Affine Deligne–Lusztig varieties of higher level and the local Langlands correspondence for GL2

Advances in Mathematics 299 (2016) 640–686

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Affine Deligne–Lusztig varieties of higher level and the local Langlands correspondence for GL2 A.B. Ivanov 1 Technische Universität München, Boltzmannstr. 3, 85748 Garching, Germany

a r t i c l e

i n f o

Article history: Received 17 April 2015 Received in revised form 23 May 2016 Accepted 23 May 2016 Available online 9 June 2016 Communicated by Roman Bezrukavnikov Keywords: Affine Deligne–Lusztig varieties Local Langlands correspondence Supercuspidal representations Cuspidal types

a b s t r a c t In the present article we define coverings of affine Deligne– Lusztig varieties attached to a connected reductive group over a non-archimedean local field. In the case of GL2 and positive characteristic, the unramified cuspidal part of the local Langlands correspondence is realized in the -adic cohomology of these varieties. We show this by giving a detailed comparison with the realization of local Langlands via cuspidal types by Bushnell–Henniart. All proofs are purely local. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The classical Deligne–Lusztig theory aims for a geometric construction of representations of finite groups of Lie-type. In [9], Deligne and Lusztig constructed the so-called Deligne–Lusztig varieties attached to a connected reductive group over a finite field and could show that any irreducible representation of the group of Fq -valued points occurs

E-mail address: [email protected]. The author was partially supported by the ERC starting grant 277889 “Moduli spaces of local G-shtukas”. 1

http://dx.doi.org/10.1016/j.aim.2016.05.019 0001-8708/© 2016 Elsevier Inc. All rights reserved.

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in the -adic cohomology of these varieties. Since then one was trying to find similar constructions in the affine setting, aiming for a geometric realization of the local Langlands correspondence. However, usual geometric realizations of local Langlands make use of p-adic methods, formal schemes and adic spaces, also using the global theory. In the present article we introduce a very natural affine analog of Deligne–Lusztig varieties of arbitrary level attached to a connected reductive group over a local field F of positive characteristic. Using these varieties we realize the unramified part of the local Langlands correspondence for GL2 over F using only schemes over Fq and purely local methods. Moreover, we will give a detailed comparison of our construction with the theory of cuspidal types of Bushnell–Kutzko [6] (we use the language of Bushnell–Henniart [5]) and on the ‘algebraic’ side we will show an improvement of the Intertwining theorem [5] 15.1. To begin with, let F be a non-archimedean local field and let L denote the completion of the maximal unramified extension of F . Let OF resp. OL be the ring of integers of F resp. L. We denote by k resp. k¯ the residue field of F resp. L, and by q the cardinality of k. Let σ : k¯ → k¯ denote the k-automorphism given by x → xq . We also denote by σ ¯ the unique F -automorphism of L, lifting σ : k¯ → k. Let G be a connected reductive group over F and let G be a smooth model of G over OF . It is a central problem to realize smooth representations of the locally compact group G(F ) in the -adic cohomology of certain schemes (or formal schemes, or adic spaces, ...) over k (where  is prime to char(k)). Usually such schemes come up with an action of G(F ) × T (F ), where T is some maximal torus of G and as a consequence the representations of G(F ) occurring in their -adic cohomology are parametrized by characters of T (F ), lying in sufficiently general position. After the fundamental work of Deligne and Lusztig [9], which followed the pioneering example of Drinfeld concerning SL2 (k), and deals with representations of the finite group G(k), many generalizations of their ideas aiming at a construction of representations of G(OF /tr ) for r ≥ 2 resp. of G(F ) were made. We give some examples. In [12] Lusztig suggested such construction (without proofs) and more recently he gave proofs in [13]. (A minor variation of) this construction was worked out for division algebras by Boyarchenko [1] and Chan [7] (see also [2]). A further closely related approach was given by Stasinski in [17], who suggested a method to construct the so-called extended Deligne–Lusztig varieties attached to G(OF /tr ). The advantages of our construction are that it (i) has a quite simple definition in terms of the Bruhat–Tits building of G, (ii) establishes a direct link with affine Deligne–Lusztig varieties, which are well-studied in various contexts. In particular, this allows to use the whole combinatoric machinery developed for their study. A starting point for our construction is Rapoport’s definition of affine Deligne–Lusztig varieties in [16] Definition 4.1. We recall this definition (in the Iwahori case). Let BL be the Bruhat–Tits building of the adjoint group GL,ad . The Bruhat–Tits building of Gad over F can be identified with the σ-invariant subset of BL . Let S be a maximal L-split torus in G, which is defined over F (such a torus exists due to [4] 5.1.12). Let I ⊆ G(L) be the Iwahori subgroup attached to a σ-stable alcove in the apartment corresponding to S. Let F be the affine flag manifold of G, seen as an ind-scheme over k if F has positive

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characteristic, and seen as a perfect algebraic space in the sense of [19] otherwise. Its ¯ ˜ denote the extended affine Weyl group of can be identified with G(L)/I. Let W k-points G attached to S. The Bruhat decomposition of G(L) induces the invariant position map ¯ × F (k) ¯ →W ˜. inv : F (k) ˜ and b ∈ G(L) the affine Deligne–Lusztig variety attached to w and b is the For w ∈ W locally closed subset Xw (b) = {gI ∈ F : inv(gI, bσ(g)I) = w} of F , which is given the structure of the reduced induced sub-Ind-scheme resp. perfect algebraic subspace. Let Jb be the σ-stabilizer of b, i.e., the algebraic group over F defined by Jb (R) = {g ∈ G(R ⊗F L) : g −1 bσ(g) = b} for any F -algebra R. Then Jb (F ) acts on Xw (b). We sketch now the construction of natural covers of these varieties, which still admit an action by Jb (F ). The details are given in Section 2. Let Φ = Φ(G, S) be the relative root system. We see 0 as the ‘root’ corresponding to the centralizer T of S in G (as G is quasi-split, this is a maximal torus). After choosing a σ-stable special base point x in BL , with a concave function f on Φ ∪ {0} (for a definition cf. Section 2.1) one can associate a subgroup G(L)f ⊆ G(L). In [18], Yu defined a smooth model Gf of GL over OL , such that Gf (OL ) = G(L)f . Assume that G(L)f ⊆ I and that G(L)f is σ-stable. Then Gf descends to a smooth group scheme over OF . Further, G(L)/G(L)f is the set ¯ of k-points of an Ind-scheme resp. an Ind perfect algebraic space F f , which defines a natural cover of F , as follows from the work of Pappas and Rapoport [15] Theorem 1.4 resp. [19] Theorem 1.5. Moreover, if G(L)f is normal in I, then F f → F is a (right) principal homogeneous space under I/G(L)f . There is a map ¯ × F f (k) ¯ → DG,f , invf : F f (k) which covers the map inv. Here DG,f is a set of representatives of double cosets of G(L)f in G(L). For w ∈ DG,f , b ∈ G(L), we define the affine Deligne–Lusztig variety of level f attached to w and b as the locally closed subset ¯ : invf (¯ Xwf (b) = {¯ g = gG(L)f ∈ F f (k) g , bσ(¯ g )) = w}, endowed with its induced structure of a reduced sub-Ind-scheme resp. sub-(Ind) perfect algebraic space (for the mixed characteristic case, compare with [19] Section 0.3). In fact, in cases of interest this is a scheme resp. perfect scheme locally of finite type over k. Assume G(L)f is normal in I. Then I acts on DG,f by σ-conjugation w → i−1 wσ(i), hence we can consider the stabilizer If,w ⊆ I of w under this action. It acts on Xwf (b)

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on the right and this action commutes with the left action of Jb (F ). Moreover this If,w -action can be extended to an action of Z(F )If,w , where Z is the center of G. Thus we obtain the desired variety resp. perfect scheme Xwf (b) with an action of G(F ) ×Z(F )If,w . In the mixed characteristic case, note that perfect schemes have enough structure such that étale cohomology groups can be attached to them. We study further properties of If,w and Xwf (b) for general G elsewhere. The rest of the paper is devoted to the detailed study of G = GL2 in the equal characteristic case. Now we explain our results in this case. As the levels indexed by concave functions are cofinal, we restrict attention to special functions fm and elements w ∈ DG,fm (cf. Sections 2.1, 3.1) for integers m ≥ 0 and write I m instead of G(L)fm , Xwm (1) instead of Xwfm (b), etc. We determine the varieties Xwm (1) and the G(F )-representations in the cohomology of these varieties with Q -coefficients. Further we compare our results with the algebraic construction of the same representations in [5] using the theory of cuspidal types. We sketch our results here; for a precise treatment cf. Section 4.1. Let E/F be the unramified extension of degree 2. If the image of w in the finite Weyl group is non-trivial, then Z(F )Im,w has a natural quotient isomorphic to E ∗ , and the Z(F )Im,w -action in the -adic cohomology of Xwm (1) factors through an E ∗ -action. In this way we obtain a G(F )-representation in the spaces Hic (Xwm (1), Q )[χ], where χ goes through smooth ∗ Q -valued characters of E ∗ . It turns out that if χ is minimal of level m, lies in sufficiently general position, then there is an integer i0 , such that Hic (Xwm (1), Q )[χ] = 0 for all i = i0 and Rχ = Hic0 (Xwm (1), Q )[χ] is an unramified irreducible cuspidal representation of G(F ), of level m (we also define Rχ for χ non-minimal). Here for an irreducible cuspidal representation π of G(F ) to be unramified means essentially that π arises by an automorphic induction process from a character of an (anisotropic modulo center) torus of G(F ), which is split after an unramified extension of F . Alternatively, unramified cuspidal representations are those which contain unramified fundamental strata in terms of [5]. Cuspidal representations attached to (at least tamely) ramified tori can also be studied via Deligne–Lusztig type constructions, see for example the work of Stasinski [17] or a future work of the author [11]. Let Pnr 2 (F ) be the set of all isomorphism classes of admissible pairs over F attached to E/F (cf. [5] 18.2). Let A2nr (F ) be the set of all isomorphism classes of unramified irreducible cuspidal representations of G(F ). We defined a map nr R : Pnr 2 (F ) → A2 (F ),

(E/F, χ) → Rχ .

(1.1)

As a consequence of our trace computations in Sections 4.2–4.4, we see that this map is injective (cf. Proposition 4.26). Using the theory of cuspidal types and strata, Bushnell–Henniart attached to an admissible pair (E/F, χ) an irreducible cuspidal

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G(F )-representation πχ ([5] §19; we recall the construction briefly in Section 4.6). The tame parametrization theorem ([5] 20.2 Theorem) then shows that the map ∼

nr Pnr 2 (F ) −→ A2 (F ),

(E/F, χ) → πχ

is a bijection (also for even q). Here is our main result (which also works for even q). Theorem 4.3. Let (E/F, χ) be an admissible pair. The representation Rχ is irreducible cuspidal, unramified, has level (χ) and central character χ|F ∗ . Moreover, Rχ is isomorphic to πχ . In particular, the map (1.1) is a bijection. The proof is purely local. Two ideas in the proof follow [1,2]: it is Boyarchenko’s trace formula (cf. Lemma 4.7) and maximality of certain closed subvarieties of Xwm (1) (note that Xwm (1) itself is not maximal due to the presence of a ‘level 0 part’). The rest of the proof is independent of [1,2]. Finally, we remark that for G = GL2 and b superbasic, Jb (F ) = D∗ for D a quaternion algebra over F and the varieties Xxmm (b) seem to be very close (but unequal) to the varieties studied by Chan in [7] (cf. Section 3.6). Outline of the paper In Section 2 we define affine Deligne–Lusztig varieties for a connected reductive group G of level attached to a concave function on the roots. In Section 3 we compute these varieties for G = GL2 , b = 1 and determine their -adic cohomology. In Section 4.1 we recall the setup and state our main result for GL2 , Theorem 4.3. After performing necessary trace calculations in Sections 4.2–4.4, we compare our construction with that in [5] in Sections 4.5–4.6, and finish the proof of Theorem 4.3. 2. Coverings of affine Deligne–Lusztig varieties The goal of this section is to define coverings of affine Deligne–Lusztig varieties. 2.1. Concave functions and smooth models Let G be a connected reductive group over F . As k¯ is algebraically closed, GL is quasi-split over L. Let S ⊆ G be a maximal L-split torus, which is defined over F . Let T = ZG (S) be the centralizer of S. As GL is quasi-split, T is a maximal torus. Let Φ = Φ(GL , SL ) denote the relative root system. For a ∈ Φ, write Ua for the corresponding root subgroup and let U0 = T . Let BL be the Bruhat–Tits building of GL and let AS be the apartment corresponding to SL . We fix a σ-stable base alcove a contained in AS ˜ = R ∪ {r+ : r ∈ R} ∪ {∞} be the monoid and let x be one of its special vertices. Let R ˜ for as in [3] 6.4.1. Then x defines a filtration of Ua (L) by subgroups Ua (L)x,r (r ∈ R) a ∈ Φ (cf. [3] §6.2 and §6.4).

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Moreover, choose an admissible schematic filtration on tori in the sense of Yu [18] §4. This gives a filtration U0 (L)x,r = T (L)r on T . If G satisfies condition (T) from [18] 4.7.1, then this filtration is independent of the choice of the admissible filtration and coincide with the Moy–Prasad filtration on T (L), cf. [18] Lemma 4.7.4. Moreover, G satisfies (T) if it is either simply connected or adjoint or split over a tamely ramified extension [18] 8.1. We do not use this in the following. ˜ is called concave ([3] 6.4), if A function f : Φ ∪ {0} → R s 

s  f (ai ) ≥ f ( ai ).

i=1

i=1

˜ ≥0  {∞}. Let G(L)x,f be the subgroup of G(L) Fix a concave function f : Φ ∪ {0} → R generated by Ua (L)x,f (a) , a ∈ Φ ∪ {0}. By [18] Theorem 8.3, there is a unique smooth model Gx,f of GL over OL such that Gx,f (OL ) = G(L)x,f . Moreover, if G(L)x,f is σ-stable, then Gx,f descends to a group scheme defined over OF ([18] 9.1). We denote it again by Gx,f . Let I ⊆ G(L) be the Iwahori subgroup associated with a and let Φ+ ⊆ Φ denote the set of positive roots determined by a. Let fI be the concave function on Φ ∪ {0} defined by  fI (a) =

for a ∈ Φ+ ∪ {0}

0

fI (a) = 0+ for a ∈ Φ− .

˜ ≥0  {∞} be the concave Then G(L)x,fI = I (cf. [18] 7.3). For m ≥ 0 let fm : Φ ∪{0} → R function defined by  fm (a) =

m

if a ∈ Φ+

m+

if a ∈ Φ− ∪ {0}.

Write I m = G(L)x,fm . Lemma 2.1. For m ≥ 0, I m is normal in I and I m is σ-stable. In particular, I m admits a unique smooth model Gx,fm . This model is already defined over OF . Proof. I (resp. I m ) is generated by Ua (L)x,fI (a) (resp. Ua (L)x,fm (a) ) for a ∈ Φ ∪ {0}. To show normality, it is enough to show that for any roots a, b ∈ Φ ∪ {0}, the commutator (Ua (L)x,fI (a) , Ub (L)x,fm (b) ) is contained in I m . By [3] (6.2.1) V3 (we can treat 0 as a root), (Ua (L)x,fI (a) , Ub (L)x,fm (b) ) is contained in the subgroup generated by Upa+qb (L)pfI (a)+qfm (b) for p, q > 0 such that pa + qb ∈ Φ ∪ {0}. Now qfm (b) ≥ m, hence Upa+qb (L)pfI (a)+qfm (b) ⊆ I m can only happen if fI (a) = 0, fm (b) = m, fm (pa + qb) = m+ . This is equivalent to a ∈ Φ− ∪{0}, b ∈ Φ+ , a +b ∈ Φ− ∪{0}. This is impossible, hence Upa+qb (L)pfI (a)+qfm (b) ⊆ I m and the normality is shown. We show now the σ-stability

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of I m . On the one side, I is generated by the subgroups Uσ(a) (L)x,fI (σ(a)) (for varying a), and on the other side I = σ(I) is generated by σ(Ua (L)x,fI (a) ) = Uσ(a) (L)σ(x),fI (a) . Using parts (i), (ii) of [18] Theorem 8.3 we deduce that Uσ(a) (L)x,fI (σ(a)) = Uσ(a) (L)σ(x),fI (a) . But then the same is true also for fm instead of fI . From this the σ-stability of I m follows. 2 To have common notation for mixed and equal characteristic cases, for a k-algebra R set W(R) :=

  k OF R⊗ W (R) ⊗W (k) OF

if char(F ) > 0 if char(F ) = 0,

where W (R) denotes the p-typical Witt ring of R. In the mixed characteristic case, W behaves only well, if R is a perfect k-algebra. In any case, let  denote a uniformizer of F . Consider the loop group LG, which is the functor on the category of k-algebras, LG : R → G(W(R)[−1 ]). Assume the concave function f is such that G(L)x,f is σ-invariant. Let L+ Gx,f be the functor on the category of k-algebras defined by L+ Gx,f : R → Gx,f (W(R)), and taking perfection in the mixed characteristic case (as in [19] Section 1.1). Consider the quotient of fpqc-sheaves F f = LG/L+ Gx,f

on the category of k-algebras in the equal characteristic case resp. on the category of perfect k-algebras in the mixed characteristic case. By [15] Theorem 1.4 resp. [19] Theorem 1.5 it is represented by an Ind-k-scheme of ind-finite type over k¯ resp. by a ind per¯ = G(L)/G(L)x,f . ¯ fectly proper perfect algebraic space over k and its k-points are F f (k) Moreover, if g ≤ f are two concave functions as above, then we have a natural projection F f  F g . We write F = F fI for the affine flag manifold associated with Gx,fI , the smooth model of I and F m = F fm for m ≥ 0. 2.2. Affine Deligne–Lusztig varieties and covers We keep the notations from Section 2.1. We fix a concave function f : Φ ∪ {0} → ˜ ≥0  {∞}, such that f ≥ fI , i.e., G(L)x,f ⊆ I and s.t. G(L)x,f is σ-invariant, i.e., Gx,f R ¯ F f (k), ¯ is defined over OF . We write I f = G(L)x,f . There are natural σ-actions on F (k), which are compatible with natural projections.

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Let NT be the normalizer of T in G. Let W = NT (L)/T (L) be the finite Weyl group ˜ the extended affine Weyl group. If Γ denotes the absolute associated with S and W ˜ sits in the short exact sequence Galois group of L, then W ˜ → W → 0. 0 → X∗ (T )Γ → W Then the Iwahori–Bruhat decomposition states that G(L) =



I wI, ˙

˜ w∈W

where w˙ is any lift of w to N (L). Consider now the set of double cosets DG,f = G(L)x,f \G(L)/G(L)x,f , ˜ . If m ≥ 0, we also equipped with the natural projection map DG,f  I\G(L)/I ∼ =W write DG,m instead of DG,fm . At least for w ‘big’ enough, the fiber DG,f (w) over a ¯ ˜ can be given the structure of a finite-dimensional affine variety over k, fixed w ∈ W by parametrizing it using subquotients of (finite) root subgroups. As this seems quite technical and as in this article we only need the case G = GL2 (cf. (3.3)), we omit the corresponding result in this article. We obtain a map, which covers the classical relative position map. Definition 2.2. Define the map ¯ × F f (k) ¯ → DG,f invf : F f (k) ¯ on k-points by invf (xG(L)x,f , yG(L)x,f ) = w, where w is the double G(L)x,f -coset −1 containing x y. We come to our main definition. Definition 2.3. For f ≥ fI concave, such that G(L)x,f is σ-invariant, b ∈ G(L), and w ∈ DG,f we define the affine Deligne–Lusztig variety of level f associated with b, w as ¯ : invf (¯ Xwf (b) = {¯ g = gG(L)x,f ∈ F f (k) g , bσ(¯ g )) = w}, with its induced reduced sub-Ind-scheme resp. sub-Ind perfect algebraic space structure. We write Xwm (b) instead of Xwfm (b). As usual, Xwf (b) is equipped with two group actions. For b ∈ G(L), let Jb be the σ-stabilizer of b, i.e., the algebraic group over F defined by Jb (R) = {g ∈ G(R ⊗F L) : g −1 bσ(g) = b}

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for any F -algebra R. Then Jb (F ) acts on Xwf (b) for any f and w ∈ DG,f . If f ≥ f  and  w ∈ DG,f lies over w ∈ DG,f  , then Xwf (b) lies over Xwf  (b) and the Jb (F )-actions are compatible. To describe the second group action, assume additionally that G(L)x,f is normal in I. ˜ , we have a left and a right I/I f -action on DG,f (w) For w ¯ ∈W ¯ by multiplication. We obtain the (right) I/I f -action on DG,f (w) ¯ by (i, w) → i−1 wσ(i). ˜ and w ∈ DG,f (w). Lemma 2.4. Assume I f is normal in I. Let b ∈ G(L), w ¯∈W ¯ (i) Xwf (b) is locally of finite type over k. (ii) For every g ∈ G(L), the map (h, xI f ) → (g −1 hg, g −1 xI f ) defines an isomorphism ∼ of pairs (Jb (F ), Xwf f (b)) −→ (Jg−1 bσ(g) (F ), Xwf f (g −1 bσ(g))). ∼

(iii) For i ∈ I, the map xI f → xiI f defines an isomorphism Xwf (b) −→ Xif−1 wσ(i) (b). Proof. (ii) and (iii) are trivial computations. (i): The affine Deligne–Lusztig varieties Xw (b) are locally of finite type, F f  F is a I/I f -bundle and I/I f is of finite dimension ¯ 2 over k. By Lemma 2.4 (iii), the σ-stabilizer If,w = {i ∈ I : i−1 wσ(i) = w} of w ∈ DG,f (w) ¯ in I acts on Xwf (b) by right multiplication, and this action factors through an action of If,w /I f . Let Z denote the center of G. Note that Z(F ) ⊆ Jb (F ), and that Jb (F )-action restricted to Z(F ) can also be seen as a right action, thus extending the right If,w -action on Xwf (b) to a right Z(F )If,wf -action. If m ≥ 0, we also write Im,w instead of Ifm ,w . 3. Computations for GL2 From now on and until the end of the paper we set G = GL2 and restrict ourselves to the case of positive characteristic, i.e., F = k((t)) is the field of Laurent series with   ¯ uniformizer t, L = k((t)) and σ : L → L is given by σ( i ai ti ) = i aqi ti . In this section, we compute the associated varieties Xwm (1) and their -adic cohomology. 3.1. Some notations and preliminaries We fix the diagonal torus T and the upper triangular Borel subgroup B of G. We set K = G(OF ) and fix the Iwahori subgroup I and its subgroups I m for m ≥ 0:  Im =

pm 1 + pm+1 L L m+1 pL 1 + pm+1 L



 I=

∗ OL OL ∗ pL OL

 ⊆ G(OL ).

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Note that the groups I m coincide with those defined in Section 2.1 with respect to the valuation on the root datum, which corresponds to the vertex of the Bruhat–Tits building of G associated with the maximal compact subgroup G(OL ). The maximal torus T is split over F and hence the filtration on it does  not depend on  the choice of an admissible r 0 1+p ˜ be the affine schematic filtration. It is given by T (L)r = . Let Wa ⊆ W 0 1 + pr and the extended affine Weyl group of G. The variety Xw (1) is empty, unless w = 1 or w ∈ Wa with odd length (cf. e.g. [10] Lemma 2.4). The case w = 1 is not very interesting: X1 (1) is a disjoint union of points and the cohomology of coverings of X1 (1) contains the principal series representations of G(F ), as for classical Deligne–Lusztig varieties and as in [10] in case of level 0. Thus we restrict attention to elements of odd length in Wa . To simplify some computations, we fix once for all time an even integer n > 0 and the elements  w˙ =

0 t−n −tn 0



 ,

v˙ =



n

t2

t− 2

n

∈ NT (L) ⊆ G(L)

(3.1)

and denote by w (resp. v) the image of w˙ (resp. v) ˙ in Wa (the elements with n < 0 can be obtained by conjugation; the elements with n odd lead to similar results). We denote the image of w˙ in DG,m again by w˙ (from the context it is always clear, in which set w˙ lies). Let prm : F m → F be the natural projection. Let Cv ⊆ F denote the (open) −1 Schubert cell attached to v. We have the following parametrizations of Cvm = prm (Cv ) and DG,m (w). For m ≥ 0, let Rm denote the Weil restriction functor Res(k[t]/tm+1  )/k from  k[t]/tm+1 -schemes to k-schemes. Cv is parametrized by Rn−1 Ga → Cv , a → where a =

n−1 i=0

1a 1

v,

ai ti . Then for m ≥ 0, Cvm is parametrized by ∼

ψvm : Rn−1 Ga × Rm G2m × Rm G2a −→ Cvm = IvI/I m       1a C 1A 1 a, C, D, A, B → v˙ Im 1 D 1 tB 1 (3.2) n−1 m m We write a = i=0 ai ti , A = i=0 Ai ti and C = c0 (1 + i=1 ci ti ). Moreover, for m ≤ n, DG,m (w) is parametrized by ∼

2 ¯ 2 ¯ m m φm w : Rm Gm (k) × Rm−1 Ga (k) −→ DG,m (w) = I \IwI/I      1 C 1 (C, D), (E, B) → I m w˙ I m. tE 1 D tB 1

(3.3)

The proof that ψvm resp. φm w is an isomorphism of varieties resp. sets amounts to a simple computation. We omit the details.

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Finally, we remark the existence of the following determinant maps. Let x ∈ Wa . There is a natural k-morphism of k-varieties: detm : Cxm = IxI/I m → Rm Gm ,

yI m → det(y)

mod tm+1 .

In the same way we have the k-morphism detm : DG,m (w) → Rm Gm ,

I m yI m → det(y)

mod tm+1 .

3.2. The structure of Xwm ˙ (1) Lemma 3.1. Let m ≥ 0. There is a natural isomorphism  ∼

Im,w˙ /I m −→

 C A 0 D

m+1 ¯ ∈ G(k[t]/t ) : σ 2 (C) = C, D = σ(C) .

Proof. An easy computation using (3.3) shows the lemma. 2 r m+1 ¯ ¯  k[t]/t denote the reduction modulo tm+1 . Using For r > m, let τm : k[t]/t coordinates from (3.2), let S = τm (σ(a) − a) and let Yvm ⊆ Cvm be the locally closed subset defined by

a0 ∈ /k B=0

(3.4)

σ(C)D−1 S −1 = 1 σ(D)C −1 S = 1 Let Dv ⊆ Cv be the open subset defined by the condition a0 ∈ / k. The composition Yvm → Cvm → Cv factors through Yvm → Dv . The natural K-action on Cvm by left multiplication restricts to an action on Yvm (this will follow implicitly from the proof of Theorem 3.2). Moreover, Lemma 3.1 implies that the natural right I/I m -action on Cvm restricts to a right action of Im,w˙ /I m on Yvm . m Theorem 3.2. Let 0 ≤ m < n. Let w = φm w (C, D, E, B) ∈ DG,m (w). Then Xw (1) is  non-empty if and only if one has B = −σ(E). If this holds true, then w is I-σ-conjugate  −1 to w˙ = φm wσ(i) ˙ ∈ DG,m for some i ∈ I). In w (1, 1, 0, 0) in DG,m (w) (that is w = i m m ∼ particular, Xw (1) = Xw˙ (1), compatible with appropriate group actions. Further, there is an isomorphism equivariant for the left G(F )- and right (I/I m )w˙ -actions:

∼ Xwm ˙ (1) =

 G(F )/K

Yvm .

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Proof. In [10] it was shown that Xw (1) = g∈G(F )/K gDv is the decomposition of Xw (1) in connected components. As the natural projection F m → F restricts to a map prm : Xwm (1) → Xw (1), we have   Xwm (1) ∼ pr−1 g pr−1 = m (gDv ) = m (Dv ). G(F )/K

G(F )/K

−1 Thus it is enough to determine prm (Dv ). One sees from Lemma 3.3, that if w = m −1 (D φw (C, D, E, B) does not satisfy B = −σ(E), thenprm  v ) = ∅. On the other hand, 1 if w satisfies this, then σ-conjugating w first by ∈ I and then by a diagonal B 1   i1 2 i= ∈ I such that i−1 1 Cσ(D)σ (i1 ) = 1 (such i1 exists by Lang’s theorem) and i2 i2 = Cσ(i1 ), we deduce that w is I-σ-conjugate to w˙ in DG,m . Thus by Lemma 2.4(iii) −1 we may assume w = w. ˙ In this case Lemma 3.3 shows prm (Dv ) = Yvm , which finishes the proof. 2

Lemma 3.3 (Key computation). Let 0 ≤ m < n. Let xI ˙ m = ψvm (a, C, D, A, B) ∈ Cvm such that a0 ∈ / k. Write S = τm (σ(a) − a). Then −1 −1 invm (xI ˙ m , σ(x)I ˙ m ) = φm S , σ(D)C −1 S, −B, σ(B)). w (σ(C)D

Proof. Let

 x˙ =

t 2 t− 2 a n t− 2 n

n



 C D

 1A 1

 1 tB 1

∈ G(L).

We have to compute the (I m , I m )-double coset of x˙ −1 σ(x). ˙ By assumption S is a unit and one computes (using m < n) 

t 2 t− 2 a n t− 2 n

n

−1  σ

t 2 t− 2 a n t− 2 n

n



 =

1 t−n (σ(a) − a) 1



 ∈ Im

 S S −1

wI ˙ m,

in G(L). Thus by normality of I m in I, we obtain:       1 S 1 −A C −1 −1 m x˙ σ(x) ˙ ∈I · w˙ . . . D−1 −tB 1 S −1 1     σ(C) 1 σ(A) 1 ... I m. σ(D) 1 tσ(B) 1 Then we can pull the term containing −A to the right side of w, ˙ without changing the other terms. The corresponding term, which then appears on the right side of w˙ will lie in I m , i.e., we can cancel it by normality of I m in I. The same can be done then with

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the term containing σ(A), by pulling it to the left side of w˙ and canceling it. Computing the remaining matrices together, we obtain:      1 1 σ(C)D−1 S −1 −1 m x˙ σ(x) ˙ ∈I w˙ I m. C −1 σ(D)S −tB 1 tσ(B) 1 This finishes the proof. 2 3.3. The structure of Yvm We keep notations from Sections 3.1 and 3.2. Let k2 /k denote the subextension of ¯ of degree two. There is a natural surjection k/k   C Im,w˙  Tw,m = : C ∈ (k2 [t]/tm+1 )∗ . σ(C) Let Tw,m,0 = Tw,m ∩ SL2 (k2 [t]/tm+1 ) be the subgroup defined by the condition C −1 = r ¯ we write X = σ(C). Let f : k¯ → k¯ denote the map f (x) = xq − x. For X ∈ k[t]/t r−1 i ¯ i=0 Xi t . We denote the affine space (over k) spanned by coordinates X0 , . . . , Xr−1 by Ar (X0 , . . . , Xr−1 ) resp. by Ar (X). Recall that S = τm (σ(a) − a) is a function on Dv m+1 ∗ ¯ ¯ with values in (k[t]/t ) = (Rm Gm )(k). Proposition 3.4. Let 0 ≤ m ≤ n. (i) The variety Yvm is isomorphic to the finite covering of Dv × Am (A) which is the closed subset of Dv × Am (A) × Rm Gm cut out by the equation σ 2 (C)C −1 = σ(S)S −1

(3.5)

¯ It is a finite étale Galois covering with Galois group Tw,m . in (Rm Gm )(k). (ii) The (set-theoretic) image of detm : Yvm → Rm Gm is the disjoint union of the k-rational points, which is as a set equal to (k[t]/tm+1 )∗ . Moreover, π0 (Yvm ) = (k[t]/tm+1 )∗ . If m > 0, the map π0 (Yvm )  π0 (Yvm−1 ) induced by the projection corresponds to the reduction modulo tm map. m (iii) Let Yv,0 be the connected component of Yvm corresponding to 1 ∈ (k[t]/tm+1 )∗ . Then m Yv,0 is (isomorphic to) a finite covering of Dv × Am given by Cσ(C) = S.

(3.6)

It is a connected finite étale Galois covering with Galois group Tw,m,0 . Moreover, m−1 m Yv,0  Yv,0 × A1 (Am−1 ) is given by cqm + cm =

m−1 f (am )  q − ci cm−i . f (a0 ) i=1

(3.7)

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Proof. Note that for a point ψvm (a, C, D, A, 0) ∈ Yvm , S = τm (a), C, D are units in m+1 ¯ . From the last two equations in (3.4), we see that on Yvm , D = σ(C)S −1 is k[t]/t uniquely determined by C and S and that Yvm is indeed given by the equation (3.5). Let us from now on proceed by induction on m. We see that Yv0 → Dv is defined by 2 c0q −1 = f (a0 )q−1 , i.e., it is finite étale with Galois group isomorphic to k2∗ . Clearly, Yvm lies over Yvm−1 × A1 (Am−1 ). Bring equation (3.5) to the form σ 2 (C)S = Cσ(S). m  Expanding this expression with respect to C = c0 (1 + i=1 ci ti ), S = i f (ai )ti , shows that Yvm → Yvm−1 × A1 (Am−1 ) is defined by an equation of the form 2

cqm − cm = p(a0 , . . . , am , c0 , . . . , cm−1 ) with p some regular function on Yvm−1 ×A1 (Am−1 ). This is clearly a finite étale covering. Moreover, it is Galois and the Galois group is isomorphic to k2 , where λ ∈ k2 acts by cm → cm + λ. By induction, Yvm → Dv × Am (A) is also finite étale and has degree (q 2 −1)q 2m . Equation (3.5) shows that the automorphism group of this covering contains Tw,m . Comparing the degrees we see that Yvm → Dv × Am is Galois with Galois group Tw,m . This shows part (i). Let xI ˙ m = ψvm (a, C, D, A, 0) ∈ Yvm . As Yvm ⊆ Xwm ˙ (1), we obtain m m −1 1 = detm (φm ˙ σ(x)I ˙ m ) = (CD)−1 σ(CD). w (1, 1, 0, 0)) = det (I x

It follows that detm (xI ˙ m ) = CD ∈ (k[t]/tm+1 )∗ . This shows the claim about the image of detm . In particular, we obtain a map π0 (Yvm ) → (k[t]/tm+1 )∗ . Its surjectivity follows using the action of Tw,m on Yvm and the fact that det : Tw,m → (k[t]/tm+1 )∗ is surjective. m m Let Yv,0 be the preimage of 1 under detm : Yvm → Rm Gm . Then Yv,0 is connected: this is m a byproduct of Lemma 3.9 (i) below. The compatibility of det with changing the level is immediate. Thus it remains to prove part (iii) of the proposition. Equation CD = 1, m holding on Yv,0 , inserted into (3.4) shows the first claim of (iii). The second statement m  of (iii) is clear from parts (i), (ii). Inserting C = c0 (1 + i=1 ci ti ), S = i f (ai )ti into (3.6) shows (3.7). 2 3.4. Cohomology of Yvm We keep the notations from Sections 3.1–3.3. Fix a prime  = char(k). We are interested in the -adic cohomology with compact support of the base change of Yvm ¯ To simplify notation, for a scheme X over k, we write Hi (X) for the space to k. c i ¯ Q ). This space comes with a natural action of the Frobenius σ. Set Hc (X ×k Spec k, hic (X) = dimQ Hic (X). Further, Hic (X)(r) denotes the r-th Tate twist. Set: ¯ N− = {a0 , c0 ∈ k¯ : a0 ∈ k2  k, cq+1 = f (a0 )} ⊆ k¯ × k, 0 and let C+ , C− be affine curves over Fq defined by

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C± : xq ± x = y q+1 , where for C− we additionally require x ∈ / k. We write V± = H1c (C± ). One has dimQ V+ = q(q − 1).  i m Theorem 3.5. Let 0 ≤ m < n. Then Hic (Yvm ) = (k[t]/tm+1 )∗ Hc (Yv,0 ). Let d0 = m ) = 0 if i > d0 + 1 or i < d0 − m and d0 (n, m) = 2(n − 1) + 2m + 1. Then Hic (Yv,0 m ∼ Hdc 0 +1 (Yv,0 ) = Q (−(n + m)) m ∼ Hdc 0 (Yv,0 ) = V− (−(n + m − 1)) q2(j−1) (q−1) m ∼ Hdc 0 −j (Yv,0 )= Q

for any 1 ≤ j ≤ m.

N− m For 1 ≤ j ≤ m the action of Frobq2 on Hcd0 −j (Yv,0 ) is given as follows: it acts by permuting the blocks corresponding to elements of N− (by (a0 , c0 ) → (a0 , −c0 )) and acts as multiplication with the scalar (−1)d0 −j q d0 −j in each of these blocks.

Remark 3.6. We have chosen d0 such that Hcd0 −j (Yvm ) corresponds to a G(F )-representation of level j (cf. Definition 4.2). Proof. The first statement of the theorem follows from Proposition 3.4. We need some further notation: k− = k− (x) = {x ∈ k2 : xq + x = 0} ⊆ k2 N+ = N+ (x, y) = {x, y ∈ k¯ : x ∈ k2 , y q+1 = xq + x} ⊆ k¯ × k¯ . m, Let Yv,0 be the finite étale covering of the open subset {a0 ∈ / k} of the m +1-dimensional m+1 m affine space A (a0 , . . . , am ), which is defined by the same equations defining Yv,0 (cf. m, m m m (3.7)). There is a projection Yv,0  Yv,0 and the Im,w˙ /I -action on Yv,0 induces a m, m, m ∼ Tw,m -action on Yv,0 . We have Yv,0 = Yv,0 × An−1 (am+1 , . . . , an−1 , A0 , . . . , Am−1 ) and i−2(n−1)

m hence Hic (Yv,0 ) = Hc

m, m, (Yv,0 )(−(n − 1)). For m ≥ 0, let Z m ⊆ Yv,0 be the closed 2

subscheme defined by the equation aq0 − a0 = 0. Note that K- and Tw,m -actions on m, Yv,0 restrict to actions on Z m and that equation (3.7) defines Z m ⊆ Z m−1 × A1 (cm ) 2

as a covering of Z m−1 . The equation aq0 − a0 = 0 divides Z m into a disjoint union of m, q 2 − q components, which are given by the same equations as Yv,0 and on which a0 is a fixed constant in k2  k. Thus on Zm (for each m ≥ 1) we may change our equations 1 replacing am by am = f (a0 )− q am − cm . For a0 ∈ k2  k, one computes f (a0 )q = −f (a0 ) and equation (3.7) simplifies over the locus a0 ∈ k2  k to a,q m

+

am

=

m−1  i=1

cqi cm−i .

(3.8)

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Now we make a coordinate change: for all m ≥ 1 replace am by αm = am −  m−1 2  q ci cm−i . This coordinate change turns equation (3.8) defining Z m over Z m−1 i=1 into  m−1 2  q αm + αm =



2

(ci − cqi )cqm−i + δm cq+1 m/2 ,

(3.9)

i=1

where δm = 0 if m is odd and δm = 1 if m is even. All together, Z m is isomorphic to 2 the locally closed subset of A2m+2 (a0 , α1 . . . , αm , c0 , c1 , . . . , cm ) defined by aq0 − a0 = 0, aq0 − a0 = 0, cq+1 = aq0 − a0 and the m equations (3.9) for m = 1, 2, . . . , m. The 0 first three of these equations and the equation (3.9) for m = 1 obviously divide Z m into N− × k− (α1 ) components, which are all isomorphic, as one sees using K- and

Tw,m -actions on Z m . Thus Z m ∼ = N− ×k− (α1 ) Z0m , where Z0m is the closed subvariety of A2m−2 (α2 , . . . , αm , c1 , . . . , cm ) defined by equations (3.9) for m = 2, . . . , m. Lemma 3.7. Let m ≥ 1. Then Z0m is connected, i.e., π0 (Z m ) = N− × k− (α1 ). Proof. We proceed by induction: for m = 0, Z00 is a point, thus connected. Let m ≥ 1 and assume that Z0m−1 is connected. By Lemma 3.9(ii) below (this lemma is formulated for Z˜m instead of Z m ∼ = Z˜ m−1 × A1 (cm ) – see below in the proof of the theorem), the fibers of Z m → Z m−1 (and hence also of Z0m → Z0m−1 ) over the open subset defined by 2 cq1 − c1 = 0 are connected. Hence Z0m is connected. 2 We see that Z0m is a connected étale covering of Am (c1 , . . . , cm ). Hence Hic (Z m ) = 0  m for i > 2m and H2m c (Z ) = N− ×k− (α1 ) Q (−m). Consider now the decomposition in an open and a closed subset: m, m, Yv,0  Z m → Yv,0 ← Z m .

(3.10)

m, 0, 0, Lemma 3.9 shows that Hic (Yv,0  Z m ) = Hi−2m (Yv,0  Z 0 )(−m) and Yv,0  Z 0 can be c identified with the open subset C−  N− of the curve C− defined in the variables a0 , c0 .

Lemma 3.8. In the long exact sequence for H∗c (·) attached to (3.10) (cf. [14] III §1 Remark 1.30), the map m, m 2m+1 δm : H2m (Yv,0  Z m) = c (Z ) → Hc



Q (−m) ⊕ V− (−m)

N−

is surjective onto the first summand. Proof. By comparing the Frobenius-weights (which is possible due to Lemma 3.9) we see that the image is contained in the first summand. On the other hand, the natural projec-

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m, m−1, tion Yv,0 → Yv,0 induces a morphism between the corresponding long exact sequences ∗ for Hc (·), which induces a commutative diagram relating δm with δm−1 . Iterating this for all levels ≥ 1, we obtain a commutative diagram:

 N− ×k−

Q (−m)

 N−

m H2m c (Z )

Q

H0c (Z 0 )

δm

m, H2m+1 (Yv,0  Z m) c

δ0

H1c (C−  N− )

 N−

Q ⊕ V−

This diagram shows the lemma. 2 The long exact sequence for H∗c (·) and Lemma 3.8 implies:

m, Hic (Yv,0 )

⎧ ⎪ Hic (Z m ) ⎪ ⎪  ⎪  ⎪ α1 =0 ⎪ [ Q (−m)] ⎪  N k (α ) ⎨ − − 1 =

V− (−m) ⎪ ⎪ ⎪ ⎪ Q (−m − 1) ⎪ ⎪ ⎪ ⎩ 0

if i < 2m if i = 2m if i = 2m + 1

(3.11)

if i = 2m + 2 if i > 2m + 2,

 where α1 = 0 means that we take the subspace of sum zero elements. Let Z˜ m−1 be the closed subspace of A2m+1 (a0 , α1 , . . . , αm , c0 , c1 , . . . , cm−1 ) given by the same equations 2 as Z m : aq0 − a0 = 0, aq0 − a0 = 0, cq+1 = aq0 − a0 and equations (3.9) for m = 1, . . . , m. 0 2 Let H = {cq1 − c1 = 0}; this is a finite union of hyperplanes in the same affine space. Then Z m ∼ = Z˜ m−1 × A1 (cm ). For m ≥ 3 Lemma 3.9(ii) shows (here and until (3.15) we ignore Tate twists): Hic (Z˜ m−1  H) = Hi−2(m−2) (Z˜ 1  H) c ⎧ ⎪ 0 if i ≤ 2m − 4 ⎪ ⎪ ⎪   ⎨ [( Q ) ⊕ V+ ] if i = 2m − 3 = N− ×k− (α1 ) N+ (α2 ,c1 )  ⎪  ⎪ ⎪ ⎪ Q if i = 2m − 2, ⎩ N− ×k− (α1 )

because Z˜ 1  H ∼ = N− ×k− (α1 ) (C+  N+ ). Further, Lemma 3.10 shows Z˜ m−1 ∩ H =

m−2 m−2 ∼ = Z m−2 and the index (1) indicates the shift k− (α1 )×N+ (α2 ,c1 ) Z(1) , where Z(1) in variables given by αi → αi+2 , ci → ci+1 (for i ≥ 1) and hence Hic (Z˜ m−1 ∩ H) =  m−2 i ˜ m−1 ∩ H k− (α1 )×N+ (α2 ,c1 ) Hc (Z(1) ) and, in particular, the top cohomology group of Z is in degree 2m − 4 and is equal to

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Hc2m−4 (Z˜ m−1 ∩ H) =



m−2 Hc2m−4 (Z(1) )=

657



Q ,

k− (α1 )×N+ (α2 ,c1 ) N− ×k− (α3 )

k− (α1 )×N+ (α2 ,c1 )

as follows from Lemma 3.7 (note the index shift α1 → α3 ). All these, the long exact sequence for H∗c (·) attached to Z˜ m−1  H → Z˜ m−1 ← Z˜ m−1 ∩ H, the analog of Lemma 3.8 for this sequence and Lemma 3.10 show that for m ≥ 3 we have: ⎧ m−2 i−2 ⎪ k− (α1 )×N+ (α2 ,c1 ) Hc (Z(1) ) ⎪ ⎪   ⎪ ⎨ α3 =0 N− ×k− (α1 )×N+ (α2 ,c1 ) [ k− (α3 ) Q ] i m i−2 ˜ m−1 Hc (Z ) = Hc (Z )=  ⎪ ⎪ N− ×k− (α1 ) V+ ⎪ ⎪ ⎩ N− ×k− (α1 ) Q

if i < 2m − 2 if i = 2m − 2 if i = 2m − 1 if i = 2m. (3.12)

m Note that N+ = q 3 , k− = q. Hence for m ≥ 3, we have h2m c (Z ) = (N− )q and 2m−j m 2j 1 2 hc (Z ) = (N− )q (q − 1) for j ∈ {1, 2}. For Z , Z one computes: H2c (Z 1 ) =  i 1 N− ×k− Q and Hc (Z ) = 0 if i = 2 and

Hic (Z 2 )

=

˜1 Hi−2 c (Z )

=

⎧ ⎪ ⎪ ⎨0

if i ≤ 2 or i ≥ 5

N− ×k− (α1 ) V+ ⎪ ⎪ ⎩ Q N− ×k− (α1 )



if i = 3

(3.13)

if i = 4.

   Let now m ≥ 3. For j > 0, write j = 2 j−1 2  + j , where j = 1 if j odd, j = 2 j−1 otherwise. Iterating (3.12)  2  times, we get for all 0 < j < m:

hc2m−j (Z m ) = q 4

j−1 2 

 2(m−2 j−1 2 )−j

hc

m−2 j−1 

(Z( j−1 )2 ) = (N− )q 2j (q − 1),

(3.14)

2

m ∼ m where Z(l) = Z using the index shift as above l times. Thus for all m ≥ 1, j > 0:

hc2m+1−j (Z m )

⎧ ⎪ ⎪(N− )q ⎨ =

if j = 1

(N− )q 2(j−1) (q − 1) if 1 < j ≤ m ⎪ ⎪ ⎩0 otherwise.

(3.15)

Combined with (3.11), this implies the dimension formula in the theorem. It remains to m compute the Frobenius action. The Tate twists in the two top cohomology groups of Yv,0 m, m can be deduced easily by relating Yv,0 with Yv,0 . To prove the claim about Frobq2 -action in degrees i ≤ d0 − 1, note that Frobq2 acts on N− by (a0 , c0 ) → (a0 , −c0 ). Further, let

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Z1m be the subvariety of A2m (c1 , . . . , cm , a1 , . . . , am ) defined by m equations (3.8) for m = 1, . . . , m, i.e., Z m = N− × Z1m , where N− is seen as a discrete variety. Lemma 3.11 shows that Z1m is a maximal variety over Fq2 (for a definition cf. the paragraph preceding Lemma 3.11), i.e., Frobq2 acts on Hic (Z1m ) by (−1)i (q 2 )i/2 = (−q)i for any i ∈ Z. Further we have for all 2 ≤ j ≤ m: m, m Hdc 0 −j (Yv,0 ) = Hc2m+1−j (Yv,0 )(−(n − 1)) = Hc2m+1−j (Z m )(−(n − 1))

(note that for j = 1, this remains true if one replaces the second equality by an inclusion, cf. (3.11)). This implies the last statement of the theorem. 2 Lemma 3.9. With notations as in the proof of Theorem 3.5, we have: m, m−1, (i) Let m ≥ 1. The fibers of the natural projection π : Yv,0  Z m → Yv,0  Z m−1 are 1 isomorphic to A . We have: m, m−1, Hic (Yv,0  Z m ) = Hi−2 (Yv,0  Z m−1 )(−1).

(ii) Let m ≥ 3. The fibers of the natural projection Z˜ m−1  H → Z˜ m−2  H are isomorphic to A1 . We have: ˜ m−2  H)(−1). Hic (Z˜ m−1  H) = Hi−2 c (Z m, Proof. Let us prove part (i). The scheme Yv,0  Z m is the closed subspace of m−1, m−1 2 (Yv,0 Z ) × A (am , cm ) defined by the equation (3.7). Letting x be a point of m−1, Yv,0  Z m−1 , we see that the fiber of π over x is given by the equation

cqm + cm = f (a0 (x))−1 (aqm − am ) + λ(x), with a0 (x) ∈ / k2 the a0 -coordinate of x and λ(x) ∈ k¯ depending on x. Using the substi1  tution am = f (a0 (x))− q am − cm , this equation can be rewritten as q −1 a q −1 )c a,q m − λ(x). m − f (a0 ) m = (1 + f (a0 (x)) 1

1

As a0 (x) ∈ / k2 we have f (a0 (x)) q −1 = 0, −1 and hence the fiber of π over x is isomorphic ¯ to the Artin–Schreier covering of A1 (cm ), hence is itself isomorphic to the affine (over k) line. This shows the first statement of the lemma. For the second statement, note that as the fibers of π are ∼ = A1 , we have Rc2 π∗ Q ∼ = j Q (−1) and Rc π∗ Q = 0 for j = 2. This together with the spectral sequence 1

m−1, m, m Hic (Yv,0  Z m−1 , Rcj π∗ Q ) ⇒ Hi+j c (Yv,0  Z )

implies the second statement of part (i). Part (ii) of the lemma has a similar proof, using (3.9) instead of (3.7). 2

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Lemma 3.10. With notations as in the proof of Theorem 3.5, for m ≥ 3, we have Z˜ m−1 ∩



Z m−2 . H∼ = k− (α1 ) N+ (α2 ,c1 ) 2

Proof. On the union of hyperplanes H, the term (c1 − cq1 )cqm−1 in the equation (3.9) defining Z˜ m−1 over Z˜ m−2 cancels, leaving the free variable cm and the equation arising from it (after renaming the variables by αi → αi−2 for i ≥ 3, ci → ci−1 for i ≥ 2) is simply the equation defining Z m−2 over Z m−3 . The lemma follows from this observation. 2 We recall the definition of maximal varieties from the introduction of [2], where it appears in a similar setup. Let X be a scheme of finite type over a finite field FQ with Q elements. Let FrobQ denote the Frobenius over FQ . By [8] Theorem 3.3.1, for each i and each eigenvalue α of FrobQ in Hic (X), there exists an integer i ≤ i, such that all complex  conjugates of α have absolute value Qi /2 . Hence by Grothendieck–Lefschetz formula we get an upper bound on the number of points on X: X(FQ ) =



(−1)i tr(FrobQ ; Hic (X)) ≤

i∈Z



Qi/2 hic (X),

(3.16)

i∈Z

where equality holds if and only if FrobQ acts on Hic (X) by the scalar (−1)i Qi/2 for each i ∈ Z. If this is the case, then X/FQ is called maximal. Lemma 3.11. Let Z1m be as in the proof of Theorem 3.5. For m ≥ 1, Z1m is a maximal variety over Fq2 . Proof. Frobq2 acts on Hic (Z1m ) as an endomorphism with eigenvalues being Weil numbers  with absolute value (q 2 )i /2 ≤ (q 2 )i/2 = q i . From (3.16) we obtain the upper bound u(Z1m , q 2 ) for the number of Fq2 -points on Z1m : Z1m (Fq2 ) ≤ u(Z1m , q 2 ) =

2m 

q i hic (Z1m ).

i=0

Using equation (3.15), we see that u(Z1m , q 2 ) = q 3m . On the other hand, let pi (c) = i−1 q q j=1 cj ci−j and let cj ∈ Fq 2 for j = 1, . . . , m be given. Then we have pi (c) = pi (c), i−1 q i.e., pi ((cj )j=1 ) ∈ Fq for all 1 ≤ i ≤ m. But the equation x + x = λ ∈ Fq has precisely q solutions in Fq2 . Thus for each given point (c1 , . . . , cm ) ∈ Am (Fq2 ), there are exactly q m points in Z1m lying over it (cf. equation (3.8)). Thus Z1m (Fq2 ) = q 3m , which finishes the proof. 2 3.5. Character subspaces We keep notations from Sections 3.1–3.4 and deduce some corollaries from Theorem 3.5.

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Lemma 3.12. Let m ≥ 0. The Im,w˙ /I m -action on Hic (Yvm ) factors through a Tw,m -action. Proof. This is immediate as the action of ker(Im,w˙ /I m  Tw,m ) on Yvm = Yvm, × m, An−1 (am+1 , . . . , an−1 , A0 , . . . , Am−1 ) (where Yvm, is defined analogously to Yv,0 in the n−1 proof of Theorem 3.5) comes from an action on A , which contributes to the cohomology of Yvm only via a dimension shift. 2 For an abelian (locally compact) group A, let A∨ denote the group of (smooth) characters of A. By Lemma 3.12 we have a decomposition

∗ Q -valued

Hic (Yvm ) =



Hic (Yvm )[χ]

(3.17)

∨ χ∈Tw,m

into isotypical components with respect to the action of Tw,m . ∗

Corollary 3.13. Let m ≥ 1 and 1 ≤ j ≤ m. Let χ : Tw,m → Q be a character. Then Frobq2 acts on Hcd0 −j (Yvm )[χ] by multiplication with the scalar χ(−1)(−1)d0 −j q d0 −j .  m Proof. We have Hic (Yvm ) = (k[t]/tm+1 )∗ Hic (Yv,0 ) and Frobq2 acts trivially on the index m set of the direct sum, so it is enough to study its action on Hic (Yv,0 ). With notations as in the proof of Theorem 3.5, we have for 1 < j ≤ m: m, m Hcd0 −j (Yv,0 ) = Hc2m+1−j (Yv,0 )(−(n − 1)) = Hc2m+1−j (Z m )(−(n − 1)) =( Q ) ⊗ Hc2m+1−j (Z1m )(−(n − 1)), N−

as Z m = N− × Z1m , where N− is seen as a disjoint union of points (for j = 1 this remains true if we replace the second equality by an inclusion, cf. (3.11)). Now, Frobq2 acts on N− by (a0 , c0 ) → (a0 , −c0 ) and in Hc2m+1−j (Z1m ) by the scalar (−1)2m+1−j q 2m+1−j . Note that −1 ∈ Tw,m acts on N− in the same way as Frobq2 and trivially in Hc2m+1−j (Z1m ). Thus the eigenspaces for −1 and Frobq2 coincide. There are only two such eigenspaces U1 and U−1 , and Frobq2 acts on U±1 by the scalar (±1)(−1)2m+1−j q 2m+1−j . Now let χ be the restriction of χ to μ2 ⊆ Tw,m . Then Hic (Yvm )[χ] ⊆ Hic (Yvm )[χ] = Uχ(−1) , which proves the corollary. 2 i denote the subgroup of Tw,m of elements which are congruent 1 modulo ti . Let Tw,m ∨,gen m Let Tw,m denote the set of all characters of Tw,m , which are non-trivial on Tw,m,0 ∩Tw,m . We also need the following purity result.

Corollary 3.14. Let m ≥ 1. Let d0 = d0 (n, m). The finite étale morphism Yvm → Yvm−1 × A1 (Am−1 ) induces an isomorphism

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m−1 m−1 m ∼ i Hic (Yv,0 ) = Hc (Yv,0 × A1 (Am−1 )) ∼ = Hi−2 c (Yv,0 )(−1) ∨,gen for all i = d0 − m. If χ ∈ Tw,m , then

for all i = d0 − m.

Hic (Yvm )[χ] = 0

∨ ∨,gen Conversely, if χ ∈ Tw,m  Tw,m , then Hcd0 −m (Yvm )[χ] = 0.

Proof. The first statement follows directly from Theorem 3.5 by comparing dimensions. Let N = ker((k[t]/tm+1 )∗  (k[t]/tm )∗ ). The finite étale covering Yvm → Yvm−1 ×

A1 (Am−1 ) factors as Yvm → N Yvm−1 × A1 (Am−1 ) → Yvm−1 × A1 (Am−1 ), where the m first morphism has Galois group Tw,m,0 ∩Tw,m . The first statement of the corollary implies that the first morphism in this factorization induces an isomorphism in the cohomology for all i = d0 − m. The second statement of the corollary follows from it. If χ is trivial m on Tw,m,0 ∩ Tw,m , then Hdc 0 −m (Yvm )[χ] ⊆ Hdc 0 −m ( =





Yvm−1 × A1 (Am−1 )) =

N



Hcd0 −m−2 (Yvm−1 )

N

Hcd0 (n,m−1)−m (Yvm−1 )

N

and the last group is 0 by Theorem 3.5. Hence the third statement of the corollary. 2 3.6. Superbasic case m Beforegoingon, we make  a digression  and study the varieties Xxm (b) in the superbasic 1 t−n case b = with even n > 0 and let x resp. x˙ be the image . Let x˙ = n+1 t t ˜ resp. in DG,m (x). Let v, of x˙ in W ˙ v be as in (3.1). The group Jb (F ) is the group of units ∗ D∗ of the quaternion algebra D over F . If OD are the integers of D, then UD = OD is ∗ ∗ ∼ a maximal compact subgroup of D and D /UD = Z. Then [10] Theorem 3.3(i) shows

Xx (b) =



Cv .

D ∗ /UD

The same arguments as used in the proof of Theorem 3.2 show that for m < 2k one has: Xxm ˙ (b) =

 D ∗ /U

Yvm (b), D

where Yvm (b) ⊆ Cvm is the closed subscheme given by equations

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D−1 σ(C)(1 − taσ(a))−1 = 1 C −1 σ(D)(1 − taσ(a)) = 1 B = 0. Again after eliminating D, it is defined in the coordinates a, C, A by Cσ(1 − taσ(a)) = (1 − taσ(a))σ 2 (C).

(3.18)

An explicit comparison with results of Boyarchenko [1], who carried out the closely related construction of Lusztig for a division algebra over F of invariant n1 (for levels m = 1, 2, with a suggestion of how one can continue for higher levels) and Chan [7] (who then extended Boyarchenko’s results to all levels for the quaternion algebra) shows that the varieties Xh defined in the quoted papers are very similar to varieties Xxm ˙ (b) defined by (3.18), but do not coincide completely, at least due to the presence of the additional coordinate A in our approach. Also note that level h ≥ 2 in the quoted papers corresponds to level m = h − 1 ≥ 1 in the present article. 4. Representation theory of GL2 (F ) We continue to assume G = GL2 throughout this section and keep the notations from Section 3.1 and the beginning of Section 3.2. Let us collect some further important notation here. We try to keep it consistent with the notation in [5]. The only major difference is that we write K (and not U = UM ) for the maximal compact subgroup G(OF ) of G(F ). For λ ∈ X∗ (T ) we write tλ ∈ T (L) for the image of the uniformizer t m+1 ¯ under λ. For an element x ∈ k[t]/t , we mean by its t-adic valuation vt (x) the largest μ ¯ integer μ ≥ 0, such that x ∈ t · k[t]/tm+1 . Moreover: Z is the center of G(F ) E = k2 ((t)) ⊂ L is the unramified degree two extension of F m UM (resp. UM for m ≥ 1) denote the units (resp. the m-units) of a local field M M = M22 (OF ); it is an OF -algebra K = G(OF ) = M∗ ; it is a maximal compact subgroup of G(F ) K i = 1 + ti M for i ≥ 0, and (K i )∞ i=0 defines a descending filtration of K by open normal subgroups • Km = K/K m+1 ∼ = G(OF /tm+1 ) i i m+1 • Km = K /K = 1 + ti M/1 + tm+1 M define a filtration on Km i ∨,gen • Tw,m , Tw,m,0 are as in the beginning of Section 3.3 and Tw,m , Tw,m as in the paragraph preceding Corollary 3.14 • • • • • •

For a locally compact abelian group A, a Q -vector space W with a right A-action ∗ and a Q -valued character χ of A, we let W [χ] be the maximal quotient of W , on which A acts by χ (if A is compact, W finite dimensional, W [χ] is canonically isomorphic to

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the maximal χ-isotypical subspace of W ). A left G(F )-action on W , which commutes with the A-action, induces a left G(F )-action on W [χ]. 4.1. Definitions and results Let ιE : E → M22 (L) be the embedding of F -algebras given by e → diag(e, σ(e)). We have ιE (UE )/ιE (UEm+1 ) = Tw,m . Inflating the Tw,m -action to ιE (UE ) and pulling back ∗ via ιE , we obtain an UE -action on Xwm ˙ (1). The center Z of G(F ) is F , thus (as in the ∗ ∗ last lines of Section 2.2), the action of UE on Xwm ˙ (1) extends to an action of E = F UE , which commutes with the left G(F )-action. Let χ be a non-trivial character of E ∗ . The level (χ) of χ is the least integer m ≥ 0, such that χ|U m+1 is trivial. Moreover, the pair (E/F, χ) is said to be admissible ([5] 18.2) E if χ does not factor through the norm NE/F : E ∗ → F ∗ . Two pairs (E/F, χ), (E/F, χ ) are F -isomorphic, if there is some γ ∈ GE/F such that χ = χ ◦ γ. Let Pnr 2 (F ) denote the set of all isomorphism classes of admissible pairs over F attached to E/F . An admissible pair (E/F, χ) is called minimal if χ|UEm does not factor through NE/F , where m is the level of χ. Definition 4.1. Let χ be a character of E ∗ , such that (E/F, χ) is admissible. The essential level ess (χ) of χ is the smallest integer m ≥ 0, such that there is a character φ of F ∗ with (φE χ) = m , where φE = φ ◦ NE/F . Clearly, ess (χ) ≤ (χ). Moreover, an admissible pair (E/F, χ) is minimal if and only if (χ) = ess (χ). Using the geometric constructions from last sections, we associate to any admissible pair (E/F, χ) a G(F )-representation. Recall the element w˙ ∈ NT (L) introduced in (3.1), which depends on an even integer n > 0. Recall from the beginning ¯ Q ) for a k-scheme X. of Section 3.4, that we write Hic (X) instead of Hic (X ×k Spec k, Definition 4.2. Let (E/F, χ) be admissible. Let (χ) = m ≥ ess (χ) = m . We take n > 0 even such that 0 ≤ m < n and let d0 (n, m) be as in Theorem 3.5. Define Rχ to be the G(F )-representation 

Rχ = Hdc 0 (n,m)−m (Xwm ˙ (1))[χ]. One easily sees that this definition is independent of the choice of n (cf. Theorem 3.2 and the definition of Yvm ). To state our main result, we need some terminology from [5], which we will freely use here. In particular, the level (π) ∈ 12 Z of an irreducible G(F )-representation is defined in [5] 12.6. Moreover, in [5] 20.1, 20.3 Lemma it is explained when an irreducible cuspidal representation π of G(F ) is called unramified. We denote by A2nr (F ) the set of all isomorphism classes of irreducible cuspidal unramified representations of G(F ). This is a subset of the set A20 (F ) of the isomorphism classes of all irreducible cuspidal representations of G(F ) ([5], §20). The (unramified part of

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the) tame parametrization theorem ([5] 20.2 Theorem) states the existence of a certain bijection (also for even q): ∼

nr Pnr 2 (F ) −→ A2 (F ),

(E/F, χ) → πχ

(4.1)

where πχ is a certain G(F )-representation constructed in [5] §19. Below, in Section 4.6 we briefly recall this construction. Here is our main result. Theorem 4.3. Let (E/F, χ) be an admissible pair. The representation Rχ is irreducible cuspidal, unramified, has level (χ) and central character χ|F ∗ . Moreover, Rχ is isomorphic to πχ , i.e., the map nr R : Pnr 2 (F ) → A2 (F ),

(E/F, χ) → Rχ

(4.2)

is a bijection and coincides with the map from the tame parametrization theorem (4.1). The theorem will be proven at the end of Section 4.6, after the necessary preparations in Sections 4.2–4.6 are done. We wish to point out here, that the injectivity of (4.2) follows from Proposition 4.26 and does not use the theory developed in [5], whereas to prove surjectivity of (4.2), we use the full machinery of [5]. In the rest of this section, we only deal with the central character, reduce to the minimal case and introduce some further notation. For a character φ of F ∗ and a representation π of G(F ), we write φπ for the G(F )-representation given by g → φ(det(g))π(g) and we let φE = φ ◦ NE/F be the corresponding character of E ∗ . If φ is a character of F ∗ and (E/F, χ) an admissible pair, then (4.1) satisfies: the central character of πχ is χ|F ∗ and φπχ = πχφE . We have an analogous statement for Rχ . Lemma 4.4. Let (E/F, χ) be admissible. Then the central character of Rχ is χ|F ∗ . If φ is a character of F ∗ , then φRχ = RχφE . Proof. The first statement follows from the definition of Rχ as the χ-isotypic component of some cohomology space, and the fact that the actions of F ∗ ∼ = Z ⊆ G(F ) and F ∗ ⊆ E ∗ in this cohomology space coincide as they already coincide on the level of varieties. The second statement follows by unraveling the definition of Rχ , using the natural isomord (n,m)−m d (n,λ)−m m ∼ λ phism Hc 0 (Yv,0 ) = Hc 0 (Yv,0 ) for m ≥ λ ≥ m from Theorem 3.5 and χ|ker(NE/F ) = φE χ|ker(NE/F ) . 2 We fix some notations for the rest of Section 4. Let (E/F, χ) be a minimal pair, let m be the level of χ and write i0 = d0 (n, m) − m. Let Y˜vm be the (disjoint) union of all m ˜m Z-translates of Yvm inside Xwm ˙ (1). Note that Yv is fixed by K, hence Yv is fixed by ZK. Define the ZK-representation Ξχ by Ξχ = Hic0 (Y˜vm )[χ].

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Then Theorem 3.2 shows G(F )

Rχ = c-IndZK Ξχ . Moreover, let ξχ be the restriction of Ξχ to K, i.e., Ξχ is the unique extension of ξχ to ZK such that t(1,1) acts as χ(t(1,1) ). Let Vχ denote the space in which Ξχ (resp. ξχ ) acts. Note that ξχ is inflated from a representation of the finite group Km = K/K m+1 , as K m+1 acts trivially in the cohomology of Yvm . Lemma 4.5. ξχ ∼ = Hic0 (Yvm )[χ|UE ]. Proof. Let W be a Q -vector space in which K acts on the left, UE on the right such that these actions commute. Let ⎧ ⎫ ⎪ ⎪ ⎨  ⎬ i W [t] = wi t : wi ∈ W ⎪ ⎪ ⎩ ⎭ i∈Z −∞i∞

with obvious K and UE -actions. Extend them to ZK- resp. E ∗ -actions by letting t act   as t.( i wi ti ) = i wi−1 ti . Then one checks that (W [t])[χ|UE ] = (W [χ|UE ])[t] and that the composed map W [χ|UE ] → (W [χ|UE ])[t]  (W [t])[χ] is a bijection. Apply this to i0 ˜ m W = Hic0 (Yvm ) and W [t] ∼ = c-IndZK K W = Hc (Yv ). 2 4.2. Trace computations I: preliminaries Let (E/F, χ) be a minimal pair of level m ≥ 0. Via ιE , χ|UE induces a character of Tw,m . We denote this character of Tw,m also by χ. The context excludes any ambiguity. ∨,gen m , i.e., χ is non-trivial on Tw,m,0 ∩ Tw,m . Lemma 4.6. We have χ ∈ Tw,m

Proof. As (E/F, χ) is minimal, χ|ker(NE/F : UEm →UFm ) is non-trivial. The level of χ is m, m i.e., χ is trivial on UEm+1 . Now we have UEm /UEm+1 ∼ via ιE , the norm map induces = Tw,m m+1 m m+1 ∗ ¯ ¯ where det : T m → the map N : UE /UE → (k[t]/t ) , and moreover, det ◦ιE = N, w,m m m → (k[t]/tm+1 )∗ ) = Tw,m,0 ∩ Tw,m (k[t]/tm+1 )∗ is the determinant. Now ker(det : Tw,m ∨,gen m and χ ∈ Tw,m if and only it is non-trivial on Tw,m,0 ∩ Tw,m . This shows the lemma. 2 In Section 4.1 we defined the Km -representation ξχ . Our goal in Sections 4.2–4.4 will be to compute the trace of ξχ on some important subgroups of Km . We will use the following trace formula from [1], which is similar to [9] Theorem 3.2 and is adapted to cover the situation with wild ramification. Lemma 4.7 ([1] Lemma 2.12). Let X be a separated scheme of finite type over a finite field FQ with Q elements, on which a finite group A acts on the right. Let g : X → X

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be an automorphism of X, which commutes with the action of A. Let ψ : A → Q be a character of A. Assume that Hic (X)[ψ] = 0 for i = i0 and FrobQ acts on Hic0 (X)[ψ] by a ∗ scalar λ ∈ Q . Then Tr(g ∗ , Hic0 (X)[ψ]) =

(−1)i0  ψ(τ ) · Sg,τ , λ · A τ ∈A

where Sg,τ = {x ∈ X(Fq ) : g(FrobQ (x)) = x · τ }. We adapt this to our situation. Recall from (3.4) and Proposition 3.4(i) that Yvm n−1 m+1 ∗ m+1 ¯ ¯ ¯ was parametrized by coordinates a ∈ k[t]/t , C ∈ (k[t]/t ) , A ∈ k[t]/t with 2 a0 = a mod t ∈ / k. We use Lemma 4.7 with Q = q . Lemma 4.8. Let (E/F, χ) be a minimal pair and let g ∈ K. Assume g acts on Yvm such m+1 ∗ ¯ that there is some rational expression p(g, a) ∈ (k[t]/t ) in a, such that g.(a, C, A) = (g.a, p(g, a) · C, g.A) on coordinates. Let τ ∈ Tw,m . Then Tr(g ∗ ; Hic0 (Yvm )[χ]) =

1 q m+1



 χ(τ )Sg,τ ,

τ ∈Tw,m

 m+1 ¯ where Sg,τ is the set of solutions in the variable a = a mod tm+1 ∈ k[t]/t (with a0 ∈ k¯  k) of the equations

σ(σ(a) − a)(σ(a) − a)−1 = −p(σ 2 (a), g)−1 τ

(4.3)

2

g.σ (a) = a. Proof. The contribution to the cohomology of the affine space Am (A) is just a shift in degree, the character χ is trivial on ker(Im,w˙ /I m  Tw,m ) and a computation shows that g acts on A by g.A = A + r(g, a, C) for some rational expression r depending only on g, a, C. If we write Yvm = Y  × Am (A), where Y  is given in coordinates a, C by the same equations as Yvm , we are reduced to apply Lemma 4.7 to Y  , on which also the  finite group Tw,m acts. We claim that for this scheme Sg,τ = (q 2 − 1)q 2(n−1) Sg,−τ . We  ¯ observe that a point (a, C) ∈ Y (k) lies in Sg,τ if and only if it satisfies the following equations: σ 2 (C)C −1 = σ(σ(a) − a)(σ(a) − a)−1

mod tm+1

g.σ 2 (a) = a p(σ 2 (a), g) · σ 2 (C) = C · τ The coordinates am+1 , . . . , an−1 occur only in the second equation and hence each of them contributes a factor q 2 to the number of solutions. Finally, for a fixed a, the third m+1 ∗ ¯ equation has exactly (q 2 − 1)q 2m different solutions in C ∈ (k[t]/t ) , and we can

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eliminate C by putting the first and the third equations together. Thus Sg,τ is equal to m+1 ¯ (q 2 − 1)q 2(n−1) times the number of solutions of equations (4.3) in a ∈ k[t]/t with τ replaced by −τ . This shows the claim. Now, Corollaries 3.13 and 3.14 show that the conditions of Lemma 4.7 are satisfied and the lemma follows from an easy computation involving the above claim. 2 Definition 4.9. For x ∈ G(OL /tm+1 )  {1} the level of x is the maximal integer (x) ≥ 0, such that x ≡ 1 mod t(x) . The level of 1 is m + 1. This definition is auxiliary and will be used in the next two sections. Note that the level is invariant under conjugation in G(OL /tm+1 ). 4.3. Trace computations II: Nm -action on Vχ We keep notations from Sections 4.1, 4.2. Let further Nm ⊂ Km denote the subgroup  Nm =



 1u 1

: x ∈ k[t]/tm+1

.

i For 0 ≤ i ≤ m + 1, let Nm denote the subgroup of Nm consisting of elements congruent i ∨,gen to 1 modulo t and let Nm denote the set of characters of Nm , which are non-trivial m on Nm .

Proposition 4.10. As Nm -representations one has m m ξχ = IndN 1 − IndN m 1 = 1 Nm



ψ.

∨,gen ψ∈Nm

In particular, dimQ Vχ = (q − 1)q m . Proof. We claim that for g ∈ Nm we have ⎧ m+1 ⎪ − qm ⎪ ⎨q Tr(g ∗ , Vχ ) = −q m ⎪ ⎪ ⎩0

if g = 1 m if g ∈ Nm  {1}

if g ∈ /

(4.4)

m . Nm

The proposition follows from this claim by comparing the traces of the Nm -represen tations on the left and the right sides. We need the following lemma. Let Sg,τ be as in Lemma 4.8.  Lemma 4.11. Let g ∈ Nm of level (g) ≤ m + 1. Then Sg,τ = ∅, unless vt (1 − τ ) = (g) m+1 and τ · σ(τ ) = 1 in k2 [t]/t . If both are satisfied, then

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 Sg,τ

=

 (q − 1)q 2m+1 q

m+1+(g)

if g = 1 (and hence τ = 1), if g ∈ Nm  {1}.

Proof. As both, the Tw,m - and the Km -actions on Yvm have their origin in matrix multiplication, one sees easily that S g,τ =∅, unless det(τ ) = det(g). Thus we can assume 1x this, i.e., τ σ(τ ) = 1. Write g = ∈ Nm with x ∈ k[t]/tm+1 . Then vt (x) = (g). 1  The action of g can be described by g.(a, C) = (a + x, C). By definition, Sg,τ is the set of solutions of σ(σ(a) − a)(σ(a) − a)−1 = −τ

(4.5)

2

σ (a) + x = a m+1 m+1 ∗ ¯ ¯ in a ∈ k[t]/t with a0 ∈ / k. Let s = σ(a) − a. For a fixed s ∈ (k[t]/t ) , the equation m+1 σ(a) − a = s in a has exactly q solutions. After adding −σ(a) to both sides of the second equation in (4.5), this second equation gets σ(s) + x = −s. Thus we are reduced to solve the equations

σ(s) + x = −s −1

σ(s)s

(4.6)

= −τ

m+1 ∗ ¯ in the variable s ∈ (k[t]/t ) . Putting σ(s) = −(x + s) into the second equation, we −1 obtain 1 − τ = −xs and one checks that equations (4.6) are equivalent to

σ(s) + x = −s

(4.7) −1

1 − τ = −xs

 From the second equation of (4.7) and since s must be a unit, we see that either Sg,τ =∅ or vt (1 − τ ) = vt (x). Assume the second holds and let μ = vt (x). If μ = m + 1, then g = 1, τ = 1 and the lemma follows. Assume now 0 < μ < m + 1. Then x = tμ x ˜ for m+1−μ ∗ ¯ some x ˜ ∈ (k[t]/t ) and τ = 1 + τ˜tμ for some τ˜ ∈ (k2 [t]/tm+1−μ )∗ . The condition τ σ(τ ) = 1 is equivalent to

τ˜ + σ(˜ τ ) + τ˜σ(˜ τ )tμ ≡ 0

mod tm+1−μ .

(4.8)

The second equation of (4.7) is equivalent to s ≡ τ˜−1 x ˜ mod tm+1−μ , i.e., s is uniquely m+1−μ −1 determined modulo t . Moreover, if s ≡ τ˜ x ˜ mod tm+1−μ we have σ(s) + x + s ≡ σ(˜ τ −1 x ˜) + x + τ˜−1 x ˜=x ˜(tμ + σ(˜ τ )−1 + τ˜−1 ) ≡ 0 mod tm+1−μ , where the last equation follows from (4.8). Thus s ≡ τ˜−1 x ˜ mod tm+1−μ is the unique m+1−μ solution of equation (4.7) modulo t . One easily sees that over any solution of the

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first equation of (4.7) modulo tλ lie precisely q solutions of it modulo tλ+1 . This shows that (4.7) has precisely q μ solutions if μ > 0. It remains to handle the case μ = 0, but this is done similarly to the case μ > 0. 2   = ∅ unless τ = 1 and S1,1 = q m (q − 1). The claim (4.4) For g = 1, we have S1,τ follows immediately from Lemma 4.8. Let now g ∈ Nm  {1} of level  = (g) ≤ m. Then Lemmas 4.8 and 4.11 show

Tr(g ∗ , Vχ ) =



1 q m+1

= q ·

 χ(τ )Sg,τ

 τ ∈Tw,m,0 ∩(Tw,m 



 τ ∈Tw,m,0 ∩(Tw,m

+1 Tw,m )

χ(τ ) = −q  · 

+1 Tw,m )



χ(τ ),

+1 τ ∈Tw,m,0 ∩Tw,m

 the last equation being true as χ is a non-trivial character on Tw,m,0 ∩ Tw,m . Unless  = m, the sum in the last expression vanishes, as χ is still a non-trivial character on +1 +1 Tw,m,0 ∩ Tw,m . If  = m, we have Tw,m = {1}, and (4.4) follows. 2

Corollary 4.12. (ξχ , Vχ ) is irreducible as B(OF /tm+1 )-representation and hence also as Km -representation. ∨,gen , let Vχ [ψ] denote the ψ-isotypic component of Vχ . By PropoProof. For ψ ∈ Nm sition 4.10, Vχ [ψ] is one-dimensional. For x ∈ T (OF /tm+1 ), let ψ x be the character of Nm defined by ψ x (g) = ψ(x−1 gx). Then x.Vχ [ψ] ⊆ Vχ [ψ x ]. Let 0 = W ⊆ Vχ be a B(OF /tm+1 )-invariant subspace. Then W decomposes as the sum of its Nm -isotypical ∨,gen components W [ψ] (ψ ∈ Nm ) and W [ψ] ⊆ Vχ [ψ]. As Vχ [ψ] is one-dimensional, W [ψ] is either 0 or equal to Vχ [ψ]. But as W = 0, there is a ψ, such that W [ψ] = Vχ [ψ]. Note ∨ ∨,gen that the natural action of T (OF /tm+1 ) on Nm restricts to a transitive action on Nm . ∨,gen This transitivity implies that W [ψ] = Vχ [ψ] for all ψ ∈ Nm , i.e., W = Vχ . 2

4.4. Trace computations III: Hm -action on Vχ We keep notations from Sections 4.1 and 4.2. Let Hm ⊂ Km be a non-split torus, i.e., a subgroup which is conjugate to Tw,m inside G(OL /tm+1 ). Let Zm be the center of ∼ Km . One has Zm ⊆ Hm . We fix an isomorphism cs : Tw,m → Hm , given by conjugation i i with s ∈ G(k2 [[t]]), and let Hm = cs (Tw,m ). Let χ ˜ = χ ◦ c−1 and χ ˜σ = χσ ◦ c−1 s s , where σ χ = χ ◦ σ. is Note that if s ∈ G(L) is another matrix conjugating Tw,m into Hm , then cs c−1 s either identity or σ. In particular, up to σ-action, χ ˜ does not depend on the choice of the element s. ∨ , let i(ψ) ∈ {0, . . . , m + 1} be the smallest integer such that For a character ψ ∈ Hm i(ψ) σ ψ coincides with χ ˜ or χ ˜ on the subgroup Hm (in particular, i(ψ) = 0 if and only if ψ=χ ˜ or χ ˜σ ).

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Theorem 4.13. Let ψ be a character of Hm . Then ψ, ξχ Hm = 0 unless ψ|Zm = χ| ˜ Zm . Assume ψ|Zm = χ| ˜ Zm . Then  ψ, ξχ Hm =

1

if m − i(ψ) odd

0

if m − i(ψ) even.

The proof of Theorem 4.13 is given at the end of this section. To prepare it, we have to compute the traces of elements x ∈ Hm in Vχ . This is done in Proposition 4.20 below, for which Lemma 4.17 is the main technical tool. We have an immediate consequence of Theorem 4.13. Corollary 4.14. Let m > 0. The character χ ˜ (and hence also χ) is up to σ-conjugacy uniquely determined by ξχ among all characters of Hm as follows: it is the unique (up ∨ to σ-conjugacy) character ψ ∈ Hm , such that ψ|Zm is the central character of ξχ and (i) if m is odd: ψ occurs in ξχ , and all characters ψ  = ψ, which coincide with ψ on 1 Zm Hm do not occur in ξχ . (ii) if m is even: ψ does not occur in ξχ , and all characters ψ  = ψ, which coincide with 1 ψ on Zm Hm occur in ξχ . Moreover, the map (E/F, χ) → Ξχ from Pnr 2 (F ) to the set of isomorphism classes of ZK-representations is injective. Proof. The first statement is immediate from Theorem 4.13. We show injectivity of χ → Ξχ . From the first statement of the corollary, χ|UE is uniquely determined by the Km -representation ξχ , hence also by its inflation to K, which is equal to Ξχ |K . Moreover, by Lemma 4.4, χ|F ∗ is equal to the restriction of Ξχ to the center Z ∼ = F ∗ of ZK. This ∗ ∗ finishes the proof, as E = F UE . 2 Definition 4.15. We say that x is maximal if (x) ≥ (zx) for all z ∈ Zm .  Lemma 4.16. Let x =

 x1 x 2 x3 x4

∈ Hm  {1}. Then x maximal if and only if vt (x3 ) =

(x). Proof. Assume first  = (x) > 0. Consider τx = c−1 s (x) ∈ Tw,m and write τx = 1 + τx, t + · · · + τx,m tm . One sees immediately that maximality of an element is invariant under conjugation, hence x is maximal if and only if τx is, i.e., if and only if τx, ∈ / k. A computation (using the fact that all entries of s must be units) shows that x3 = m+1 ∗ ¯ ) . The lemma follows in the case (x) > 0. t u(τx − σ(τx )) with some unit u ∈ (k[t]/t The case (x) = 0 is similar. 2

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We introduce the following version of the characteristic polynomial of an element x ∈ Km . Let  = (x) be the level of x. Let x ˜ be some lift of x to K. Then the characteristic polynomial of x ˜ can be seen as the function px˜ : OL → OL

λ → px˜ (λ) = det(λ · Id − x).

Note that px˜ (UL ) ⊆ t2 OL . Let now λ ∈ UL and let x ˜1 , x ˜2 be two lifts of x to K. m++1 m++1 Then px˜1 (λ) − px˜2 (λ) ∈ t OL , i.e., px˜ (λ) modulo t depends only on x, not on the lift x ˜. This gives a well-defined map px : UL → t2 OL /tm++1 OL . Moreover, one immediately computes that this induces the following map defined as the composition: p

x px : UL /ULm+1 −→ t2 OL /tm++1 OL → OL /tm−+1 OL ,

(4.9)

−2 where  thesecond arrow is multiplication by t . Explicitly, if  > 0 and x = 1 + y1 y2 , then t y3 y4

px (1 + τ˜t ) = (˜ τ − y1 )(˜ τ − y4 ) − y2 y3 .  m+1 ∗ ¯ We identify Tw,m with (k2 [t]/tm+1 )∗ ⊆ (k[t]/t ) by sending

 τ σ(τ )

to τ . In

  m−+1 ¯ particular, for x ∈ Km and τ ∈ Tw,m we have the element px (τ ) ∈ k[t]/(t ), where   is the level of x. Let Sx,τ be as in Lemma 4.8.

Lemma 4.17. Let x ∈ Hm be maximal of level  = (x) ≤ m. Let τ ∈ Tw,m . Then    Sx,τ = ∅, unless τ ∈ Tw,m and det(τ ) = det(x). For τ ∈ Tw,m with det(τ ) = det(x) we have:

 Sx,τ

⎧ ⎪ q m+ ⎪ ⎪ ⎪ ⎨q m++1 = ⎪ 0 ⎪ ⎪ ⎪ ⎩ (q + 1)q m+

if px (τ ) = 0 and m −  even if px (τ ) = 0 and m −  odd if vt (px (τ )) < ∞ is odd if vt (px (τ )) < ∞ is even.

Proof. Let x ∈ Hm be maximal of level  ≤ m and let τx = c−1 s (x). Let τ ∈ Tw,m . From   the definition of Sx,τ one immediately deduces that Sx,τ = ∅, unless det(x) = det(τ ),   x 1 x2 i.e., τ ∈ τx Tw,m,0 = Tx (0). Hence we can assume τ σ(τ ) = det(x). Write x = . x3 x4 A point of Yvm is parametrized by the coordinates a, C and A as above. One computes: x.(a, C) = (x.a, x.C) = (

x1 a + x2 det(x)C , ). x3 a + x4 x3 a + x4

(4.10)

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 By Lemma 4.8, Sx,τ is the number of solutions of equations (4.3) in the variable a ∈ m+1 ¯ (satisfying a0 ∈ / k). Explicitly, these equations are: k[t]/t

x1 σ 2 (a) + x2 = a(x3 aσ 2 (a) + x4 ) (σ 2 (a) − σ(a))σ(τ ) = −(x3 σ 2 (a) + x4 )(σ(a) − a). Inserting the first equation into the second and applying σ −1 to the result, we see that the equations are equivalent to x3 aσ 2 (a) − x1 σ 2 (a) + x4 a − x2 = 0

(4.11)

x3 aσ(a) + (τ − x1 )σ(a) − (τ − x4 )a − x2 = 0.

(4.12)

Sublemma 4.18. For i ≥ 1, there are precisely q 2 solutions of equation (4.11) modulo ti+1 lying over a given solution (satisfying a0 ∈ / k) of (4.11) modulo ti . i i Proof. Write a = j=0 aj tj , xλ = j=0 xλj . The coefficient of ti on the right side of (4.11) modulo ti+1 is 2

2

(x30 a0 − x10 )aqi + (x30 aq0 + x40 )ai + R,

(4.13)

where R ∈ k¯ depends only on a0 , . . . , ai−1 and x and not on ai . As a0 ∈ / k and x ∈ G(k), q2 it is clear that x30 a0 −x10 = 0 and x30 a0 +x40 = 0. Thus (4.13) is a separable polynomial in ai of degree q 2 , i.e., it has exactly q 2 different roots. 2  Now we concentrate on the case  > 0, i.e., x = 1 + t

 y1 y2 y3 y4

. Equation (4.12)

modulo t shows (τ − 1)σ(a) = (τ − 1)a mod t . If τ ≡ 1 mod t , then this forces a0 ∈ k,    which contradicts (a, c, A) ∈ Yvm . Hence Sx,τ = ∅, unless τ ∈ Tw,m . Assume τ ∈ Tw,m ,  m+1− and τ = 1 + τ˜t , with some τ˜ ∈ k2 [t]/t . Note that the condition det(x) = det(τ ) satisfied by x, τ is equivalent to y1 + y4 + (y1 y4 − y2 y3 )t ≡ τ˜ + σ(˜ τ ) + τ˜σ(˜ τ )t

mod tm−+1 .

(4.14)

Equations (4.11) and (4.12) transform to t (y3 aσ 2 (a) − y1 σ 2 (a) + y4 a − y2 ) = σ 2 (a) − a

(4.15)

τ − y1 )σ(a) − (˜ τ − y4 )a − y2 ≡ 0 mod tm−+1 y3 aσ(a) + (˜

(4.16)

Sublemma 4.18 shows that the number of solutions of (4.15), (4.16) is equal to q 2 times the number of solutions of (4.15) and (4.16) mod tm−+1 . Let us write Q = px (τ ) with px as in (4.9). A computation involving (4.14) implies

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τ˜ + σ(˜ τ ) − y1 − y4 ≡ t τ −1 Q

mod tm−+1 .

673

(4.17)

1 Sublemma 4.16 allows us to make the linear change of variables a = b − τ˜−y and y3 m−+1 equations (4.15), (4.16) modulo t take the following form (using (4.17) and the fact that σ 2 (a) − a = σ 2 (b) − b):

  t y3 bσ 2 (b) − τ˜σ 2 (b) − (t τ −1 Q − σ(˜ τ ))b + y3−1 Q ≡ σ 2 (b) − b

mod tm−+1 (4.18)

y3 bσ(b) − t τ −1 Qb + y3−1 Q ≡ 0 mod tm−+1 .

(4.19)

m i Write b = i=0 bi t . We have three cases: vt (Q) = ∞, vt (Q) < ∞ odd, vt (Q) < ∞ even. Assume first vt (Q) = ∞, i.e., Q = 0. Then (4.19) is equivalent to b0 = b1 = m− · · · = b m−  = 0. As b = 0 is also a solution of (4.18) mod t 2 +1 , it follows from 2 Sublemma 4.16 that the number of solutions of (4.18) and (4.19) mod tm−+1 is exactly m− (q 2 )m−− 2  and the lemma follows in this case, once we have shown that no of these solutions lies in the ‘forbidden’ subset, determined by a0 ∈ k. This is done in Sublemma 4.19 below. Now assume vt (Q) < ∞. Equation (4.19) shows that we must have vt (Q) = 2vt (b).  In particular, Sx,τ = ∅ if vt (Q) is odd. Assume vt (Q) = 2j < ∞ is even and write Q = m−+1 ¯ solves (4.19) if and only if b = tj b (i.e., b0 = · · · = bj−1 = 0) t2j Q . Then b ∈ k[t]/t m−+1 i−j  and b = i=j bi t solves y3 b σ(b ) − t+j τ −1 Q b + y3−1 Q ≡ 0 mod tm−−2j+1 .

(4.20)

Note that such a solution b is necessarily a unit. Using this, we can express σ(b ) in terms of b , apply σ to it, and then insert again the expression of σ(b ) in (4.20). This shows: σ 2 (b ) ≡

Q b

σ(Q ) + y3−1 t+j σ(τ −1 Q ) (mod tm−−2j+1 ), − y3 t+j τ −1 Q

which multiplied by tj gives an expression of σ 2 (b) mod tm−−j+1 in terms of b . Now a (very ugly, but straightforward) computation shows that if we put this expression for σ 2 (b) into equation (4.18) modulo tm−−j+1 , we obtain the tautological equation 0 = 0. This simply means that any solution b of (4.19) mod tm−−j+1 is a solution of (4.18) mod tm−−j+1 . Similarly as in Sublemma 4.18, one checks that (4.19) modulo tm−−j+1 has precisely (q + 1)q m−−2j solutions (q + 1 corresponds to the freedom of choosing bj and q m−−2j corresponds to the freedom of choosing bj+1 , . . . , bm−−j ). Again by Sublemma 4.16, the lemma also follows in this case, once we have shown that no of these solutions lie in the ‘forbidden’ subset, determined by a0 ∈ k. This is done in Sublemma 4.19. In the case (x) = 0, the lemma can be proven in the same way. 2

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 Sublemma 4.19. With notations as in the proof of Lemma 4.17, assume τ ∈ Tw,m and det(τ ) = det(x). Let a be a solution of equations (4.11), (4.12), then a0 ∈ / k. r ¯ Proof. For any r ≥ 1 and an element X ∈ k[t]/(t ), denote by X0 ∈ k¯ the reduction of X −1 modulo t. Write τx = cs (x) ∈ Tw,m . We handle the case  > 0 first. Write τ = 1 + τ˜t , 1 τx = 1 + τx, t + . . . . As a is a solution of (4.11), (4.12), b = a + τ˜−y is a solution of y3 τ˜0 −y10 (4.18), (4.19). We have a0 = b0 − y30 . Assume first vt (Q) > 0. Maximality of x (and hence of τx ) implies τx, ∈ / k. Now, vt (Q) = vt (px (τ )) > 0 is equivalent to τ˜0 ≡ τx, or ≡ σ(τx, ) mod t. Hence τ˜0 ∈ / k. On the other hand, the solution b must satisfy b0 = 0 and we have y10 , y30 ∈ k. As τ˜0 ∈ /k / k. we obtain a0 ∈ Now assume vt (Q) = 0 and suppose that a0 ∈ k, i.e., aq0 = a0 . Then for b0 we must have:

bq0 = b0 +

τ˜0q − τ˜0 . y30

(4.21)

Putting this into equation (4.19) mod t, we deduce that b0 must satisfy b20 +

τ˜0q − τ˜0 Q0 b0 + 2 = 0, y30 y30

(4.22)

where Q0 = (˜ τ0 − τx, )(˜ τ0 − σ(τx, )). By assumption we have det(τx ) = det(x) = +1 det(τ ) mod t , hence τ˜0 + σ(˜ τ0 ) = τx, + σ(τx, ).

(4.23)

Assume first char(k) > 2. A computation shows that the discriminant of equation −2 (4.22) is D = y30 (σ(τx, ) − τx, )2 and hence the solutions of it are b0,± = −

σ(˜ τ0 ) − τ˜0 σ(τx, ) − τx, ± . 2y30 2y30

Putting any of these solutions into equation (4.21) shows τx, = σ(τx, ), which is a contradiction to maximality of x. This finishes the proof in the case char(k) > 2. q τ0 Assume now char(k) = 2. Let μ = τ˜0y+˜ . Then μ ∈ k. Further, (4.23) shows μ = 0 30 (otherwise, τx, ∈ k, which is a contradiction to maximality of x). Set also δ = y2Q0μ2 . 30 Note that by (4.23), Q0 ∈ k and hence also δ ∈ k. Make the change of variables b0 = μs, i.e., b0 satisfies (4.22), (4.21) if and only if s satisfies sq + s + 1 = 0 s2 + s + δ = 0.

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The second of these equations implies sq = s + Trk/F2 (δ). This together with the first τ˜0 +τx, equation implies Trk/F2 (δ) = 1. On the other hand, let R = τx, q . Using (4.23), we +τx, see that 2

R+R =

q (˜ τ0 + τx, )(˜ τ0 + τx, ) q 2 (τx, + τx, )

= δ.

This implies Trk/F2 (δ) = 0, which is a contradiction. This proves the lemma in the case char(k) = 2. Let us now consider the case (x) = 0. By maximality of x, we have x3 ∈ (k[t]/tr )∗ . If char(k) > 2, the variable change a = b − x−1 3 (τ − x1 ) analogous to the one in the case (x) > 0 leads to a very similar proof. Assume char(k) = 2. We have to show that equations (4.12) mod t and aq0 = a0 do not have a common solution. Let −1 q λ = x−1 30 (x10 + x40 ) = x30 (τx,0 + τx,0 ).

Then λq = λ and λ = 0 by maximality of x. Make the change of variables given by q q a0 = λr + x−1 30 x10 . Then a0 = a0 transforms into r = r and (4.12) gets (after using rq = r and canceling) x30 λ2 r2 + x30 λ2 r + x−1 30 det(x) = 0, or equivalently r2 + r +

q+1 τx,0 = 0. + τx,0 )2

q (τx,0

We have to show that this equation has noq solution in k. But observe that the two τ τ solutions of it are given by τ q x,0 and τ q x,0 and they lie in k2  k (note that they x,0 +τx,0 x,0 +τx,0 are different by maximality of x). This finishes the proof also in this case. 2 Proposition 4.20. Let x ∈ Hm be maximal of level (x) ≤ m. Then tr(x; Vχ ) = (−1)m−(x)+1 q (x) (χ(x) ˜ +χ ˜σ (x)).   Proof. Let τx = c−1 s (x) ∈ Tw,m . For j ∈ {0, 1, . . . , m − , ∞}, let   Tx (j  ) = {τ ∈ τx Tw,m,0 ∪ σ(τx )Tw,m,0 : τ ≡ τx or σ(τx )



mod t+j }

     = τx ker(Tw,m,0  Tw,+j  −1,0 ) ∪ σ(τx ) ker(Tw,m,0  Tw,+j  −1,0 ) ⊆ Tw,m    be the union of the two ker(Tw,m,0  Tw,+j  −1,0 )-cosets inside Tw,m in which τx and σ(τx ) lie (note that these cosets are disjoint if j  > 0 and equal if j  = 0). Note that  τ ∈ Tx (0) if and only if det(τ ) = det(x) and τ ∈ Tw,m .

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 Sublemma 4.21. For τ ∈ Tw,m with det(τ ) = det(x) we have: τ ∈ Tx (j  ) ⇔ vt (px (τ )) ≥ j  .  ˜x . The characteristic Proof of Sublemma 4.21. Write τ = 1 +t τ˜ and τx = c−1 s (x) = 1 +t τ polynomial is invariant under conjugation, hence vt (px (τ )) = vt (pτx (τ )). Write τx = 1 + t τ˜x . As x (and hence also τx ) is maximal, τ˜x − σ(˜ τx ) is a unit. We have pτx (τ ) =   (˜ τ − τ˜x )(˜ τ − σ(˜ τx )). Thus vt (px (τ )) ≥ j  ⇔ τ˜ ≡ τ˜x mod tj or τ˜ ≡ σ(˜ τx ) mod tj . The sublemma follows. 2

By Lemma 4.8 and the first statement of Lemma 4.17 we have: tr(x; Vχ ) =



1 q m+1

 χ(τ )Sx,τ

(4.24)

 τ ∈Tw,m

det(τ )=det(x)

=

1

q

( m+1



 χ(τ )Sx,τ +

m−  j  =0

τ ∈Tx (∞)



 χ(τ )Sx,τ ). 

τ ∈Tx (j ) τ ∈T / x (j  +1)

  Write Tw,m,0 = Tw,m,0 ∩ Tw,m . Lemma 4.17 implies for 0 < j  < m − :





 χ(τ )Sx,τ = (χ(x) ˜ +χ ˜σ (x)) · (const) · 

τ ∈Tx (j ) τ ∈T / x (j  +1)

χ(τ )

  τ ∈ker(Tw,m,0 Tw,l+j  −1,0 )   τ ∈ker(T / w,m,0 Tw,l+j  ,0 )



= (χ(x) ˜ +χ ˜σ (x)) · (const) ·

χ(τ ) = 0

  τ ∈ker(Tw,m,0 Tw,l+j  ,0 )

and similarly



τ ∈Tx (0) τ ∈T / x (1)

  χ(τ )Sx,τ = 0 as χ is non-trivial on ker(Tw,m,0  Tw,l+j  ,0 )

 is empty for τ ∈ (one has to apply this twice). Further, if m −  is odd, then Sx,τ Tx (m − )  Tx (∞), hence in this case Lemma 4.17 implies:

tr(x; Vχ ) =



1 q m+1

 χ(τ )Sx,τ = q  (χ(τx ) + χσ (τx )).

τ ∈Tx (∞)

If m −  is even, then ⎛ tr(x; Vχ ) =

1 q m+1

⎝



χ(τ )q m+ + (χ(τx ) + χσ (τx ))

τ ∈Tx (∞)

= −q  (χ(τx ) + χσ (τx )). This finishes the proof of Proposition 4.20. 2

⎞

 m τ ∈Tw,m,0

χ(τ )(q + 1)q m+ ⎠  {1}

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Proof of Theorem 4.13. If ψ|Zm = χ| ˜ Zm , then ψ, ξχ Hm = 0 by Lemma 4.4. Note that for x ∈ Hm , z ∈ Zm we have tr(zx; Vχ ) = χ(z)tr(x; Vχ ). Let ψ be a character of Hm with ψ|Zm = χ|Zm . Note that {x ∈ Hm : max (zx) = } =

  +1  Zm Hm Zm Hm

z∈Zm

if  ≤ m if  = m + 1.

Zm

As ψ|Zm = χ|Zm and tr(z; Vχ ) = (q − 1)q m χ(z) for z ∈ Zm by Lemma 4.11, we have ψ, ξχ Hm =

m   1 1 2 2m ψ(x)tr(x; V ) = ((q − 1) q + S ), χ (q 2 − 1)q 2m (q 2 − 1)q 2m x∈Hm

=0

(4.25) where 

S =

ψ(x)tr(x; Vχ )

  Z H +1 x∈Zm Hm m m

= (−1)m−+1 q 



ψ(x)(χ(x) ˜ +χ ˜σ (x))

  Z H +1 x∈Zm Hm m m

 +1 = (Zm Hm )ψ, χ ˜+χ ˜σ Zm Hm )ψ, χ ˜+χ ˜σ Zm Hm  − (Zm Hm +1  for 0 ≤  ≤ m, by Proposition 4.20. For  ≥ 1 we have (Zm Hm ) = (q − 1)q 2m−+1 , and we compute: ⎧ m+1 2m+1 ⎪ q (q − 1) if i(ψ) = 0 ⎪ ⎨(−1)

S0 =

(−1)m (q − 1)q 2m ⎪ ⎪ ⎩0

if i(ψ) = 1 otherwise,

and for 0 <  < m: ⎧ m−+1 ⎪ (q − 1)2 q 2m ⎪ ⎨(−1) S = (−1)m− (q − 1)q 2m ⎪ ⎪ ⎩0

if i(ψ) ≤  if i(ψ) =  + 1 otherwise,

and Sm =

 (−1)q 2m (q − 1)(q − 2) if i(ψ) ≤ m 2(q − 1)q 2m

if i(ψ) = m + 1.

Here one uses that if i(ψ) < m + 1, then ψ coincides with precisely one of the characters i(ψ) m χ, ˜ χ ˜σ on Hm and does not coincide with the other even on the last filtration step Hm σ m (because χ = χ on Tw,m,0 ). The theorem follows if we put these values into (4.25). 2

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4.5. Relation to strata We will freely use the terminology of intertwining from [5] §11 and of strata and cuspidal inducing data from [5] Chapter 4. From results of Section 4.3 we deduce that Rχ is irreducible, cuspidal and contains an unramified stratum. First we have the following general result. Proposition 4.22. Let m ≥ 0 and let Ξ be a ZK-representation, whose restriction to K is the inflation of an irreducible Km -representation ξ, which does not contain the G(F ) m trivial character on Nm . Then the G(F )-representation ΠΞ = c-IndZK Ξ is irreducible, cuspidal and admissible. If m > 0, it contains an unramified simple stratum (M, m, α) for some α ∈ t−m M. Moreover, (ΠΞ ) = m and ΠΞ does not contain an essentially scalar stratum. In particular, for any character φ of F ∗ , one has 0 < (ΠΞ ) ≤ (φΠΞ ). Corollary 4.23. Let (E/F, χ) be a minimal pair, such that χ has level m ≥ 0. The representation Rχ is irreducible, cuspidal and admissible. Assume m > 0. Then the representation Rχ contains an unramified simple stratum. In particular, (Rχ ) = m and Rχ is unramified. Moreover, for any character φ of F ∗ , one has 0 < (Rχ ) ≤ (φRχ ). Proof. All assumptions of Proposition 4.22 are satisfied for the ZK-representation Ξχ and the corresponding Km -representation ξχ by Corollary 4.12 and Proposition 4.10. 2 Proof of Proposition 4.22. Irreducibility and cuspidality of ΠΞ follow from [5] Theorem 11.4, whose assumptions are satisfied due to irreducibility of Ξ and Lemma 4.24. Then admissibility follows from irreducibility (cf. e.g. [5] 10.2 Corollary). Now assume m > 0. To contain a stratum is a priori defined with respect to a choice of an additive character. So fix some ψ ∈ F ∨ of level 1 (i.e., ψ|OF non-trivial, ψ|tOF trivial). Then [5] 12.5 Proposition gives us an isomorphism (here we use m > 0): ∼

m ∨ t−m M/t−m+1 M −→ (K m /K m+1 )∨ = (Km ) ,

a + t−m+1 M → ψa |K m ,

where ψa is given by ψ M is thetrace map M → OF . a (x) =ψ(trM (a(x − 1))), where tr a1 a 2 x1 x 2 m Explicitly, if a = t−m ∈ t−m M and x = 1 + tm ∈ Km , then a3 a4 x3 x4 ψa (x) = ψ(a1 x1 + a2 x3 + a3 x2 + a4 x4 ).

(4.26)

We show that ΠΞ contains an unramified simple stratum. Therefore, note that ΠΞ contains the inflation to K of the Km -representation ξ. Thus it is enough to show that m for any α ∈ t−m M, such that ψα is contained in ξ on Km , the stratum (M, m, α) is −m unramified simple. As in [5] 13.2, for α ∈ t M we can write α = t−m α0 with α0 ∈ M and let fα (T ) ∈ OF [T ] be the characteristic polynomial of α0 . Let f˜α (T ) be its reduction

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modulo t. By definition, (M, m, α) is unramified simple if and only if f˜α (T ) is irreducible in k[T ], or equivalently, if and only if α0 mod t ∈ G(k) is not triangularizable. m Let now α = t−m α0 be arbitrary such that ξ contains ψα on Km . It is enough to show that α mod t is not triangularizable. Suppose it is. Then there is some β =  0 β1 β2 ∈ t−m M such that gαg −1 ≡ β mod t−m+1 M, i.e., ψgαg−1 and ψβ coincide t−m 0 β4 m m m is the on Km . By Lemma 4.25, ψβ also occurs in ξ on Km and (4.26) shows that ψβ |Nm m trivial character of Nm . This is a contradiction to our assumption that ξ does not contain m the trivial character on Nm . This contradiction shows that ΠΞ contains an unramified simple stratum. As an unramified simple stratum is fundamental, [5] 12.9 Theorem shows that (ΠΞ ) = m. Suppose now (M, m , α ) is some essentially scalar stratum contained in ΠΞ . It has to intertwine with the previously found unramified simple stratum (M, m, α) contained in ΠΞ (cf. [5] 12.9). As essentially scalar strata are fundamental, [5] 12.9 Lemma 2 implies m = m. But in this case the above argumentation shows that (M, m, α ) is unramified simple and hence not essentially scalar. Finally, Theorem [5] 13.3 implies the last statement of the proposition. 2 Lemma 4.24. Let Ξ, Ξ be two ZK-representations, whose restrictions to K are inflations m of Km -representations ξ, ξ  . Assume that ξ does not contain the trivial character on Nm . An element g ∈ G(F )  ZK never intertwines Ξ with Ξ . Proof. The property of intertwining only depends on the double coset ZKgZK of g. By Cartan decomposition, a set of representatives of these cosets is given by the diagonal mα matrices {mα = t(0,α) : α ∈ Z≥0 } (cf. e.g. [5] 7.2.2).  Assumeα > 0. Then ZK∩ (ZK) = 1 tm OF m ZK∩mα ZKm−1 = , on which Ξ does not contain α contains the subgroup N 1 the trivial character, and on the other hand we have  mα



Ξ



 1 g 1



=Ξ

(m−1 α



 1g 1



mα ) = Ξ

1 tα g 1

 ,

i.e., mα Ξ restricted to N m is the trivial representation (as α > 0 and Ξ is trivial on K m+1 ). Hence HomZK∩mα (ZK) (Ξ, mα Ξ ) ⊆ HomN m (Ξ, mα Ξ ) = 0.

2

m Lemma 4.25. Let a ∈ t−m M, g ∈ K. If ψa occurs in ξ on Km , then ψgag−1 occurs in ξ m on Km .

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m Proof. For x ∈ Km one has:

ψgag−1 (x) = ψ(trM (gag −1 (x − 1))) = ψ(trM (ag −1 (x − 1)g)) = ψ(trM (a(g −1 xg − 1))) = ψa (g −1 xg). Let V denote the space in which ξ acts. For simplicity we write x.v instead of ξ(x)(v) m for x ∈ Km , v ∈ V . Let v ∈ V , such that x.v = ψa (x)v for all x ∈ Km . Then for all m x ∈ Km we have: g −1 x.(g.v) = g −1 xg.v = ψa (g −1 xg)v = ψgag−1 (x)v. m Thus x.(g.v) = ψgag−1 (x)(g.v), i.e., on the linear span of g.v any element x ∈ Km acts m as the scalar ψgag−1 (x). In particular, ψgag−1 occurs in ξ on Km . 2 nr Proposition 4.26. The map R : Pnr 2 (F ) → A2 (F ) from Theorem 4.3 is injective.

Proof. Let (E/F, χ1 ), (E/F, χ2 ) be two non-isomorphic admissible pairs. By Corollary 4.23 we may assume that χ1 , χ2 have the same level and by Lemma 4.4 we may assume that χ1 , χ2 coincide on F ∗ . Twisting by a central character, we may assume that both pairs are minimal. The last statement of Corollary 4.14 shows Ξχ1 ∼ Ξχ2 . It = remains to show that this implies Rχ1 ∼ Rχ2 . Frobenius reciprocity and Mackey formula = show that

HomG(F ) (Rχ1 , Rχ2 ) =

HomZK∩g (ZK) (Ξχ1 , g Ξχ2 )

g∈ZK\G(F )/ZK

= HomZK (Ξχ1 , Ξχ2 ), where the second equality follows from Lemma 4.24. As Ξχ1 , Ξχ2 are irreducible (by Corollary 4.12) and unequal, the Hom-space is zero. 2 4.6. Relation to cuspidal inducing data Now we want to compare our construction with the construction in [5] §19 of representations attached to minimal pairs. For the convenience of the reader and to have appropriate notations, we briefly recall their setup ([5] §15, §19). Let ψ be some fixed (additive) character of F of level one. Let ψE = ψ ◦trE/F , ψM = ψ ◦trM . Let (E/F, χ) be a minimal pair. Let m > 0 be the level of χ. Let α ∈ p−m be such that χ(1 +x) = ψE (αx) E  m +1

. Choose an F -embedding E → M22 (F ) such that E ∗ ⊆ ZK (not to be for x ∈ pE 2 confused with ιE from the beginning of Section 4.1). Then (M, m, α) is an unramified simple stratum. Let then Jα = E ∗ K 

m+1 2 

;

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this is an open subgroup of ZK. Moreover, via the embedding of E into M22 (F ), α defines m ([5] 12.5) a character ψα of K  2 +1 , which is trivial on K m+1 (thus inducing a character m  +1 of Km2 ). Let C(ψα , M) be the set of isomorphism classes of all irreducible represenm tations Λ of Jα , such that Λ contains the character ψα on K  2 +1 , or equivalently (by [5] 15.3 Theorem), Λ|K  m2 +1 is a multiple of ψα . For any Λ ∈ C(ψα , M), the triple (M, Jα , Λ) is a (in our case, unramified) cuspidal type in G(F ) in the sense of [5] 15.5 Definition. An equivalent reformulation is given in terms of cuspidal inducing data ([5] 15.8): the cuspidal inducing datum attached to (M, Jα , Λ) is the pair (M, Ξ), where Ξ = IndZK Jα Λ. The G(F )-representation G(F ) G(F ) c-IndJα Λ = c-IndZK Ξ attached to (M, Jα , Λ), resp. to (M, Ξ) is then irreducible and cuspidal. Out of the given minimal pair (E/F, χ) one constructs now the representation Λ of Jα , and thus gets a corresponding cuspidal type. We have two different cases. m+1 Case m odd. ([5] 19.3) Then  m 2  + 1 =  2 . Let Λ be the character of Jα defined by Λ|

K

m+1  2

= ψα ,

Λ|E ∗ = χ

(4.27)

(this is a consistent definition, as one sees from trM |OE = trE |OE , E ∗ ∩ K 

 m+1  UE 2 ).

m+1 2 

=

Case m > 0 even. ([5] 15.6, 19.4) Let Jα1 = Jα ∩ K 1 = UE1 K  2  , Hα1 = UE1 K  2 +1 . Then Jα1  Hα1 . Let θ be the character of Hα1 defined (as in the odd case) by m+1

θ(ux) = χ(u)ψα (x),

m

u ∈ UE1 , x ∈ K  2 +1 . m

Let η be the unique irreducible (q-dimensional) Jα1 -representation containing θ. Let μM denote the group of roots unity of a field M and let η˜ be the unique irreducible representation of μE /μF  Jα1 such that η˜|Jα1 ∼ = η and tr η˜(ζu) = −θ(u) for all u ∈ Hα1 , ζ ∈ μE /μF  {1}. Then η˜ factors through a representation of μE /μF  Jα1 / ker(θ). Let ν be the representation of E ∗  Jα1 / ker(θ) which arises by inflation from η˜ via the surjection induced by E ∗  E ∗ /F ∗ UE1 ∼ ˜ be the character of E ∗  Jα1 / ker(θ), which = μE /μF . Let χ ∗ 1 ˜ = χ⊗ν. ˜ is χ on E and trivial on Jα / ker(θ). Define the E ∗ Jα1 / ker(θ)-representation Λ ∗ 1 It factors through the surjection E  Jα / ker(θ)  Jα / ker(θ), (e, j) → ej mod ker(θ), hence it is an inflation of a representation Λ1 of Jα / ker(θ). Take Λ to be the inflation of Λ1 to Jα . Let then in both cases (M, Θχ ) be the corresponding cuspidal inducing datum, i.e., Θχ = IndZK Jα Λ.

(4.28)

Thus we attached a cuspidal inducing datum to χ and now the G(F )-representation πχ from (4.1) is defined in [5] 19.4.2 as G(F )

G(F )

πχ = c-IndZK Θχ = c-IndJα

Λ.

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Proposition 4.27. Let (E/F, χ) be a minimal pair. Then Rχ ∼ = πχ . Using Proposition 4.27, we can prove our main result. Proof of Theorem 4.3. By Lemma 4.4 we can assume that (E/F, χ) is minimal in the first statement of the theorem. If (χ) = 0, then the first statement follows essentially from [10] Theorem 1.1(i). If (χ) > 0, then the first statement follows from Corollary 4.23 and the part about the central character follows from Lemma 4.4. To show Rχ ∼ = πχ we can assume by Lemma 4.4 (along with the fact that φπχ = πφE χ ) that (E/F, χ) is minimal. Then Rχ ∼ = πχ follows from Proposition 4.27. Now bijectivity of (4.2) follows from bijectivity of (4.1). 2 Proof of Proposition 4.27. Let m be the level of χ. If m = 0, the proposition follows essentially from [10] Theorem 1.1(i) and [5] 19.1. Assume m > 0. The unramified representation Rχ is induced from the cuspidal inducing datum (ZK, Ξχ ). As the map (4.1) in the tame parametrization theorem is surjective, there is some character χ such that (E/F, χ ) is minimal and Rχ ∼ = πχ . By Corollary 4.23, (χ ) = m. One deduces ∼ Ξχ = Θχ (e.g. by the same reasoning as in the proof of Lemma 4.24). We have to show that χ = χ or χ = (χ )σ . A comparison of the central characters shows χ|F ∗ = χ |F ∗ . Thus it remains to show that χ|UE = χ |UE or χ|UE = (χ )σ |UE . The K-representation Ξχ |K is inflated from the Km -representation ξχ . Note that the image of UE in Km is a non-split torus Hm , as considered in Theorem 4.13. Thus χ|UE , χσ |UE are the unique characters among all UE -characters of level m, which satisfy condition (i) resp. (ii) of Corollary 4.14 if m odd resp. even. Thus it is enough to show that Θχ |K characterizes χ |UE in the same way. This is the content of Lemma 4.28. 2 Lemma 4.28. Let χ be a character of E ∗ of level m > 0 such that (E/F, χ) is a minimal pair. (i) If m is odd, the representation Θχ = IndZK Jα Λ (cf. (4.28)) contains the character χ ∗ on E (exactly once) and does not contain all the characters χ of E ∗ , which satisfy χ |UE = χ|UE and χ |F ∗ UE1 = χ|F ∗ UE1 . (ii) If m is even, the representation Θχ = IndZK Jα Λ (cf. (4.28)) does not contain the ∗ character χ on E and it contains all the characters χ of E ∗ , which satisfy χ |UE = χ|UE and χ |F ∗ UE1 = χ|F ∗ UE1 . Proof. Let first m > 0 be arbitrary and let χ be a character of E ∗ , satisfying χ |F ∗ UE1 = χ|F ∗ UE1 . Mackey formula and Frobenius reciprocity show: HomE ∗ (χ , Θχ ) =

g∈E ∗ \ZK/Jα

HomE ∗ ∩g Jα (χ , g Λ).

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Let g ∈ ZK. We claim that HomE ∗ ∩g Jα (χ , g Λ) = 0, unless g ∈ Jα . Indeed, we have  m +1 E ∗ ∩ g Jα ⊇ UE 2 and Λ|K  m2 +1 is a multiple of ψα , hence g Λ|K  m2 +1 is a multiple of  ψg−1 αg . Moreover, χ |  m2 +1 = χ|  m2 +1 = ψα . Thus if HomE ∗ ∩g Jα (χ , g Λ) = 0, then g UE

UE  m +1

normalizes the character ψα of UE 2 . Thus Proposition 4.29 shows our claim.  ∗ The claim implies that HomE (χ , Θχ ) = HomE ∗ (χ , Λ). In particular, if m is odd, we are done, because then Λ is one-dimensional and Λ|E ∗ = χ. Assume m is even. By construction, Λ arises by an inflation process from the E ∗  Jα1 / ker(θ)-representation ˜ =χ Λ ˜ ⊗ ν, where χ ˜ agrees with χ on E ∗ and is trivial on Jα1 / ker(θ). So, it is enough to prove the following claim: ν|E ∗ does not contain the trivial character of UE , but it contains all non-trivial characters of UE , which are trivial on UF UE1 . The restriction of ν to E ∗ is the inflation via E ∗  E ∗ /F ∗ UE1 ∼ = μE /μF of the restriction to μE /μF of the μE /μF  Jα1 -representation η˜θ . In particular, ν|UF UE1 is trivial. Now [5] 19.4 Proposition shows that η˜θ |μE /μF = RegμE /μF −1μE /μF , and the claim follows. 2 The following proposition is an improvement of a part of the Intertwining theorem [5] 15.1. Also Lemma 4.30 below improves [5] Lemma 16.2.  m +1

Proposition 4.29. Let g ∈ ZK. Then g normalizes the character ψα of UE 2 only if g ∈ Jα .

if and

Proof. We can assume g ∈ K. Let X be the appropriate quotient of t−m M/t− 2  M such that the following diagram commutes m

t−m M/t− 2  M m

∼

 m +1 ∨

(Km2

)

(4.29) X

∼

 m +1

(UE 2

/UEm+1 )∨

where the upper horizontal map is α → ψα with ψα as in [5] 12.5, and the right vertical map is restriction of characters. Let Y ⊆ M be such that t−m Y ⊆ t−m M is the preimage in t−m M of the kernel of the left vertical map. Then g normalizes ψα |  m2 +1 if and only UE

if the images of g −1 αg and α in X coincide, i.e., if the following equation holds true in t−m M: g −1 αg ≡ α

mod t− 2  M + t−m Y. m

Then the result follows from Lemma 4.30 applied to k =  m+1 2 .

2

Lemma 4.30. Write α = t−m α0 . With notations as in the proof of Proposition 4.29, for any 1 ≤ k ≤  m+1 2 , we have

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g −1 α0 g ≡ α0

mod tk M + Y

(4.30)

in M if and only of g ∈ UE + tk M. Proof. The ‘if’ part is immediate. To prove the other part, we use induction on k (as in [5] 16.2 Lemma). Let k ≥ 2 and assume (4.30). By induction hypothesis, g ∈ UE +tk−1 M. We can write g = g1 (1 +tk−1 g0 ) with g1 ∈ UE . Thus (as α0 ∈ OE ) we obtain from (4.30): tk−1 α0 g0 ≡ tk−1 g0 α0

mod tk M + Y.

Thus tk−1 (α0 g0 − g0 α0 ) = y + tk m ∈ M for some y ∈ Y , m ∈ M. We deduce y = tk−1 y  with y  ∈ M and α0 g0 − g0 α0 = y  + tm. We claim that y  ∈ Y + tM. Indeed, this claim is equivalent to ψt−m y |U m /U m+1 ≡ 1. But for u ∈ M we have: E

E

ψt−m y (1 + tm u) = ψ(trM (y  u)) = ψt−m y (1 + tm−(k−1) u) = 1, m+1 where the last equality holds as long as m −(k−1) ≥  m 2  +1, or equivalently, k ≤  2 , which is satisfied by assumption of the lemma. This shows our claim. From it we deduce α0 g0 ≡ g0 α0 mod Y + tM, i.e., by induction hypothesis, g0 ∈ OE + tM. Thus we are reduced to the case k = 1. We handle this case explicitly. The result remains unaffected if we replace the embedding j : E → M2 (F ) by a conjugate one. As all such embeddings are G(F )-conjugate, we can  assume that j(OE ) mod t ⊆ M/tM is generated as a −b k-algebra by a matrix β = for some a, b ∈ k such that the characteristic 1 −a polynomial T 2 + aT + b is irreducible in k[T ] (cf. e.g. [5] 5.3). Then α = t−m α0 with α0 mod t = x + yβ for some x, y ∈ k and j(OE ) = OF [α0 ]. After adding and multiplying by some central elements (which does not affect the condition (4.30)),we can  assume 1 that either char(k) > 2 and there is a D ∈ k∗  k ∗,2 such that α0 = or that D 2 char(k)  = 2and there is a D ∈ k such that T + T + D ∈ k[T ] is irreducible and D α0 = . We have to show that if g ∈ K and (4.30) holds for g, α0 and k = 1, then 1 1 g ∈ OE + tM. Assume first char(k) > 2. The upper horizontal map in diagram (4.29) induces the isomorphism ∼

m ∨ t−m M/t−m+1 M −→ (Km ) ,

which shows that Y + tM/tM = {tm β : β ∈ t−m M and ψ(trM (β(e − 1))) = 1 for all e ∈ 1 + tm OE }/tM   B1 B2 ={ : B1 , B2 ∈ k} −B2 D −B1

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 g1 g2 g3 g4

(the last equality is an easy computation). Now let g =

685

∈ G(k) (we can work

modulo t). Then condition (4.30) translates into 1 det(g)



g3 g4 − g1 g2 D −g22 D + g42 g12 D − g32 g1 g2 D − g3 g4



 =g

−1

 1

!

α0 g =

D



 B2 B1 −B2 D −B1

+

for some B1 , B2 ∈ k. In particular, we must have 

1 2 2 det(g) (g4 − g2 D) 1 2 2 det(g) (g1 D − g3 )

= 1 + B2 = (1 − B2 )D.

Computing B2 from the first equation and inserting it into the second, gives us 1 (g 2 − g32 − g22 D2 + g42 D) = 2D, det(g) 1 which is equivalent to D(g1 − g4 )2 = (g3 − g2 D)2 . If both sides are non-zero, on the left side we have a non-square in k∗ and on the right side we have a square, which is a contradiction. Thus both sides are zero, i.e. g1 = g4 , g3 = g2 D, i.e., g ∈ UE mod t, finishing the proof in the case char(k) > 2. Assume now char(k) = 2. Analogously to the previous case we deduce  Y + tM/tM =

 B1 B1 + B3 D B1 B3

: B1 , B3 ∈ k

.

A similar computation as above implies that for g ∈ G(k) satisfying condition (4.30) we must have det(g)−1 (g1 g2 + g2 g3 + g3 g4 D) = B1 det(g)−1 (g12 + g1 g3 + g32 D) = 1 + B3 det(g)−1 (g22 + g2 g4 + g42 D) = B1 + B3 D + D, with some B1 , B3 ∈ k. Putting the first and the second equation into the third and bringing some terms together shows g22 + g32 D2 = (g12 + g42 )D + g2 (g4 + g1 + g3 ) + g3 D(g4 + g1 ). Add 2g32 D = 0 to the right side of this equation and let A = g2 +g3 D and B = g1 +g3 +g4 . The equation is then equivalent to

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A2 + B 2 D + AB = 0. Suppose B = 0. Dividing by B 2 , we obtain (A/B)2 + (A/B) + D = 0, which is a contradiction to irreducibility of T 2 + T + D ∈ k[T ], as A/B ∈ k. Thus B = 0 and we deduce also A = 0, which finishes the proof also in the case char(k) = 2. 2 Acknowledgments The author is especially grateful to Eva Viehmann, Christian Liedtke, Stephan Neupert and to a referee for very helpful comments concerning this work. Also he is grateful to Paul Hamacher, Bernhard Werner and other people for interesting discussions concerning this work. Further, the author wants to thank Michael Rapoport and Ulrich Görtz for patiently introducing him to the field of affine Deligne–Lusztig varieties some years ago. References [1] M. Boyarchenko, Deligne–Lusztig constructions for unipotent and p-adic groups, preprint, arXiv: 1207.5876, 2012. [2] M. Boyarchenko, J. Weinstein, Maximal varieties and the local Langlands correspondence for GL(n), J. Amer. Math. Soc. 29 (1) (2016) 177–236. [3] F. Bruhat, J. Tits, Groupes réductifs sur un corps local, Publ. Math. Inst. Hautes Études Sci. 41 (1972) 5–251. [4] F. Bruhat, J. Tits, Groupes réductifs sur un corps local II. Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math. Inst. Hautes Études Sci. 60 (1984) 197–376. [5] C.J. Bushnell, G. Henniart, The Local Langlands Correspondence for GL2 , Springer, 2006. [6] C.J. Bushnell, P.C. Kutzko, The admissible dual of GL(N ) via compact open subgroups, in: Ann. of Math. Studies, vol. 129, Princeton University Press, 1993. [7] C. Chan, Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, Adv. Math. 294 (2016) 332–383. [8] P. Deligne, La conjecture de Weil II, Publ. Math. Inst. Hautes Études Sci. (52) (1980) 137–252. [9] P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1) (1976) 103–161. [10] A. Ivanov, Cohomology of affine Deligne–Lusztig varieties for GL2 , J. Algebra 383 (2013) 42–62. [11] A. Ivanov, Ramified automorphic induction and zero-dimensional affine Deligne–Lusztig varieties, preprint, arXiv:1512.07530, 2015. [12] G. Lusztig, Some remarks on the supercuspidal representations of p-adic semisimple groups, in: Automorphic Forms, Representations and L-functions, part 1, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, in: Proc. Sympos. Pure Math., vol. XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 171–175. [13] G. Lusztig, Representations of reductive groups over finite rings, Represent. Theory 8 (2004) 1–14. [14] J. Milne, Étale Cohomology, Princeton University Press, 1980. [15] G. Pappas, M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (1) (2008) 118–198. [16] M. Rapoport, A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005) 271–318. [17] A. Stasinski, Extended Deligne–Lusztig varieties for general and special linear groups, Adv. Math. 226 (3) (2011) 2825–2853. [18] J.-K. Yu, Smooth models associated to concave functions in Bruhat–Tits theory, preprint, 2002. [19] X. Zhu, Affine Grassmannians and the geometric Satake in mixed characteristic, preprint, arXiv: 1407.8519, 2014.