aa r X i v : . [ m a t h . R T ] M a y A REMARK ON THE KOTTWITZ HOMOMORPHISM
MOSHE ADRIAN
Abstract.
We prove that for any split, almost simple, connected reductive group G over a p -adic field F ,the Kottwitz homomorphism κ : G ( F ) → Ω exhibits a homomorphic section Ω ֒ → G ( F ). We then extendthis result to certain additional split connected reductive groups. Introduction
Let G be a connected reductive group over a p -adic field F . In [Kot97], Kottwitz defined a canonicalhomomorphism κ : G ( F ) ։ X ∗ ( Z ( b G ) I ) Fr . This homomorphism is surjective and, in the case that G is split, simplifies to a homomorphism κ : G ( F ) ։ X ∗ ( Z ( b G )) ∼ = X ∗ ( T ) /Q ∨ . In this note, we show that the map κ has a homomorphic section in the case that G is split and almostsimple, as well as for certain additional split groups. More specifically, fix a fundamental alcove in thebuilding of G corresponding to a maximal split torus T , and let Ω be the subgroup of the extended affineWeyl group W that stabilizes C . We show that there is a homomorphic section of the canonical projection N G ( T ) ։ Ω, where N G ( T ) is the normalizer of a maximal torus T in G . If G is almost-simple, thenthis section can be described as follows: it is known (see Proposition 3.1) that Ω may be identified witha collection of elements { , ǫ i ⋊ w i } ⊂ W = X ∗ ( T ) ⋊ W ◦ , where ǫ i are certain fundamental coweights and W ◦ is the finite Weyl group. By [Spr98, § N ◦ : W ◦ → N G ( T ) (denoted φ in loc. cit. ) that is compatible with the projection N G ( T ) → W ◦ . We may then consider the map ι : Ω → N G ( T ) ǫ i w i ǫ i ( ̟ − ) N ◦ ( w i ) , where ̟ is a uniformizer in F . The map ι is a section of the projection N G ( T ) ։ Ω, and it turns out that ι is a homomorphism in all cases except the adjoint group of type D l where l is odd, and some cases in type A l (see Theorem 3.5 and Remark 3.6). Nonetheless, we can still use ι to construct a homomorphic sectionfor all almost-simple p -adic groups (see Theorem 3.5). We then show that for certain split connected groupswith connected center, the Kottwitz homomorphism exhibits a homomorphic section (see Proposition 4.3).We would like to remark that if G is any split connected reductive group with simply connected derivedgroup, then κ has a homomorphic section. This follows from the fact that Ω is a free abelian group of finiterank, isomorphic to a free quotient of X ∗ ( T ). Then one constructs a section by taking a homomorphic sectionof X ∗ ( T ) → Ω and then composing that section with the map X ∗ ( T ) → T , λ λ ( ̟ − ). In particular, theimage of this section lies in T , not just N G ( T ). The situation where Ω is finite is much more subtle, whichis what this paper is about.1.1. Acknowledgements.
This paper was written in response to a question that Karol Koziol asked me;I wish to thank him for asking the question. I thank the referees for their comments, especially one of thereferees for very valuable comments and suggestions; in particular, ideas on how to expand the results beyondsplit almost-simple groups, and for the proof of Proposition 4.3. I thank Karol Koziol and Sean Rostami forhelpful conversations. Support for this project was provided by a grant from the Simons Foundation Preliminaries
Let G be a split connected reductive group over a p -adic field F . Fix a pinning ( B, T, { X α } ) for G . Thisgives rise to a set of non-zero roots Φ of G with respect to T , a set of positive roots Π in Φ, and a basis∆ = { α , α , ..., α l } of the set of positive roots, so that l is the rank of G . We recall that for each α ∈ Φ, thereexists an isomorphism u α of F onto a unique closed subgroup U α of G such that tu α ( x ) t − = u α ( α ( t ) x ), for t ∈ T, x ∈ F [Spr98, § X ∗ ( T ), X ∗ ( T ) be the character, cocharacter lattices of T , respectively. Let Q be the lattice generatedby Φ, and P ∨ the coweight lattice. Namely, P ∨ is the Z -dual of Q relative to the standard pairing ( · , · ) : X ∗ ( T ) × X ∗ ( T ) → Z . We let Φ ∨ be the system of coroots, Q ∨ the lattice generated by Φ ∨ , and P the weightlattice. Then P ∨ is spanned by the l fundamental coweights, which are denoted ǫ , ǫ , ..., ǫ l . We recall thatthe ǫ i are defined by the relation ( ǫ i , α j ) = δ ij . If α is a root, we denote its associated coroot by α ∨ .We now let W ◦ = N G ( T ) /T be the Weyl group of G relative to T . For each α ∈ Φ, we let s α ∈ W ◦ be the simple reflection associated to α . Then the u α may be chosen such that for all α ∈ R , n α = u α (1) u − α ( − u α (1) lies in N G ( T ) and has image s α in W ◦ (see [Spr98, § N ◦ : W ◦ → N G ( T ) [Spr98, § φ in loc.cit.), defined by N ◦ ( w ) = n β n β · · · n β m for a reduced expression w = s β s β · · · s β m .2.1. The map N ◦ . In this section, we recall a result about the map N ◦ from [Ros16]. Definition 2.1.
For u, v ∈ W ◦ , we define F ( u, v ) = { α ∈ Π | v ( α ) ∈ − Π , u ( v ( α )) ∈ Π } . The following proposition describes the failure of N ◦ to be a homomorphism. Proposition 2.2. [Ros16, Proposition 3.1.2]
For u, v ∈ W ◦ , N ◦ ( u ) · N ◦ ( v ) = N ◦ ( u · v ) · Y α ∈F ( u,v ) α ∨ ( − . Definition 2.3.
For w ∈ W ◦ , i ∈ N , we define F w ( i ) = { α ∈ Π | w i ( α ) ∈ − Π , w i +1 ( α ) ∈ Π } . Corollary 2.4. If w ∈ W ◦ and n ∈ N , then N ◦ ( w ) n = N ◦ ( w n ) · n − Y m =1 Y α ∈F w ( m ) α ∨ ( − . Proof.
By Proposition 2.2, N ◦ ( w ) = N ◦ ( w ) · Y α ∈F w (1) α ∨ ( − . Multiplying by N ◦ ( w ) on the left and usingProposition 2.2 again, we get N ◦ ( w ) = N ◦ ( w ) · Y α ∈F w (2) α ∨ ( − · Y α ∈F w (1) α ∨ ( − . Continuing in this way,the claim follows. (cid:3) Embedding Ω into G Let G be a split, almost-simple p -adic group. We set W = N G ( T ) /T ◦ , where T ◦ is the maximal boundedsubgroup of T . The group W is the extended affine Weyl group, and we note that we have a semidirectproduct decomposition W = X ∗ ( T ) ⋊ W ◦ . We also set Ω = W/W ◦ , where W ◦ = Q ∨ ⋊ W ◦ is the affine Weylgroup. We therefore have a canonical projection N G ( T ) → Ω. This projection is exactly the restriction of κ to N G ( T ).The group Ω can be identified with the subgroup of W that stabilizes a fundamental alcove C . Moreover,it is known that Ω acts on the set { − α , α , α , ..., α l } , where α is the highest root in Φ. The action of Ωon this set can be found in [IM65, p. 18-19]. We let Ω ad be the analogous group for the adjoint group G ad .It is known that there exists in W ◦ an element w ∆ such that w ∆ (∆) = − ∆. The element w ∆ is unique andsatisfies w = 1. Moreover, if we denote the subset ∆ − { α i } by ∆ i , then the subgroup W i of W ◦ generated REMARK ON THE KOTTWITZ HOMOMORPHISM 3 by s α , ..., ˆ s α i , ..., s α l (ˆ s α i means that s α i is omitted) contains an element w ∆ i such that w ∆ i (∆ i ) = − ∆ i and w i = 1.We recall the following result from [IM65]. Proposition 3.1. [IM65, Proposition 1.18]
The mapping from the set { } ∪ { ǫ i : ( α , ǫ i ) = 1 } onto Ω ad defined by , ǫ i ǫ i w ∆ i w ∆ is bijective. The notation ρ i (and sometimes ρ ) is used in [IM65] to denote the element ǫ i w Π i w ∆ . We will adopt thesame notation. We will also let S ad denote the set { } ∪ { ǫ i : ( α , ǫ i ) = 1 } . We note that every latticebetween Q ∨ and P ∨ arises as h Q ∨ , S i , for some subset S ⊂ S ad . If S is such a subset, we will talk of thealmost-simple p -adic group G that is determined by the lattice h Q ∨ , S i . We note in particular that if Ω G denotes the omega group for G , then one can see that Ω G = W/W ◦ ∼ = h , ρ i : ǫ i ∈ S i .We now assume that G is not simply connected. For otherwise, Ω = 1, so the claim that κ : G → Ω hasa homomorphic section is vacuous.Let ̟ be a uniformizer of F . There is a natural map X ∗ ( T ) → N G ( T ) given by λ λ ( ̟ − ) (see [Tit79,p. 31]). We also have the map N ◦ : W ◦ → N G ( T ). Coupling these maps together, we obtain a natural map W → N G ( T )( λ, w ) λ ( ̟ − ) N ◦ ( w )for λ ∈ X ∗ ( T ) , w ∈ W ◦ . Most of the time, we will write λw instead of ( λ, w ). Proposition 3.1 gives us aset-theoretic embedding Ω ֒ → W . We can then consider the composite map Ω ֒ → W → N G ( T ), which givesus a section of the canonical projection N G ( T ) → Ω: ι : Ω → N G ( T ) ω = ǫ i w ∆ i w ∆ ǫ i ( ̟ − ) N ◦ ( w ∆ i w ∆ )That ǫ i ( ̟ − ) is well-defined follows from the fact that ǫ i ∈ X ∗ ( T ) by our definition of G earlier. That ι is a section follows from Proposition 3.1. In particular, ι is injective. We will sometimes identify ǫ i with ǫ i ( ̟ − ) for ease of notation.We will show that ι is a homomorphic embedding for all types except A l , and the specific case when G isadjoint of type D l where l is odd. Nonetheless, we will still produce a homomorphic embedding Ω ֒ → N G ( T )which is a section of N G ( T ) → Ω, in these two outlier cases.Suppose ω = ǫ i w ∆ i w ∆ is a generator of Ω, whose order is r . Propositions 3.2 and 3.3 will be dedicatedto showing that ι ( ω ) also has order r . Let w i = w ∆ i w ∆ for convenience of notation. We compute ι ( ω ) r = ( ǫ i ( ̟ − ) N ◦ ( w i )) r = ǫ i ( ̟ − ) · (cid:0) N ◦ ( w i ) ǫ i ( ̟ − ) N ◦ ( w i ) − (cid:1) · (cid:0) N ◦ ( w i ) ǫ i ( ̟ − ) N ◦ ( w i ) − (cid:1) · · · (cid:0) N ◦ ( w i ) r − ǫ i ( ̟ − ) N ◦ ( w i ) − r (cid:1) N ◦ ( w i ) r = ( ǫ i + w i ( ǫ i ) + w i ( ǫ i ) + · · · + w r − i ( ǫ i ))( ̟ − ) N ◦ ( w i ) r . We will now show that ǫ i + w i ( ǫ i ) + w i ( ǫ i ) + · · · + w r − i ( ǫ i ) = 0 and N ◦ ( w i ) r = 1. Proposition 3.2. ǫ i + w i ( ǫ i ) + w i ( ǫ i ) + · · · + w r − i ( ǫ i ) = 0 .Proof. We compute w ji ( ǫ i ) for j = 1 , , ..., r −
1. The tables on pages 18-19 of [IM65] give the values of i for each type, and the explicit action of w i on the set {− α , α , α , ..., α l } . We also note that the orderof ω equals the order of w i (see [IM65, p. 18]). We begin with type B l and end with type A l (since,computationally, A l is the most intricate). • In type B l , we have that r = 2 and i = 1, so we wish to show that ǫ + w ( ǫ ) = 0. Note that w ( α ) = − α and w fixes the other simple roots. To compute w ( ǫ ), we pair w ( ǫ ) with all ofthe simple roots. By Weyl-invariance of the inner product ( · , · ) and the fact that w = 1, we have( w ( ǫ ) , α j ) = (cid:26) j = 1( ǫ , − α ) if j = 1But since α = α + 2( α + ... + α l ), we have that ( ǫ , − α ) = −
1. Therefore, w ( ǫ ) = − ǫ , so that ǫ + w ( ǫ ) = 0. • In type C l , we have that i = l , and the same argument as in type B l holds. Indeed, w l ( α l ) = − α , w l permutes the simple roots other than α l , and α = 2( α + ... + α l − )+ α l . Thus, ( w l ( ǫ l ) , α l ) = − w l ( ǫ l ) = − ǫ l , and the result follows. MOSHE ADRIAN • We consider type D l . We first consider the case that l is odd and G is adjoint. In this caseΩ ∼ = Z / Z and it is enough to consider i = l . The claim is that ǫ l + w l ( ǫ l ) + w l ( ǫ l ) + w l ( ǫ l ) = 0.One can see from the table on [IM65, p. 19] that w l permutes α , α , ..., α l − , and also acts by − α α l α α l −
7→ − α . We therefore conclude that ( w l ( ǫ l ) , α j ) = ( ǫ l , w l ( α j )) = 0if j = 2 , , ..., l −
2. Moreover, since w l ( α ) = α l , w l ( α l − ) = α , w l ( α l ) = − α , we concludethat ( w l ( ǫ l ) , α ) = 1 , ( w l ( ǫ l ) , α l − ) = 0 , and ( w l ( ǫ l ) , α l ) = −
1. Therefore, w l ( ǫ l ) = ǫ − ǫ l . One cancompute similarly that w l ( ǫ l ) = ǫ l − − ǫ and w l ( ǫ l ) = − ǫ l − . Therefore, ǫ l + w l ( ǫ l )+ w l ( ǫ l )+ w l ( ǫ l ) =0. We now consider the case where l is odd and G is neither adjoint nor simply connected. We havethat ρ l = ρ generates Ω. Thus, we need to show that ǫ + w ( ǫ ) = 0. First, we note that w fixes α j , for j = 2 , , ..., l −
2, it exchanges − α and α , and it exchanges α l − and α l . Since w has order2, we compute that ( w ( ǫ ) , α j ) = ( ǫ , w ( α j )) = 0if j = 2 , ..., l . We also have ( w ( ǫ ) , α ) = ( ǫ , − α ) = −
1. Thus, w ( ǫ ) = − ǫ , so the result follows.We now consider the case that l is even and G is adjoint. In this case, Ω ∼ = Z / Z × Z / Z . In thenotation of [IM65, p. 19], the generators of Ω are ρ , ρ l − , ρ l . It is straightforward to compute that w ( ǫ ) = − ǫ , w l ( ǫ l ) = − ǫ l , and that w l − ( ǫ l − ) = − ǫ l − , proving the claim for G .If l is even and G is neither adjoint nor simply connected, the result follows readily from theadjoint case. • We now consider type E . Then r = 3 , i = 1, and w acts by α α
7→ − α . Since w has order3, we compute that ( w ( ǫ ) , α j ) = ( ǫ , w ( α j )) = j = 1 , − j = 11 if j = 6which implies that w ( ǫ ) = − ǫ + ǫ . Similarly one may compute that w ( ǫ ) = − ǫ . Therefore, ǫ + w ( ǫ ) + w ( ǫ ) = 0. • Type E is analogous to types B l and types C l . Just note that in this case we have i = 1 and w ( α ) = − α , and from [IM65, p. 19] we see that the coefficient of α in α is 1. • We finally consider type A l . We may identify roots and co-roots, fundamental weights and funda-mental co-weights. Recall that we may take ∆ = { α , α , ..., α l } to be α i = α ∨ i = e i − e i +1 for1 ≤ i ≤ l . The corresponding fundamental coweights are ǫ i = ǫ ∨ i = (cid:8) l +1 [( l + 1 − i )( e + e + ... + e i ) − i ( e i +1 + e i +2 + ... + e l +1 )] if i ≤ l We recall that the isogenies of type A l are in one to one correspondence with the subgroups ofΩ ad = Z / ( l + 1) Z . The element ρ generates Ω ad . Let a, b ∈ N such that l + 1 = ab . Let ω = ρ a , sothat ω b = 1. Let G be the group of type A l that is given by the subgroup h ω i of Ω ad . In particular,its associated cocharacter lattice, which we denote by X ∗ ( A al ), is given by h Q ∨ , ǫ a i . For ease ofnotation, let n = l + 1. Then ω = ǫ a w a , where w a is the a-th power of the n -cycle (1 2 · · · n ). Weneed to show that ǫ a + w a ( ǫ a ) + ... + w b − a ( ǫ a ) = 0. A computation shows that this sum is1 n [( n − a )( e + e + ... + e a ) − a ( e a +1 + ... + e n )]+ 1 n [( n − a )( e a +1 + e a +2 + ... + e a ) − a ( e a +1 + ... + e n + e + e + ... + e a )]+ ... + 1 n [( n − a )( e n − a +1 + e n − a +2 + ... + e n ) − a ( e + e + ... + e n − a )] , which equals zero. (cid:3) Proposition 3.3. N ◦ ( w i ) r = 1 . REMARK ON THE KOTTWITZ HOMOMORPHISM 5
Proof.
We again assume that G is not simply connected. We proceed on a type by type basis, beginningwith B l and ending again with type A l . To compute N ◦ ( w i ) r , we use Corollary 2.4. We remind the readeragain that the order of ǫ i w i equals the order of w i (see [IM65, p. 18]). • Suppose that G is of type B l and adjoint. We recall that the roots may be identified with thefunctionals ± e i (1 ≤ i ≤ l ) and ± e i ± e j (1 ≤ i < j ≤ l ). The corresponding coroots may be identified(in the obvious way) with the functionals ± e i , ± e i ± e j . The fundamental coweights correspondingto the standard choice of simple roots are given by ǫ i = e + ... + e i , for 1 ≤ i ≤ l .We note that the cocharacter lattice of type B l adjoint is h Q ∨ , ǫ i . The action of w exchanges − α and α . Therefore, F w (1) is the set of positive roots that contain α . In other words, F w (1) = { e + e j : j = 2 , , ..., l } ∪ { e − e j : j = 2 , , ..., l } ∪ { e } . One may therefore compute that X α ∈F w (1) α ∨ = X j> ( e + e j ) ∨ + X j> ( e − e j ) ∨ + e ∨ = X j> ( e + e j ) + X j> ( e − e j ) + 2 e , where we have identified e ∨ with 2 e in the usual way. Writing e + e j and e − e j as sums of simplecoroots, one may compute that X j> ( e + e j ) + X j> ( e − e j ) + 2 e = 2 lα ∨ + 2 lα ∨ + ... + 2 lα ∨ l − + lα ∨ l . Noting that ǫ = α ∨ + α ∨ + ... + α ∨ l − + α ∨ l , we have that X α ∈F w (1) α ∨ = 2 lǫ . Therefore, N ◦ ( w ) = ( ǫ )( − l = 1. • We now turn to type C l adjoint. We recall that the roots may be identified with the functionals ± e i (1 ≤ i ≤ l ) and ± e i ± e j (1 ≤ i < j ≤ l ). The corresponding coroots may be identified (in theobvious way) with the functionals ± e i , ± e i ± e j . The fundamental coweights corresponding to thestandard choice of simple roots are ǫ i = e + ... + e i , for 1 ≤ i < l , and ǫ l = ( e + e + ... + e l ).We note that the cocharacter lattice of type C l adjoint is h Q ∨ , ǫ l i . The action of w l exchanges − α and α l . Therefore, F w l (1) is the set of all positive roots that contain α l , so that F w l (1) = { e i + e j : 1 ≤ i < j ≤ l } ∪ { e i : 1 ≤ i ≤ l } . Therefore, one may compute that X α ∈F wl (1) α ∨ = X i 1) + 1) α l − + (cid:18) ( l − l − l − l (cid:19) α l . Modulo 2 X ∗ ( D adl ), γ is equivalent to α l − + α l . But α l − + α l ≡ ǫ (mod 2 X ∗ ( D adl )), so N ◦ ( w l ) =1 as needed.We now consider the group G , of type D l , with l odd, that is neither simply connected nor adjoint.We need to show that N ◦ ( w l ) = 1. First, we recall that w l fixes α i , for i = 2 , , ..., l − − α and α , and exchanges α l − and α l . We must therefore count the positive roots thatcontain α . But this has already been computed in the D l cases with l even, and our results thereimply that N ◦ ( w l ) = 1, noting that the cocharacter lattice in the current case is given by h Q ∨ , ǫ i . REMARK ON THE KOTTWITZ HOMOMORPHISM 7 • We now turn to the group G of type E and adjoint. We follow here [Bou02, Plate V], whichhas different conventions than [IM65]. The Weyl element w in question acts by α α 7→ − α .Therefore, we need to compute the sum of all roots α that contain α , together with all roots thatcontain α that also do not contain α . One computes that this sum is X α ∈F w (1) α ∨ + X β ∈F w (2) β ∨ = 16 α + 16 α + 24 α + 32 α + 24 α + 16 α ∈ Q ∨ . Therefore, N ◦ ( w ) = 1. • We now turn to E adjoint. We need to show that N ◦ ( w ) = 1, where w is the Weyl element inquestion. We follow here [Bou02, Plate VI], which has different conventions than [IM65]. Using thefact that w exchanges α and − α , one counts that the sum of all of the positive roots that contain α is X α ∈F w (1) α ∨ = 18 α + 27 α + 36 α + 54 α + 45 α + 36 α + 27 α . But this sum is exactly equal to 18 ǫ , so N ◦ ( w ) = ( ǫ )( − = 1. • We finally consider type A l . We re-adopt our conventions and notation from the proof of Proposition3.2 in the case of type A l . That is, we let a, b ∈ N such that l + 1 = ab . Let ω = ρ a , so that ω b = 1,and for ease of notation, let n = l + 1. Then ω = ǫ a w a , where w a is the a-th power of the n -cycle(1 2 · · · n ).A computation then shows that b − X m =1 X α ∈F wa ( m ) α ∨ = ( n − a )[ e + e + ... + e a ] + ( n − a )[ e a +1 + e a +2 + ... + e a ]+( n − a )[ e a +1 + ... + e a ] + ... + ( a − n )[ e n − a +1 + e n − a +2 + ... + e n − + e n ] . We denote this sum by γ . We recall that the cocharacter lattice of this isogeny is given by X ∗ ( A al ) = h Q ∨ , ǫ a i , where ǫ a = 1 n [( n − a )( e + e + ... + e a ) − a ( e a +1 + ... + e n )] . Suppose first that n is odd, so that a is also odd. Therefore, n − a, n − a, n − a, ..., a − n are alleven, so one can see that γ ∈ Q ∨ , which implies that N ◦ ( w a ) b = 1. Suppose now that n is even.Then γ − nǫ a = ( n − a )[ e a +1 + e a +2 + ... + e a ] + ( n − a )[ e a +1 + ... + e a ]+ ... + (2 a − n )[ e n − a +1 + e n − a +2 + ... + e n − + e n ] . One can see that γ − nǫ a =: η ∈ Q ∨ . Therefore, γ = η + nǫ a is twice a cocharacter, so N ◦ ( w a ) b = 1. (cid:3) Remark 3.4. The previous argument in the case of type A l depends on the group not being simply con-nected. Otherwise, it may not be that N ◦ ( w a ) b = 1. Indeed, if n is even, we relied on the fundamentalcoweight ǫ a being contained in the cocharacter lattice in order to conclude that N ◦ ( w a ) b = 1. If n is odd,however, it was automatic that N ◦ ( w a ) b = 1. Indeed, this does not conflict with a basic known example; if G = SL ( n ) and w is the long Weyl element, then N ◦ ( w ) n = (cid:26) − n is even1 if n is odd Theorem 3.5. For G a split, almost simple, p -adic group, there exists an embedding Ω ֒ → N G ( T ) that isalso a section of the canonical map N G ( T ) → Ω .Proof. Suppose that Ω is cyclic of order n . We have shown that if ω = ǫ i w ∆ i w ∆ ∈ Ω is a generator, then ι ( ω ) n = 1. But in fact ι ( ω ) has order n . To see this, note that if m < n , then ι ( ω ) m = ( ǫ i + w i ( ǫ i ) + ... + w m − i ( ǫ i ))( ̟ − ) · N ◦ ( w i ) m . But N ◦ ( w i ) m has a nontrivial projection to W ◦ since w i has order n . Therefore, ι ( ω ) m has a nontrivial projection to W ◦ as well, so in particular must be nontrivial. Since ι ( ω ) has order n , MOSHE ADRIAN we may define a homomorphism Ω → N G ( T ) by sending ω j to ι ( ω ) j , and one may check that this map is infact a section of the map N G ( T ) → Ω.It remains to consider the case where G is adjoint of type D l with l even, since its fundamental group isnot cyclic. We denote the associated cocharacter lattice by X ∗ ( D adl ). We show that ι is a homomorphismin this case. Recall that in Proposition 3.3, we showed that ι ( ω ) = 1 for each ω ∈ Ω. We need toshow that ι ( ρ ρ l ) = ι ( ρ ) ι ( ρ l ) , ι ( ρ ρ l − ) = ι ( ρ ) ι ( ρ l − ) , and ι ( ρ l − ρ l ) = ι ( ρ l − ) ι ( ρ l ). We will carry outthe case ι ( ρ ρ l ) = ι ( ρ ) ι ( ρ l ), noting that the other cases are similar. First note that ι ( ρ ρ l ) = ι ( ρ l − ) = ǫ l − N ◦ ( w l − ) and ι ( ρ ) ι ( ρ l ) = ǫ N ◦ ( w ) ǫ l N ◦ ( w l ) = ǫ N ◦ ( w ) ǫ l N ◦ ( w ) − N ◦ ( w ) N ◦ ( w l ). One can computethat ǫ N ◦ ( w ) ǫ l N ◦ ( w ) − = ǫ l − , so it suffices to show that N ◦ ( w ) N ◦ ( w l ) = N ◦ ( w l − ). By Corollary 2.2,we need to show that Y α ∈F ( w ,w l ) α ∨ ( − 1) = 1 . One computes that F ( w , w l ) = { α ∈ Π : w l ( α ) ∈ − Π , w w l ( α ) ∈ Π } = { α ∈ Π : α contains α l and α does not contain α l − } . This last set, by [Bou02, Plate IV], is the set { e i + e l : 1 ≤ i < l } . Adding these roots togethergives γ := α + 2 α + 3 α + ... + ( l − α l − + ( l − α l . But X ∗ ( D adl ) contains ǫ l − , ǫ l , and we see that γ = (2 − l ) ǫ l − + lǫ l , which lives in 2 X ∗ ( D adl ) since l is even. The result follows. (cid:3) Remark 3.6. (1) It is not difficult to show that ι is a homomorphism in the case that G is adjoint of type E . Sincewe cannot claim this for all types, we do not include the computation.(2) In the case that G is adjoint of type D l where l is odd, one can show that ι is not a homomorphism.In fact, one can show that ι ( ρ l ) = ι ( ρ l ), but it turns out that ι ( ρ l ) = ι ( ρ l ). This boils downto computing that the sum of all (co)roots in F w l (2) equals le − ( e + e + ... + e l ), which whenevaluated at − G is type A l , it turns out that ι is sometimes a homomorphism and sometimes not.For example, if a = 1 (in the notation of Proposition 3.3), then the group in consideration if P GL n (recall that in our notation, n = l + 1 = ab ), and one can show that ι ( ρ ) = ι ( ρ ). On the otherhand, if both n and a are even, then ι is a homomorphism.4. Beyond split almost-simple groups One may ask about generalizing Theorem 3.5 to more general connected reductive groups. The biggestobstacle to generalizing the result, using the methods in this paper, revolves around the fact that if W ◦ (Ω)denotes the projection of Ω onto the finite Weyl group, then N ◦ | W ◦ (Ω) : W ◦ (Ω) → N G ( T ) is not necessarilya homomorphism. This problem occurred in some A l types, as well as adjoint D l with l odd. But in thesecases, we were able to skirt this issue by adjusting ι as in Theorem 3.5, using the fact that Ω is cyclic.On the other hand, we are able to extend our result to certain additional split connected reductive groups.Note first that since G ad is a product of split, almost-simple groups, Theorem 3.5 gives a section s G ad of κ G ad : G ad ( F ) → Ω G ad . Definition 4.1. Call a homomorphic section s G of κ G good if it is compatible with the one constructed for G ad . In other words, the following diagram commutes: G ( F ) G ad ( F )Ω G Ω ad κ G κ G ad s G s G ad Remark 4.2. Recall that when G der = G sc , there is an easy way to produce a homomorphic section withvalues in T ( F ). However, this will not generally make the diagram commute, so it is not good. Proposition 4.3. Let G be a split connected reductive group over F . Let C be an alcove in the apartmentcorresponding to a split maximal torus T , with associated extended affine Weyl group W = X ∗ ( T ) ⋊ W ◦ .Then: (1) If Z = Z ( G ) is connected, then the induced map G ( F ) /Z ( O F ) → Ω G has a good homomorphicsection (the analogue of the diagram above commutes). REMARK ON THE KOTTWITZ HOMOMORPHISM 9 (2) If Z is connected and Ω G ∼ = Z (e.g. G = GSp (2 n ) ), then κ G has a good homomorphic section. (3) If Z is connected, Ω G ∼ = Z n , with n > , and ( | Ω G ad | , q ( q − , where q is the cardinality of theresidue field, then κ G has a good homomorphic section.Proof. We start with (1). It follows from Theorem 3.5 that κ G ad has a homomorphic section s G ad , since G ad is known to be a product of almost-simple groups. Moreover, if κ Z denotes the Kottwitz homomorphismfor Z ( F ), then κ Z also has a homomorphic section, which we denote s Z . As H ( F, Z ) = 1, we have acommutative diagram of exact sequences1 Z ( F ) G ( F ) G ad ( F ) 11 Ω Z Ω G Ω G ad κ Z κ G pr κ G ad pr We naturally have Z ( F ) /Z ( O F ) ∼ = X ∗ ( Z ), therefore obtaining another diagram1 Z ( F ) /Z ( O F ) G ( F ) /Z ( O F ) G ad ( F ) 11 Ω Z Ω G Ω G ad κ Z κ G pr κ G ad pr where κ Z , κ G are the induced maps. Let κ G ad denote the map induced from κ G ad on s ad (Ω ad ). Then wehave a commutative diagram of groups:1 Z ( F ) /Z ( O F ) pr − ( s ad (Ω ad )) s ad (Ω ad ) 11 Ω Z Ω G Ω G ad κ Z κ G pr κ G ad pr We have that κ Z , κ G ad are isomorphisms, so by the five lemma, κ G is an isomorphism, and thus the map κ G : G ( F ) /Z ( O F ) → Ω G has a homomorphic section.We now prove (2). Make an initial choice of a homomorphic section s Z of κ Z . Given σ ∈ Ω G , let s ( σ )be any lift in G ( F ) of s G ad (pr( σ )) ∈ N G ad ( T ad )( F ); it automatically lies in N G ( T )( F ). It might happenthat s is not a section of κ G . However, for all σ ∈ Ω G , we have pr( κ G ( s ( σ ))) = κ G ad (pr( s ( σ ))) = κ G ad ( s G ad (pr( σ ))) = pr( σ ). Thus, the difference between σ and κ G ( s ( σ )) belongs to Ω Z . Since κ Z issurjective, we may alter each s ( σ ) by an element z σ ∈ Z ( F ) in such a way that σ s ( σ ) z σ is a section of κ G .So we may assume s is a set-theoretic section of κ G , taking values in N G ( T )( F ). Because s G ad ishomomorphic, the map ( σ , σ ) s ( σ ) s ( σ ) s ( σ σ ) − is a 2-cocycle of Ω G with values in Z ( F ), with Ω G acting trivially on Z ( F ). Therefore, we get an elementof H (Ω G , Z ( F )). This group parameterizes isomorphism classes of extensions of Ω G by Z ( F ) where theinduced action of Ω G on the normal subgroup Z ( F ) is trivial (i.e. Z ( F ) is central in the extension group).We claim that the extension corresponding to the 2-cocycle is the direct product Z ( F ) × Ω G . This followsbecause Ω G = Z and H ( Z , A ) = 1 for any abelian group A with trivial Z -action.The fact that the extension is trivial means that the 2-cocycle defining it is a 2-coboundary. This meansthat we may alter our initial choice of set-theoretic section s to give a homomorphism s : Ω G → G ( F ),taking values again in N G ( T )( F ).The problem now is that s might not be a section of κ G , which we take care of as before. By construction, σ − κ G ( s ( σ )) ∈ Ω Z for every σ ∈ Ω G . So we may define z σ := s Z ( σ ( κ G ( s ( σ ))) − ) ∈ Z ( F ), for σ ∈ Ω G . Notethat σ z σ is a homomorphism Ω G → Z ( F ). Now define s G ( σ ) := z σ s ( σ ) . Then s G is the desired homomorphic section of κ G in case (2). In case (3), the same argument works, as long as we can prove that the 2-cocycle defined by s is still a2-coboundary. But when n > H ( Z , A ) always vanishes for abelian groups A withtrivial Z n -action. Nevertheless, we will show that the extension corresponding to the given 2-cocycle is stilltrivial. Write ˙ e i = s ( e i ), where e i corresponds to a standard basis vector in Ω G ∼ = Z n . Then the extensionis the exact sequence 1 → Z ( F ) → Z ( F ) h ˙ e , · · · , ˙ e n i κ G −−→ Ω G → . Write N := | Ω G ad | . As pr( ˙ e j ) ∈ im( s G ad ) ∼ = Ω G ad , we have pr( ˙ e j ) N = 1 and hence ˙ e Nj ∈ Z ( F ). Moreover,˙ e i ˙ e j ˙ e − i ˙ e − j ∈ Z ( F ). We may write a ˙ e j = ˙ e i ˙ e j ˙ e − i , for some a ∈ Z ( F ). Raising to the N -th power, we get a N ˙ e Nj = ˙ e Nj , and hence a N = 1. Therefore, a ∈ Z ( O F ). Moreover, since N is coprime to the pro-order of the profinitegroup Z ( O F ), we conclude that a = 1, and therefore the elements ˙ e i pairwise commute. Therefore, theextension is an abelian group. But then the extension is trivial, since Ω G ∼ = Z n .This concludes the proof of the proposition. But we make one additional comment. By construction, themap s G | Ω Z has image in Z ( F ) and so gives a homomorphic section s Z of κ Z . This section might be differentfrom the initial choice s Z . But now we have a commutative diagram1 Z ( F ) G ( F ) G ad ( F ) 11 Ω Z Ω G Ω ad κ Z κ G κ G ad s Z s G s G ad (cid:3) References [Bou02] N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6 Elements of Mathematics (Berlin), Springer-Verlag, Berlin,2002.[IM65] N. Iwahori and H. Matsumoto On some Bruhat Decomposition and the structure of the Hecke rings of p -adic Chevalleygroups. IHES Publ. Math. 25 (1965), 5-48.[Kot97] R. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255-339.[Ros16] S. Rostami, On the canonical representatives of a finite Weyl group , arXiv:1505.07442.[Spr98] T. Springer, Linear algebraic groups, Reductive Groups over Local Fields, Proceedings of Symposia in Pure Mathematics, Vol. 33 (1979), part 1, pp.29-69. Department of Mathematics Queens College, CUNY 65-30 Kissena Blvd., Queens, NY 11367-15971 E-mail address ::