides with the building of its semisimplification. However, there is a notion of extended buildings which takes into account the difference between these two groups.

Part I of the paper under review develops the fundamental theory of buildings from two different axiomatic angles. The application to reductive groups is postponed to part II. Let us describe in more detail the content of part I. In the first five sections a building (“immeuble”) is associated to an affine Tits system, which is the first axiomatic approach to buildings. In sections six to nine the second axiomatic approach via valued root data is developed. Both theories overlap to a certain extent, but not completely.

The first section recalls some facts on Coxeter groups, Tits systems and affine Weyl groups, relying on the Bourbaki volume Lie groups and Lie algebras, chapter 4 to 6. A Tits system is a quadruplet \((G,B,N,S)\) consisting of a group \(G\) together with two subgroups \(B\) and \(N\) generating \(G\) such that \(B \cap N\) is a normal subgroup in \(N\). The quotient group \(W = N / (B \cap N)\) is also called Weyl group. \(S\) is a set of involutions generating the Weyl group such that the two following conditions hold.

i) \(sBw \subset BwB \cup BswB\) for all \(s \in S\) and \( w \in W\),

ii) \(sBs \neq B\) for all \( s \in S\).

To avoid confusion note that \(G\) here is a true group and not a group scheme. The group \(G = \mathcal{G}(K)\) of \(K\)-rational points of a reductive group \(\mathcal{G}\) over \(K\) carries the structure of a Tits system. As an example, consider the general linear group \(G\) of rank \(n\) together with the subgroup \(B\) of upper triangular matrices and the subgroup \(N\) of matrices with precisely one non-zero entry in every line and column. Here \(B \cap N\) is equal to the subgroup of diagonal matrices, and \(W\) can be identified with the symmetric group of \(n\) elements. Under this identification the set \(S\) corresponds to the set of transpositions of \(i\) and \(i+1\) for all \(i = 1, \ldots, n-1\).

Tits systems with finite Weyl group were studied before. They give rise to the so-called spherical buildings or Tits buildings, which can be defined as the complex of all parabolic subgroups. This theory was developed by Jacques Tits in order to find geometric interpretations for linear algebraic groups over arbitrary fields, see [J. Tits, “Buildings of spherical type and finite \(BN\)-pairs.” Lecture Notes in Mathematics 386, Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0295.20047)]. It does not take into account the non-archimedean absolute value. In fact, for a reductive group over a field with a suitable non-archimedean absolute value, there exist two buildings: the Tits (or spherical) building and the Bruhat-Tits building. They are related since the Bruhat-Tits building can be compactified by attaching a Tits building “at infinity”.

In the case of ground fields with a non-archimedean valuation, one has to consider affine Tits systems, where the Weyl group \(W\) is an infinite group of affine reflections in a Euclidean space \(A\). The corresponding affine hyperplane arrangement defines a decomposition of \(A\) into faces. Faces of maximal dimension are called chambers.

In section two the building \(\mathcal{I}\) associated to an affine Tits system is defined. It is a polysimplicial complex, i.e., a product of simplicial complexes, whose faces correspond to the so-called parahoric subgroups of \(G\). It can be described as a union of Euclidean spaces, which are called apartments, and it carries a metric and an action of the group \(G\).

In the third section an important fixed-point theorem is proven, stating that the set-wise stabilizer of a bounded subset of the building \(\mathcal{I}\) has a fixed point. This is used to show that the bounded subgroups of \(G\) are cum grano salis the stabilizers of faces in the building. Since the notion of Tits systems does not a priori contain topological data, the definition of bounded subgroups is more involved. In the fourth section Iwasawa and Cartan decompositions of \(G\) are proven with the help of the building. Section five analyses double Tits systems, leading to an affine and to a spherical Tits system. This reflects the fact that there are two buildings associated to a reductive group over a non-archimedean field, as was explained above.

In section six the second approach to buildings is prepared by the definition of a valued root datum. Here we fix a root system \(\Phi\) in the dual space of a vector space \(V\). A root datum in a group \(G\) consists of a subgroup \(T\) together with data \(({U}_a, M_a)\) for every root \(a \in \Phi\) such that the \(U_a\) are subgroups satisfying certain commutator conditions. Each \(M_a\) is a coset of \(T\) satisfying among other conditions the important relation \(U_{-a} \backslash \{1\} \subset U_a M_a U_a\). The full list of axioms can be found in Definition (6.1.1). From these axioms a list of further properties of \(G\) is derived. As a motivating example consider the group \(G = SL_2(K)\) over any field \(K\), together with the subgroup \(T\) of diagonal matrices and the root system \(\Phi= \{a,-a\}\) of type \(A_1\). Let \(U_a\) be the unipotent subgroup of upper triangular matrices with entries \(1\) on the diagonal, and let \(U_{-a}\) be the unipotent subgroup of lower triangular matrices with entries \(1\) on the diagonal. Put

\[ m = \left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right) \]

and \(M_a = M_{-a} = Tm\). These data form a root datum. In particular, by a straightforward calculation one can check that \(U_{-a} \backslash \{1\} \subset U_a M_a U_a\) holds in this case.

Bruhat and Tits then define the notion of a valuation on a root datum. This is a family of maps \(\varphi_a: U_a \rightarrow \mathbb{R} \cup \{\infty\}\) which are compatible with the structure of a root datum. The full list of axioms is stated in Definition (6.2.1). In particular, the truncated subsets \(U_{a,r} = \varphi_a^{-1} ([r, \infty])\) are required to form a subgroup of \(U_a\).

Let us again consider the example \(SL_2(K)\) and assume that \(K\) is endowed with a non-trivial valuation map \(\omega: K \rightarrow \mathbb{R} \cup \{\infty\}\). Then we can define a valuation on the root datum defined above by setting

\[ \varphi_a \left( \left(\begin{matrix} 1 & u \\ 0 & 1 \end{matrix} \right) \right) = \varphi_{-a} \left( \left(\begin{matrix} 1 & 0 \\ u & 1 \end{matrix} \right) \right) = \omega (u). \]

More generally, it is shown that valued root data exist for all groups \(\mathcal{G}(K)\), where \(\mathcal{G}\) is a split reductive \(K\)-group.

Now Bruhat and Tits fix a root datum and a valuation on it. In (6.2.5) they define an affine space \(A\) under the vector space \(V\) (coming from the root datum) as the set of all valuations on the root datum depending on the fixed one in an affine-linear way. The precise notion here is “valuations équipollentes”. This affine space carries a natural action by the group \(N\) generated by \(T\) and the union of the \(M_a\). Note that in our example \(G = SL_2(K)\) the group \(N\) is the normalizer of \(T\), i.e., the group of monomial matrices in \(G\). Now one can define an arrangement of affine hyperplanes in \(A\). The hyperplane directions are given by the roots, and the set of translations leading to a family of parallel hyperplanes is basically given by the image of the valuation map.

Note that in our example \(G= SL_2(K)\), the space \(A\) is one-dimensional, and the affine hyperplane arrangement simply corresponds to the image of the valuation map \(\omega(K^\ast)\).

In section 6.5 it is shown that if all valuation maps \(\varphi_a\) have discrete image, then the notion of a valuation on a root datum gives rise to an affine Tits system. This provides the link to the theory developed in the first five sections. Hence in the discrete case there is an associated building.

Also in the general case of non-discrete valuation maps, one can define a building. This is the topic of section seven. For every point \(x\) in \(A\) and every root \(a\) one determines the minimal affine halfspace with direction \(a\) in our affine hyperplane arrangement which contains \(x\). This corresponds to an element \(r\) in the image of the valuation map, and hence to a subgroup \(U_{a,r}\) of \(U_a\). Now let \(P_x\) be the subgroup of \(G\) generated by all these \(U_{a,r}\) together with the elements in \(N\) fixing \(x\). Then in (7.4.1) the building \(\mathcal{I}\) is defined as the quotient of \(G \times A\) after the following equivalence relation: \((g,x) \sim (h,y)\) if and only if there exists an \(n \in N\) such that \(nx=y\) under the action of \(N\) on \(A\) and \(g^{-1} hn \in P_x\).

This space carries a natural \(G\)-action by multiplication in the first factor. Besides, the polysimplicial structure on \(A\) can be extended to a polysimplicial structure on \(\mathcal{I}\). To be precise, all faces in \(\mathcal{I}\) are of the form \(g F\), where \(g\) is an element in \(G\) and \(F\) is a face in \(A\). Every subset of the form \(gA\) in \(\mathcal{I}\) is called an apartment. It is shown in Theorem (7.4.18) that any two faces are contained in one apartment. Note that there is a distance function on \(A\) given by the choice of a scalar product associated to the root system. We transfer it in a \(G\)-invariant way to the other apartments. Since any two points in \(\mathcal{I}\) are contained in one apartment, this defines a \(G\)-invariant distance function on the whole building. It is well-defined, and gives a metric on the building \(\mathcal{I}\). The building is contractible in the associated topology, see Proposition (7.4.20).

Moreover, Iwasawa and Bruhat decompositions of \(G\) are proven in section 7.3. In section eight it is shown that for dense valuations the maximal bounded subgroups of \(G\) in a suitable sense are precisely the stabilizers of points. Section nine provides descent results for valued root data which will be used in part II of the paper. In section ten, classical groups over a field with a complete non-archimedean absolute value are investigated. It is shown how their natural root datum can be equipped with a valuation in the above sense. Note that more details on the buildings associated to classical groups may be found in the articles “Schémas en groupes et immeubles des groupes classiques sur un corps local” [Bull. Soc. Math. Fr. 112, 259–301 (1984; Zbl 0565.14028)] and “Schémas en groupes et immeubles des groupes classiques sur un corps local. II: Groups unitaires” [Bull. Soc. Math. Fr. 115, 141–195 (1985; Zbl 0636.20027)] by the same authors.

The purpose of part II of the paper under review is to show that under rather general assumptions the group \(\mathcal{G}(K)\) of \(K\)-rational points of a reductive connected group \(\mathcal{G}\) over a field \(K\) with a non-trivial non-archimedean absolute value actually possesses a valued root datum which is compatible with the valuation on \(K\). Hence the results of part I can be applied to these groups, thereby proving the existence of Bruhat-Tits buildings. In particular, the fixed point and decomposition results shown in part I hold in this case.

The construction of a valued root datum is achieved in a double descent process, already prepared by section nine of part I. The first descent step is used to proceed from split groups to quasi-split groups, the second descent step deals with the passage from strictly henselian to henselian fields. This is useful since a result of Steinberg states that a reductive group over a discretely valued henselian field \(K\) with perfect residue field becomes quasi-split after base-change with the strict henselisation.

An important technical tool in part II is the construction of group schemes over the ring of integers \(R\) in \(K\) such that their generic fiber is \(\mathcal{G}\) or one of its subgroups. We refer to a scheme over \(R\) also as a model of its generic fiber.

Part II starts with a section recalling general facts in algebraic geometry which are used in the following sections.

In section two, Bruhat and Tits investigate the properties of models for root groups and their products. In section three, this leads to the definition of a “donnée radicielle schématique” or schematic root datum. Let \(\mathcal{Z}\) be the centralizer of the maximal split torus in \(\mathcal{G}\), and denote by \(\mathcal{U}_a\) the root group associated to a root \(a\). Then a schematic root datum is a collection of models of the groups \(\mathcal{Z}\) and \(\mathcal{U}_a\) satisfying a list of conditions mainly stating that some natural maps extend from the generic fibers to the whole models. For details see Definition (3.1.1).

In section four quasi-split groups \(\mathcal{G}\) are considered. In this case there exists a Borel group in \(\mathcal{G}\) which is defined over \(K\), and the base change \(\mathcal{G}_{\tilde{K}}\) of \(\mathcal{G}\) to a Galois extension \(\tilde{K}\) of \(K\) is a split group. By a careful analysis of the effect of the Galois group of \(\tilde{K}/K\) it is shown that the valued root datum for \(\mathcal{G}_{\tilde{K}}(\tilde{K})\) constructed in part I can be descended to a valued root datum for \(\mathcal{G}(K)\). Section five deals with a reductive group \(\mathcal{G}\) over a henselian (for example complete) field \(K^\natural\). If the base change \(\mathcal{G}_K\) of \(\mathcal{G}\) to the strict henselisation \(K\) is quasi-split (and two other conditions hold, see (5.1.1)), then the valued root datum on the group \(\mathcal{G}_{K}(K)\) can be descended to \(\mathcal{G}(K^\natural)\). In this case the Bruhat-Tits building associated to \(\mathcal{G}\) can be identified with the fixed-point set of the building associated to \(\mathcal{G}_{K}\) with respect to the action of the Galois group of \(K/K^\natural\). The required list of conditions is fulfilled if the henselian valuation on \(K^\natural\) is discrete with perfect residue field. Hence in this case every reductive group over \(K^\natural\) gives rise to a Bruhat-Tits building.

Bruhat-Tits buildings are an indispensable tool for many different questions on non-archimedean reductive groups. In particular, one can use the action of such a group on its building to prove results on the structure of interesting subgroups. Let us only mention two important early papers in this direction, which were followed by numerous other results. H. Garland used buildings to prove his vanishing statement for the cohomology of discrete subgroups [“\(p\)-adic curvature and the cohomology of discrete subgroups of \(p\)-adic groups,” Ann. Math. (2) 97, 375–423 (1973; Zbl 0262.22010)], and A. Borel and J.-P. Serre used them to investigate the cohomology of \(S\)-arithmetic groups [“Cohomologie d’immeubles et de groupes \(S\)-arithmétiques,” Topology 15, 211–232 (1976; Zbl 0338.20055)]. Bruhat-Tits buildings are also very useful for questions in representation theory of reductive groups. For an overview of this topic see the introductory article [P. Schneider, “Gebäude in der Darstellungstheorie über lokalen Zahlkörpern,” Jahresber. Dtsch. Math.-Ver. 98, No.3, 135–145 (1996; Zbl 0872.11028)]. They also occur in various contexts in arithmetic geometry, for example in relation to Drinfeld’s \(p\)-adic upper half-spaces [V.G. Drinfeld, “Elliptic Modules,” Math. USSR, Sb. 23, 561–592 (1976); translation from Mat. Sb., n. Ser. 94(136), 594–627 (1974; Zbl 0321.14014)]. Recently it has been proven that Bruhat-Tits buildings can be realized inside Berkovich spaces. For split groups this has been shown by V. Berkovich in [“Spectral theory and analytic geometry over non-Archimedean fields.” Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013)], the general theory is contained in [A. Thuillier, B. Rémy, A. Werner, “Bruhat-Tits buildings from Berkovich’s point of view. I: Realizations and compactifications of buildings,” Ann. Sci. Éc. Norm. Supér. (4) 43, No.3, 461–554 (2010; Zbl 1198.51006)].

The theory initiated by Bruhat and Tits features an attractive interplay of arithmetic geometry, group theory and discrete geometry. Meanwhile, buildings have been generalized in various directions. All these applications and generalizations are based on the fundamental ideas of Bruhat and Tits, which even after decades continue to be very much alive in mathematical research.

The two articles under review contain the foundations of the theory of Bruhat-Tits buildings.

When Bruhat and Tits developed the theory of the buildings now bearing their names, the structure theory of reductive algebraic groups over an arbitrary field (as an algebraic avatar of the theory of Lie groups) was already quite well understood. The foundations of this theory were laid by Armand Borel, Claude Chevalley, Jacques Tits and many others.

Bruhat and Tits embarked on the project to understand reductive algebraic groups over a field \(K\) with a non-archimedean absolute value. Their goal was to define a new geometric object taking into account the valuation on the ground field, which might be seen as a \(p\)-adic avatar of the Riemannian symmetric space \(G/K\) associated to a semisimple real Lie group \(G\) and a maximal compact subgroup \(K\).

Note that a non-archimedean absolute value satisfies the triangle inequality in the following strong form

\[ |a+b| \leq \max\{|a|, |b|\}. \]

One example of a field with a non-archimedean absolute value is the field of formal Laurent series \(k((X))\) over an arbitrary ground field \(k\), which is endowed with the absolute value \(|\sum_{n \geq n_0} a_n X^n| = e^{-n_0}\) if \(a_{n_0} \neq 0\). The field \(K = k((X))\) is discretely valued, which means that the value group \(|K^\times|\) is a discrete subgroup of \(\mathbb{R}\).

Other prominent examples of fields with a non-archimedean absolute value are the completions \(\mathbb{Q}_p\) of \(\mathbb{Q}\) with respect to the \(p\)-adic absolute value. These fields are local, i.e., they are locally compact in the topology induced by the absolute value. The fields \(\mathbb{Q}_p\) and their extensions are important for local questions in number theory.

However, the topology induced by a non-archimedean absolute value has disadvantages. The fields \(k((X))\) and \(\mathbb{Q}_p\) are for example totally disconnected, i.e., the only connected subsets are the empty set and the one-point sets. These topological flaws create difficulties if one seeks for non-archimedean analogies of archimedean constructions. In the case of Bruhat-Tits buildings this explains why a new object had to be constructed: The quotient space of a semisimple group by a maximal compact subgroup, which works fine in the archimedean world, is not an interesting topological space in the non-archimedean world.

Now we consider a reductive algebraic group \(\mathcal{G}\) over a field \(K\) with a non-archimedean valuation. Recall that a linear algebraic group is a group variety over \(K\) which can be embedded in the \(K\)-variety given by a general linear group. It is called reductive if it does not contain any non-trivial connected unipotent normal subgroup. If it does not even contain any non-trivial connected solvable normal subgroup, then \(\mathcal{G}\) is called semisimple. The general linear group is reductive. The special linear group, the symplectic group or the special unitary group are examples of semisimple groups. Since for many interesting questions it is crucial to work with fields \(K\) which are not algebraically closed, one has to work with group schemes rather than groups. A group scheme over \(K\) can be thought of as an object encoding information on all groups \(\mathcal{G}(L)\), where \(L\) runs through the extension fields of \(K\).

Before Bruhat and Tits developed their general theory, O. Goldman and N. Iwahori had constructed a space of non-archimedean norms which afterwards turned out to be the Bruhat-Tits building associated to the general linear group [“The space of \(p\)-adic norms,” Acta Math. 109, 137–177 (1963; Zbl 0133.29402)]. Besides, N. Iwahori and H. Matsumoto [“On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups,” Publ. Math. IHES 25, 5–48 (1965; Zbl 0228.20015)] had investigated split groups before Bruhat and Tits embarked on the general theory.

The Bruhat-Tits building associated to a reductive group \(\mathcal{G}\) over a field with a non-archimedean absolute value is a metric space endowed with a continuous action by the \(K\)-rational points \(\mathcal{G}(K)\). If the valuation on \(K\) is discrete, then it carries a simplicial structure. Its existence depends on rather general hypotheses. For example, Bruhat-Tits buildings exist if the ground field \(K\) is discretely valued and henselian with a perfect residue field. Moreover, they always exist for split groups, i.e., for reductive groups \(\mathcal{G}\) such that the maximal split torus is defined over \(K\). Note that the Bruhat-Tits building of a reductive group coincides with the building of its semisimplification. However, there is a notion of extended buildings which takes into account the difference between these two groups.

Part I of the paper under review develops the fundamental theory of buildings from two different axiomatic angles. The application to reductive groups is postponed to part II. Let us describe in more detail the content of part I. In the first five sections a building (“immeuble”) is associated to an affine Tits system, which is the first axiomatic approach to buildings. In sections six to nine the second axiomatic approach via valued root data is developed. Both theories overlap to a certain extent, but not completely.

The first section recalls some facts on Coxeter groups, Tits systems and affine Weyl groups, relying on the Bourbaki volume Lie groups and Lie algebras, chapter 4 to 6. A Tits system is a quadruplet \((G,B,N,S)\) consisting of a group \(G\) together with two subgroups \(B\) and \(N\) generating \(G\) such that \(B \cap N\) is a normal subgroup in \(N\). The quotient group \(W = N / (B \cap N)\) is also called Weyl group. \(S\) is a set of involutions generating the Weyl group such that the two following conditions hold.

i) \(sBw \subset BwB \cup BswB\) for all \(s \in S\) and \( w \in W\),

ii) \(sBs \neq B\) for all \( s \in S\).

To avoid confusion note that \(G\) here is a true group and not a group scheme. The group \(G = \mathcal{G}(K)\) of \(K\)-rational points of a reductive group \(\mathcal{G}\) over \(K\) carries the structure of a Tits system. As an example, consider the general linear group \(G\) of rank \(n\) together with the subgroup \(B\) of upper triangular matrices and the subgroup \(N\) of matrices with precisely one non-zero entry in every line and column. Here \(B \cap N\) is equal to the subgroup of diagonal matrices, and \(W\) can be identified with the symmetric group of \(n\) elements. Under this identification the set \(S\) corresponds to the set of transpositions of \(i\) and \(i+1\) for all \(i = 1, \ldots, n-1\).

Tits systems with finite Weyl group were studied before. They give rise to the so-called spherical buildings or Tits buildings, which can be defined as the complex of all parabolic subgroups. This theory was developed by Jacques Tits in order to find geometric interpretations for linear algebraic groups over arbitrary fields, see [J. Tits, “Buildings of spherical type and finite \(BN\)-pairs.” Lecture Notes in Mathematics 386, Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0295.20047)]. It does not take into account the non-archimedean absolute value. In fact, for a reductive group over a field with a suitable non-archimedean absolute value, there exist two buildings: the Tits (or spherical) building and the Bruhat-Tits building. They are related since the Bruhat-Tits building can be compactified by attaching a Tits building “at infinity”.

In the case of ground fields with a non-archimedean valuation, one has to consider affine Tits systems, where the Weyl group \(W\) is an infinite group of affine reflections in a Euclidean space \(A\). The corresponding affine hyperplane arrangement defines a decomposition of \(A\) into faces. Faces of maximal dimension are called chambers.

In section two the building \(\mathcal{I}\) associated to an affine Tits system is defined. It is a polysimplicial complex, i.e., a product of simplicial complexes, whose faces correspond to the so-called parahoric subgroups of \(G\). It can be described as a union of Euclidean spaces, which are called apartments, and it carries a metric and an action of the group \(G\).

In the third section an important fixed-point theorem is proven, stating that the set-wise stabilizer of a bounded subset of the building \(\mathcal{I}\) has a fixed point. This is used to show that the bounded subgroups of \(G\) are cum grano salis the stabilizers of faces in the building. Since the notion of Tits systems does not a priori contain topological data, the definition of bounded subgroups is more involved. In the fourth section Iwasawa and Cartan decompositions of \(G\) are proven with the help of the building. Section five analyses double Tits systems, leading to an affine and to a spherical Tits system. This reflects the fact that there are two buildings associated to a reductive group over a non-archimedean field, as was explained above.

In section six the second approach to buildings is prepared by the definition of a valued root datum. Here we fix a root system \(\Phi\) in the dual space of a vector space \(V\). A root datum in a group \(G\) consists of a subgroup \(T\) together with data \(({U}_a, M_a)\) for every root \(a \in \Phi\) such that the \(U_a\) are subgroups satisfying certain commutator conditions. Each \(M_a\) is a coset of \(T\) satisfying among other conditions the important relation \(U_{-a} \backslash \{1\} \subset U_a M_a U_a\). The full list of axioms can be found in Definition (6.1.1). From these axioms a list of further properties of \(G\) is derived. As a motivating example consider the group \(G = SL_2(K)\) over any field \(K\), together with the subgroup \(T\) of diagonal matrices and the root system \(\Phi= \{a,-a\}\) of type \(A_1\). Let \(U_a\) be the unipotent subgroup of upper triangular matrices with entries \(1\) on the diagonal, and let \(U_{-a}\) be the unipotent subgroup of lower triangular matrices with entries \(1\) on the diagonal. Put

\[ m = \left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right) \]

and \(M_a = M_{-a} = Tm\). These data form a root datum. In particular, by a straightforward calculation one can check that \(U_{-a} \backslash \{1\} \subset U_a M_a U_a\) holds in this case.

Bruhat and Tits then define the notion of a valuation on a root datum. This is a family of maps \(\varphi_a: U_a \rightarrow \mathbb{R} \cup \{\infty\}\) which are compatible with the structure of a root datum. The full list of axioms is stated in Definition (6.2.1). In particular, the truncated subsets \(U_{a,r} = \varphi_a^{-1} ([r, \infty])\) are required to form a subgroup of \(U_a\).

Let us again consider the example \(SL_2(K)\) and assume that \(K\) is endowed with a non-trivial valuation map \(\omega: K \rightarrow \mathbb{R} \cup \{\infty\}\). Then we can define a valuation on the root datum defined above by setting

\[ \varphi_a \left( \left(\begin{matrix} 1 & u \\ 0 & 1 \end{matrix} \right) \right) = \varphi_{-a} \left( \left(\begin{matrix} 1 & 0 \\ u & 1 \end{matrix} \right) \right) = \omega (u). \]

More generally, it is shown that valued root data exist for all groups \(\mathcal{G}(K)\), where \(\mathcal{G}\) is a split reductive \(K\)-group.

Now Bruhat and Tits fix a root datum and a valuation on it. In (6.2.5) they define an affine space \(A\) under the vector space \(V\) (coming from the root datum) as the set of all valuations on the root datum depending on the fixed one in an affine-linear way. The precise notion here is “valuations équipollentes”. This affine space carries a natural action by the group \(N\) generated by \(T\) and the union of the \(M_a\). Note that in our example \(G = SL_2(K)\) the group \(N\) is the normalizer of \(T\), i.e., the group of monomial matrices in \(G\). Now one can define an arrangement of affine hyperplanes in \(A\). The hyperplane directions are given by the roots, and the set of translations leading to a family of parallel hyperplanes is basically given by the image of the valuation map.

Note that in our example \(G= SL_2(K)\), the space \(A\) is one-dimensional, and the affine hyperplane arrangement simply corresponds to the image of the valuation map \(\omega(K^\ast)\).

In section 6.5 it is shown that if all valuation maps \(\varphi_a\) have discrete image, then the notion of a valuation on a root datum gives rise to an affine Tits system. This provides the link to the theory developed in the first five sections. Hence in the discrete case there is an associated building.

Also in the general case of non-discrete valuation maps, one can define a building. This is the topic of section seven. For every point \(x\) in \(A\) and every root \(a\) one determines the minimal affine halfspace with direction \(a\) in our affine hyperplane arrangement which contains \(x\). This corresponds to an element \(r\) in the image of the valuation map, and hence to a subgroup \(U_{a,r}\) of \(U_a\). Now let \(P_x\) be the subgroup of \(G\) generated by all these \(U_{a,r}\) together with the elements in \(N\) fixing \(x\). Then in (7.4.1) the building \(\mathcal{I}\) is defined as the quotient of \(G \times A\) after the following equivalence relation: \((g,x) \sim (h,y)\) if and only if there exists an \(n \in N\) such that \(nx=y\) under the action of \(N\) on \(A\) and \(g^{-1} hn \in P_x\).

This space carries a natural \(G\)-action by multiplication in the first factor. Besides, the polysimplicial structure on \(A\) can be extended to a polysimplicial structure on \(\mathcal{I}\). To be precise, all faces in \(\mathcal{I}\) are of the form \(g F\), where \(g\) is an element in \(G\) and \(F\) is a face in \(A\). Every subset of the form \(gA\) in \(\mathcal{I}\) is called an apartment. It is shown in Theorem (7.4.18) that any two faces are contained in one apartment. Note that there is a distance function on \(A\) given by the choice of a scalar product associated to the root system. We transfer it in a \(G\)-invariant way to the other apartments. Since any two points in \(\mathcal{I}\) are contained in one apartment, this defines a \(G\)-invariant distance function on the whole building. It is well-defined, and gives a metric on the building \(\mathcal{I}\). The building is contractible in the associated topology, see Proposition (7.4.20).

Moreover, Iwasawa and Bruhat decompositions of \(G\) are proven in section 7.3. In section eight it is shown that for dense valuations the maximal bounded subgroups of \(G\) in a suitable sense are precisely the stabilizers of points. Section nine provides descent results for valued root data which will be used in part II of the paper. In section ten, classical groups over a field with a complete non-archimedean absolute value are investigated. It is shown how their natural root datum can be equipped with a valuation in the above sense. Note that more details on the buildings associated to classical groups may be found in the articles “Schémas en groupes et immeubles des groupes classiques sur un corps local” [Bull. Soc. Math. Fr. 112, 259–301 (1984; Zbl 0565.14028)] and “Schémas en groupes et immeubles des groupes classiques sur un corps local. II: Groups unitaires” [Bull. Soc. Math. Fr. 115, 141–195 (1985; Zbl 0636.20027)] by the same authors.

The purpose of part II of the paper under review is to show that under rather general assumptions the group \(\mathcal{G}(K)\) of \(K\)-rational points of a reductive connected group \(\mathcal{G}\) over a field \(K\) with a non-trivial non-archimedean absolute value actually possesses a valued root datum which is compatible with the valuation on \(K\). Hence the results of part I can be applied to these groups, thereby proving the existence of Bruhat-Tits buildings. In particular, the fixed point and decomposition results shown in part I hold in this case.

The construction of a valued root datum is achieved in a double descent process, already prepared by section nine of part I. The first descent step is used to proceed from split groups to quasi-split groups, the second descent step deals with the passage from strictly henselian to henselian fields. This is useful since a result of Steinberg states that a reductive group over a discretely valued henselian field \(K\) with perfect residue field becomes quasi-split after base-change with the strict henselisation.

An important technical tool in part II is the construction of group schemes over the ring of integers \(R\) in \(K\) such that their generic fiber is \(\mathcal{G}\) or one of its subgroups. We refer to a scheme over \(R\) also as a model of its generic fiber.

Part II starts with a section recalling general facts in algebraic geometry which are used in the following sections.

In section two, Bruhat and Tits investigate the properties of models for root groups and their products. In section three, this leads to the definition of a “donnée radicielle schématique” or schematic root datum. Let \(\mathcal{Z}\) be the centralizer of the maximal split torus in \(\mathcal{G}\), and denote by \(\mathcal{U}_a\) the root group associated to a root \(a\). Then a schematic root datum is a collection of models of the groups \(\mathcal{Z}\) and \(\mathcal{U}_a\) satisfying a list of conditions mainly stating that some natural maps extend from the generic fibers to the whole models. For details see Definition (3.1.1).

In section four quasi-split groups \(\mathcal{G}\) are considered. In this case there exists a Borel group in \(\mathcal{G}\) which is defined over \(K\), and the base change \(\mathcal{G}_{\tilde{K}}\) of \(\mathcal{G}\) to a Galois extension \(\tilde{K}\) of \(K\) is a split group. By a careful analysis of the effect of the Galois group of \(\tilde{K}/K\) it is shown that the valued root datum for \(\mathcal{G}_{\tilde{K}}(\tilde{K})\) constructed in part I can be descended to a valued root datum for \(\mathcal{G}(K)\). Section five deals with a reductive group \(\mathcal{G}\) over a henselian (for example complete) field \(K^\natural\). If the base change \(\mathcal{G}_K\) of \(\mathcal{G}\) to the strict henselisation \(K\) is quasi-split (and two other conditions hold, see (5.1.1)), then the valued root datum on the group \(\mathcal{G}_{K}(K)\) can be descended to \(\mathcal{G}(K^\natural)\). In this case the Bruhat-Tits building associated to \(\mathcal{G}\) can be identified with the fixed-point set of the building associated to \(\mathcal{G}_{K}\) with respect to the action of the Galois group of \(K/K^\natural\). The required list of conditions is fulfilled if the henselian valuation on \(K^\natural\) is discrete with perfect residue field. Hence in this case every reductive group over \(K^\natural\) gives rise to a Bruhat-Tits building.

Bruhat-Tits buildings are an indispensable tool for many different questions on non-archimedean reductive groups. In particular, one can use the action of such a group on its building to prove results on the structure of interesting subgroups. Let us only mention two important early papers in this direction, which were followed by numerous other results. H. Garland used buildings to prove his vanishing statement for the cohomology of discrete subgroups [“\(p\)-adic curvature and the cohomology of discrete subgroups of \(p\)-adic groups,” Ann. Math. (2) 97, 375–423 (1973; Zbl 0262.22010)], and A. Borel and J.-P. Serre used them to investigate the cohomology of \(S\)-arithmetic groups [“Cohomologie d’immeubles et de groupes \(S\)-arithmétiques,” Topology 15, 211–232 (1976; Zbl 0338.20055)]. Bruhat-Tits buildings are also very useful for questions in representation theory of reductive groups. For an overview of this topic see the introductory article [P. Schneider, “Gebäude in der Darstellungstheorie über lokalen Zahlkörpern,” Jahresber. Dtsch. Math.-Ver. 98, No.3, 135–145 (1996; Zbl 0872.11028)]. They also occur in various contexts in arithmetic geometry, for example in relation to Drinfeld’s \(p\)-adic upper half-spaces [V.G. Drinfeld, “Elliptic Modules,” Math. USSR, Sb. 23, 561–592 (1976); translation from Mat. Sb., n. Ser. 94(136), 594–627 (1974; Zbl 0321.14014)]. Recently it has been proven that Bruhat-Tits buildings can be realized inside Berkovich spaces. For split groups this has been shown by V. Berkovich in [“Spectral theory and analytic geometry over non-Archimedean fields.” Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013)], the general theory is contained in [A. Thuillier, B. Rémy, A. Werner, “Bruhat-Tits buildings from Berkovich’s point of view. I: Realizations and compactifications of buildings,” Ann. Sci. Éc. Norm. Supér. (4) 43, No.3, 461–554 (2010; Zbl 1198.51006)].

The theory initiated by Bruhat and Tits features an attractive interplay of arithmetic geometry, group theory and discrete geometry. Meanwhile, buildings have been generalized in various directions. All these applications and generalizations are based on the fundamental ideas of Bruhat and Tits, which even after decades continue to be very much alive in mathematical research.

Reviewer: Annette Werner (Frankfurt) (2011)

##### MSC:

14L99 | Algebraic groups |

20G25 | Linear algebraic groups over local fields and their integers |

22E99 | Lie groups |

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\textit{F. Bruhat} and \textit{J. Tits}, Publ. Math., Inst. Hautes Étud. Sci. 41, 5--251 (1972; Zbl 0254.14017)

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