Embeddings of local fields in simple algebras and simplicial structures on the Bruhat-Tits building

This article answers a question that naturally arises from the articles by Grabitz and Broussous"Pure elements and intertwining classes of simple strata in local central simple algebras"and Broussous and Lemaire"Buildings of GL(m,D) and Centralizers". For an Azumaya-Algebra A over a non-Archimedean local field F, Grabitz and Broussous have introduced embedding invariants for field embeddings, that is for pairs (E,a), where E is a field extension of F in A, and $a$ is a hereditary order which is normalised by E^x. On the other hand if we take such a field extension E and define B to be the centralizer of E in A, then G:=A^x and G_E:=B^x are sets of rational points of reductive groups defined over F and E respectively. Broussous and Lemaire have defined a map j_E: I^{E^x}\to I_E, where $I$ is the the Euclidean building of $G$, and I_E is the Euclidean building of G_E. The question which we address is to relate the embedding invariants to the behavior of the map j_E with respect to the simplicial structures of $I$ and I_E. I have to thank very much Prof. Zink from Homboldt University Berlin for his helpful remarks, the revision of the work and for giving my the interesting task.


Introduction and notation 1.First remark
This article answers a question that naturally arises from the articles by M.Grabitz and P. Broussous (see [BG00]) and P. Broussous and B. Lemaire (see [BL02]). For an Azumaya-Algebra A over a non-archimedean local field F, M. Grabitz and P. Broussous have introduced embedding invariants for field embeddings, that is for pairs (E, ), where E is a field extension of F in A, and is a hereditary order which is normalised by E × . On the other hand if we take such a field extension E and define B to be the centralizer of E in A, then G := A × are G E := B × are sets of rational points of reductive groups defined over F and E respectively. P. Broussous and B. Lemaire have defined a map j E : I E × → I E , where I is the g.r. (geometric realization) of the euclidean building of G, and I E is the g.r. of the euclidean building of G E . The question which we address is to relate the embedding invariants to the behavior of the map j E with respect to the simplicial structures of I and I E . I have to thank very much Prof. Zink from Homboldt University Berlin for his helpful remarks, the revision of the work and for giving my the interesting task.

Notation
1. The set of natural numbers starts with 1 and the set of the first r natural numbers is denoted by N r . For the set of non-negative integers we use the symbol N 0 .
2. The letter F denotes a non-archimedean local field. For the valuation ring, the valuation ideal, the prime element and the residue field of F we use the notation o F , Ô F , π F and κ F respectively. We use similar notation for other division algebras with non-archimedean valuation.
3. The letter ν denotes the valuation on F with ν(π F ) = 1 q , where q is the cardinality of κ F . 4. We assume D to be a finite dimensional central division algebra over F of index d.

5.
We fix an m dimensional right D vector space V , m ∈ N, and put A := End D (V ) . In particular V is a left A ⊗ F D op -module.
6. The letter L denotes a maximal unramified field extension of F in D and we assume that π D normalizes L, i.e. the map σ(x) := π D xπ −1 D , x ∈ D, generates Gal(L|F ). 7. For a positive integer f |d we denote by L f the subfield of degree f over F in L.

Vectors and Matrices up to cyclic permutation
Remark 1 All invariants which are considered in this aritcle are vectors or matrices modulo cyclic permutation.
Vectors: We denote by Row(s, t) the set of all vectors w ∈ N s 0 whose sum of entries is t, where s and t are natural numbers, i.e.
Two vectors w, w ′ ∈ Row(s, t) are called equivalent if w can be obtained from w ′ by cyclic permutation of the entries of w, i.e.
The equivalence class is denoted by w . Analogous we define the class for every vector, e.g. for a vector of pairs. One can represent the class w of a vector w ∈ Row(s, t) by pairs where (w ij ) 0≤j≤k is the subsequence of the non-zero coordinates. Given the same w with pairs( w ) = (a 0 , b 0 ), . . . , (a k , b k ) we define the complement of w , denoted by w c to be the class w , such that This is a bijection ( ) c : Row(s, t)→ Row(t, s).
Matrices: Given three natural numbers s, r, t the symbol M r,s (t) denotes the set of matrices with r rows and s columns such that all entries are non-negative integers and the sum of them is t. For a matrix M = (m i,j ) ∈ M r,s (t), we define the vector row(M ) ∈ Row(r + s, t) to be (m 1,1 , m 1,2 , . . . , m 1,s , m 2,1 , . . . , m 2,s , . . . , m r,s ).
Two matrices M, N ∈ M r,s (t) are said to be equivalent if row(M ) and row(N ) are. The equivalence class is denoted by M .

Hereditary orders, lattice chains and lattice functions
In this section we give a description of the euclidean building of A in terms of lattice functions and its simplicial structure in terms of hereditary orders. As references we recommand [Rei03] for hereditary orders , [Bro89] for the definition of an euclidean building and [BL02] for the discription of the euclidean building of GL m (D) in terms of lattice functions and norms. One can find the description with norms in [BT84] as well. For more details see also [Sko05].
Definition 1 Let D ′ be a central division algebra of finite index d ′ over a nonarchimedian local field F ′ , and let W be a D ′ -vector space of finite dimension.
We omit the word full.
We call an o F -order hereditary if the Jacobson radical rad( ) is a projective right-module. The set of all hereditary orders is denoted by Her(A) . For ∈ Her(A) we denote by lattices( ) the set of all o D -lattices Γ of V such that aΓ ⊆ Γ for all a ∈ .
Definition/Remark 1 1. Let R be a non-empty set, and take r ∈ N. Given non-empty subsets R i,j of R, (i, j) ∈ N 2 r , and natural numbers n 1 , . . . , n r , we denote by (R i,j ) n1,...,nr the set of all block matrices in M P r i=1 ni (R), such that for all (i, j) the (i, j)-block lies in M ni,nj (R i.j ).
2. Given r ∈ N,n = (n 1 , . . . , n r ) ∈ N r , we get a hereditary order If we say that sets are conjugate to each other, we mean conjugate by an element of A × . The proof is given in [Rei03].
Theorem 1 We fix a D-basis of V and identify A with M m (D).

Every ∈ Her(A) is conjugate to a hereditary order in standard form.
By this theorem the notion of invariant and period carries over to every element of Her(A) and they do not depend on the choice of the basis.  Definition 5 • For every lattice function Λ of V we can define a family the set of square lattice functions in A.

Embedding types
For a field extension E|F we denote by E D |F the maximal field extension in E|F, which is F -algebra isomorphic to a subfield of L. Its degree is the greatest common divsor of d and the residue degree of E|F.
Definition 6 An embedding is a pair (E, ) satisfying Two embeddings (E, ) and (E ′ , ′ ) are said to be equivalent if there is an element Remark 3 In each equivalence class of embeddings there is a pair such that the field can be embedded in L.
Notation 1 For r, s, t ∈ N, M r,s (t) denotes the set of r × s-matrices with nonnegative integer entries, such that • in every column there is an entry greater than zero, and • the sum of all entries is t.
Until the end of this section we fix a D-basis of V and identify A with M m (D).
Definition 7 Let f |d and r ≤ m. A matrix with f rows and r columns is called an embedding datum if it belongs to M f,r (m). Given an embedding datum λ, we define the pearl embedding as follows. The pearl embedding of λ (with respect to the fixed basis) is the embedding (E, ), with the following conditions: 3. is a hereditary order in standard form according to the partition m = n 1 + . . . + n r where n j :

In any class of embeddings lies a pearl embedding.
Definition 8 Let (E, ) be an embedding. By the theorem it is equivalent to a pearl-embedding. The class of the corresponding matrix (λ i,j ) i,j is called the embedding type of (E, ). This definition does not depend on the choice of the basis by the proposition of Skolem-Noether. Definition 9 A building is a triple (Ω, A, ≤), such that (Ω, ≤) is a simplicial complex and A is a set of subcomplexes of (Ω, ≤) which cover Ω, i.e.
A = Ω, (The elements of A are called apartments.) statisfying the following "Building Axioms": • B0 Every element of A is a Coxeter complex.
• B1 For faces (also called simplicies), i.e. elements, S 1 and S 2 of Ω there is an apartment Σ containing them.
• B2 If Σ and Σ ′ are two arpartments containing S 1 and S 2 then there is a poset isomorphism from Σ to Σ ′ which fixesS 1 andS 2 whereS for a face S is defined to be the set of all faces T ≤ S.
A building is said to be thick if every codimension 1 face is attached to at least three chambers.
Remark 4 The buildings considered in this article are thick.
A euclidean Coxeter comlex is a Coxeter complex (Σ, ≤) which is posetisomorphic to simplicial complex Σ(W, V ) defined by an essential irreducible infinite affine reflection group (W, V ).
For a face S of a simplicial complex (Ω, ≤) the set of all formal sums , such that for every face S ∈ Ω the restriction f :S→f (S) is a poset isomorphism. In [Bro89] the notion of non-degenerate simplicial map is used instead of morphism. A morphism f induces a map |f | between the g.r., by Definition 10 Given two buildings (Ω, A, ≤) and (Ω ′ , A ′ , ≤ ′ ) a morphism from the first to the latter is a morphism of simplicial complexes such that the image of an apartment of A is contained in an apartment of A ′ .
As descriped in [Bro89] VI.3 there is a canonical way to define a metric, up to a scalar, on the g.r. of an euclidean building by pulling back the metric from an affine reflection group to the apartment and this defines a canonical affine structure on the g.r. of the building. The map |φ| between the g.r. of two euclidean buildings induced by an isomorphism φ is affine.

The description with lattice functions
We describe the euclidean Bruhat-Tits-building Ω of Aut D (V ) in terms of lattice chains, and the g.r. I in terms of lattice functions as it is done in [BL02], section I.3.

A hereditary order is a vertex (resp. a chamber) if and only if its period
is 1 (resp. m).
A lattice chain Γ, lattice function Λ, hereditary order is split by Ê if every element of lattices(Γ), lattices(Λ), lattices( ) resp. is split by Ê. An equivalence class is split by Ê if every element of the equivalence class is split by Ê. The set of these classes split by Ê is called the apartment corresponding to Ê and is denoted by LC oD (V ) Ê , Her(A) Ê , Latt oD (V ) Ê resp.. For the set of these apartments we are isomorphic euclidean buildings via Ψ.

Ψ is
For the proof see for example [Rei03].
Remark 5 Every ∈ Her(A) has a rank as a face in the chamber complex (Her(A), ⊇), and we have period( ) = rank( ). This rank is not the o F -rank of .
for some real vectors α, α ′ . Take β ∈ R, and put Now the next propositions explain why one can replace Ω by the building of classes of lattice functions and I by Latt oD V.
Notation 2 By the two propositions above we can identify Ω with (Her(A), (Her(A)), ⊇) and I with Latt oD V and Latt 2 oF (A).

The map j E
Notation 3 For this section let E|F be a field extension in A and we set B to be the centraliser of E in A, i.e.
The building of B we denote by Ω E and its g.r. by I E .
The next results are taken from [BL02].
Theorem 4 [BL02, Thm 1.1.] There exists a unique application j E : I E × →I E such that for any x ∈ I we have j E ( (x)) = B∩ (e(E|F )x). The map j E satisfies the following properties: 1. it is a B × -equivariant bijection.

it is affine.
Moreover its inverse j −1 E is the only map I E →I such that 2. and 3. hold. We briefly give Broussous and Lemaire's description of j E in terms of lattice functions but only in the case, where E|F is isomorphic to a subextension L f |F of L|F. Then E ⊗ F L ∼ = f −1 k=0 L comming from the decomposition 1 = f −1 k=0 1 k labeled such that the Gal(L|F )-action on the second factor gives σ(1 k ) = 1 k−1 for k ≥ 1 and σ(1 0 ) = 1 f −1 . Applying it on the E ⊗ F L-module V , we get V = k V k , V k := 1 k V.

The connection between embedding types and barycentric coordinates
In this sction we keep the notation from section 5. We repeat that E D denotes the biggest extension field of E|F which can be embedded in L|F. The centralizer of E D in A is denoted by B D . We need a notion of orientation on Ω ED to order the barycentric coordimnates of a point in I ED .
Definition 13 An edge of Ω with vertices e and e ′ is oriented towards e ′ , if there are lattices Γ ∈ lattices(e) and Γ ′ ∈ lattices(e ′ ), such that Γ ⊇ Γ ′ with the quotient having κ D -dimension 1, i.e. κ F -dimension d. We write e→e ′ If x is a point in I then there is a chamber C ∈ Ω such that x lies in the closure of |C|, i.e. in S≤C |S|.
Firstly we need some lemmas. The actions of G on square lattice functions by conjugation induces maps m g : Ω→Ω, x → g.x and c g : Lemma 1 |m g | and c g induce isomorphisms on the simplicial structures of the euclidean buildings, which preserve the orientation, i.e. an oriented edge is mapped to an oriented edge such that the direction is preserved. In particular |m g | and c g are affine bijections, m g preserves the embedding type, c g the local type, and the following diagram is commutatve: The following lemma gives a geometric interpretation of the map row() T : {embedding types}→{embedding types of vertices} Lemma 2 (rank reduction lemma) Assume there is a field extension F ′ |F of degree s in E|F, where 2 ≤ s ≤ m. Let be a vertex in Ω E × such that ∩ C A (F ′ ) is a face of rank s in Ω E × F ′ and assume (E, ) has embedding type λ and (E, ∩ C F ′ (A)) has embedding type λ ′ . Then we get row(λ) ∼ row(λ ′ ), i.e. λ ∼ row(λ ′ ) T .
Proof: By lemma 1 it is enough to show the result only for one embedding equivalent to (E, ). For simplicity we can restrict ourself to the case of s = 2. The argument for s > 2 is similar. We fix a D-basis of V. It is (E, ) equivalent to the pearl embedding (λ) =: (E λ , λ ). Now we apply a permutation p on (λ) such that the odd exponents of σ in pE λ p −1 are behind all even exponents, i.e. pE λ p −1 is the image of and For the embedding (E ′ , ′ ) obtained by conjugating p(λ)p −1 with the matrix we have the following properties.
• F ′ is the image of the diagonal embedding of L 2 in M m (D) and its cen- is a herditary order in standard form with invariant n1, n2 . The positivity of the integers follows from the assumption that this intersection is a face of rank 2.
Since π ∆ F ′ := π 2 D is a prime element of ∆ F ′ which normalises L and since the powers of σ occuring in the description of E ′ are even we can read the embedding type of (E ′ , ′ ∩ M m (∆ F ′ )) directly. It is the class of Thus the result follows. q.e.d. The next lemma shows that changing the skewfield does not change the embedding type.
Lemma 3 (changing skewfield lemma) Let D ′ be a central skewfield over a local field F ′ of index d with a maximal unramified extension field L ′ normalized by a prime element π D ′ and assume that V ′ is an m dimensional right vector space over D ′ . Denote the euclidean building of GL m (D ′ ) by I ′ and let Σ, Σ ′ be an apartment of I, I ′ corresponding to a basis (v i ), (v ′ i ) respectively. Then Σ ′ is fixed by the image E ′ of the diagonal embedding of L ′ f in M m (D ′ ) . Assume further that E is the image of diagonal embedding of L f in M m (D) . Under these assumptions the map ≡ from |Σ| to |Σ ′ | defined by is the g.r. from an isomorphism φ of simplicial complexes which preserves the orientation and the embedding type. The latter means that if ′ is the image of a hereditary order under φ then the embedding types of (E, ) and (E ′ , ′ ) equal.
Proof: We only show the preserving of the embedding type. The other properties are verified easely. If Ä is a lattice chain of and Ä ′ one of ′ then by φ( ) = ′ we can assume that for all indexes j the lattices Ä j and Ä ′ j have the same exponent vectors, i.e. if and thus by applying from the left a permutation matrix P and a diagonal matrix T, T ′ , whose entries are powers of the corresponding prime element, we obtain simultanously lattice chains corresponding to hereditary orders , ′ in the same standard form. More precisely T ′ is obtained from T if π D is substituted by π D ′ . Thus (T P EP −1 T −1 , ) and (T ′ P E ′ P −1 T ′−1 , ′ ) have the same embedding type and thus by conjugating back (E, ) and (E ′ , ′ ) have the same embedding type. q.e.d.
We now fix a D-basis v 1 , . . . , v m of V and therefore a frame i.e. a diagonal matrix, |m g | induces an affine bijection of |Σ|. If g is diag(1, . . . , 1, π k D , 1, . . . , 1), with π k D in the i-th row, the map |m g | is of the form where we set Q m+1 := Q 1 .
In the next example we consider the most important special case for E, which is the only case that one has to consider for the proof of the theorem.
Example: Let us assume E is the image of the diagonal embedding of L f in M m (D), i.e. E = {(x, . . . , x)| x ∈ L f }.
Then B and j E simplify, i.e.
1. B = End ∆ W with ∆ := C D (L f ) and W := i v i ∆ 2. In terms of lattice functions j E has the form where Λ ∩ W denotes the lattice function 3. The image of j E | |Σ| is the g.r. of the apartment Σ E which belongs to the frame {v i ∆| 1 ≤ i ≤ m} and in affine coordinates the map has the form 4. The vertices of Σ E are the points of |Σ E | with affine coordinate vectors in Z m−1 . Specifically the points P i := j E (Q i ) are vertices of a chamber of Σ E .
5. The edge from P i to P i+1 is oriented to P i+1 . Proof: [example] To prove the statements of the example it is enough to calculate j E in terms of lattice functions, i.e to show 2. The statements then follow by similar and standard calculations. For 2: For an o D -lattice function Λ, whose class is E × -invariant there exists an o ∆ -lattice function Γ, such that Using the decomposition Thus by the injectivity of j E the lattice functions Λ andΛ are equivalent and therefore The appearence of j E in terms of coordinates follows now from q.e.d. Proof: [theorem] By lemma 1 and by theorem 3 we can assume that we are in the situation of the example 6 above and that there is a diagonal matrix h consisting of powers of π D with exponents in N f −1 ∪ {0} such that is the pearl embedding (λ). We consider two cases for the proof. Case 1: has period 1, i.e.
and λ is only one column. We get from Q 1 by applying m h −1 which is a composition of maps m g where g differs from the identity matrix by only one diagonal entry π k D . Now remark 9 gives Thus in barycentric coordinates j E (M ) has the form and therefore the vector µ := ( f − a m + a 1 f , a 2 − a 1 f , . . . , a m − a m−1 f ) fullfils part one of the theorem. If (λ i l ) 1≤l≤s is the subsequence of non-zero entries we define the indexes j l := λ 1 + . . . + λ i l−1 + 1 and j 1 := 1. This are the indexes where the µ j are non-zero, more precisely from we obtain for a j the following values: a j = a j l = i l − 1, j l ≤ j < j l+1 and a j = a js = i s − 1, j s ≤ j ≤ m, and thus the subsequence of non-zero entries of f µ is (f µ j l ) = (f − i s + i 1 , i 2 − i 1 , i 3 − i 2 , . . . , i s − i s−1 ).
Therefore we get for pairs( f µ ) the expression (1 − i s + i 1 , λ i1 ), (i 2 − i 1 , λ i2 ), (i 3 − i 2 , λ i3 ), . . . , (i s − i s−1 , λ is ) and this is precisely row(λ) c . Case 2: Assume the period r of is not 1. Here we want to use rank reduction. We fix an unramified field extension L ′ |F of degree r * d in an algebraic closure of F. Denote by D ′ a skewfield which is a central cyclic algebra over F with maximal field L ′ and an L ′ -normalising prime element π D ′ , i.e.
π D ′ L ′ π −1 D ′ = L ′ , and π d+r D ′ = π F . The images of L ′ r , L ′ r * f under the diagonal embedding of L ′ in M m (o D ′ ) are denoted by F ′ , E ′ respectively and the apartment of the euclidean building I ′ of GL m (D ′ ) corresponding to the standart basis is denoted by Σ ′ , i.e. we have a field tower E ′ ⊇ F ′ ⊇ F and apartments Σ ′ , Σ ′ E ′ , Σ ′ F ′ in the buildings I ′ , I ′ E ′ , I ′ F ′ respectively. We then obtain a commutative diagram of bijections, where the lines are induced by isomorphisms of chamber complexes which preserve the orientation.
The map ≡ F is given by and ≡ E analogously. Because of lemma 3 the map ≡ F preserves the embedding type and thus we can finish the proof by applying lemma 2 on More precisely, let S r be a face of rank r in Σ ′ F ′ . Its barycenter has affine coordinates in 1 r Z m−1 and therefore the preimage of it under j F ′ is a point S 1 with integer affine coeffitients, i.e. it corresponds to a vertex of I ′ . Because of j E ′ (M Sr ) = j E ′ (j F ′ (S 1 )) = j E ′ (S 1 ) the theorem follows now from the rank reduction lemma and case 1. q.e.d.