Class polynomials for some affine Hecke algebras

Journal of Algebra 452 (2016) 502–548

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Journal of Algebra www.elsevier.com/locate/jalgebra

Class polynomials for some affine Hecke algebras Zhongwei Yang Beijing International Center For Mathematical Research, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, China

a r t i c l e

i n f o

Article history: Received 31 March 2015 Available online 21 January 2016 Communicated by Gerhard Hiss Keywords: Affine Deligne–Lusztig variety Affine Hecke algebra Affine Weyl group Class polynomial

a b s t r a c t Class polynomials attached to affine Hecke algebras were first introduced by He in [13]. They play an important role in the study of affine Deligne–Lusztig varieties. Motivated by [14], we compute the class polynomials attached to an affine Hecke 2 . Using these class polynomials, algebra of type (twisted) A we prove a conjecture of Görtz–Haines–Kottwitz–Reuman for the general linear group, unitary group, and division algebra of semisimple rank 2. Furthermore, we discuss some interesting patterns of affine Deligne–Lusztig varieties. © 2016 Elsevier Inc. All rights reserved.

Introduction In this study, we consider class polynomials of affine Hecke algebras and we apply them to the investigation of affine Deligne–Lusztig varieties in some affine flag varieties. First, let’s recall the classical Deligne–Lusztig variety, which was introduced by Deligne and Lusztig in 1976 to construct linear representations of finite groups of Lie type (see [4]). Let Fq be a finite field and k be its algebraic closure. Let G be a reductive group defined over Fq with a Frobenius automorphism σ and let B be a Borel subgroup E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2015.11.049 0021-8693/© 2016 Elsevier Inc. All rights reserved.

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defined over Fq . We have the Bruhat decomposition G = w∈W BwB, where W is the Weyl group. The Deligne–Lusztig variety associated with w ∈ W is defined by Xw = {gB ∈ G/B | g −1 σ(g) ∈ BwB}. This is a locally closed subvariety of the flag variety G/B of dimension (w). The notion of an affine Deligne–Lusztig variety first introduced by Rapoport in [20] is an analogue of Deligne and Lusztig’s classical construction. For simplicity, let G be as given above and let L = k(()) be the field of the Laurent series. Again, we denote σ by the automorphism on the loop group G(L). Let I be a σ-stable Iwahori subgroup of  is the G(L). We have the Iwahori–Bruhat decomposition G(L) = w∈ ˜ where W  I wI, ˜ W Iwahori–Weyl group. By definition, the affine Deligne–Lusztig variety Xw˜ (b) associated  and b ∈ G(L) is defined as with w ˜∈W Xw˜ (b) = {gI ∈ G(L)/I | g −1 bσ(g) ∈ I wI}. ˜ The affine Deligne–Lusztig variety Xw˜ (b) plays an important role in the study of the reduction of Shimura varieties with Iwahori level structure. In particular, it is related to the intersection of the Newton stratum and the Kottwitz–Rapoport stratum. There are two important stratifications on the special fiber of a Shimura variety: the Newton stratification with strata indexed by certain σ-conjugacy classes [b] ⊂ G(L), and the Kottwitz–Rapoport stratification with strata indexed by specific elements w ˜ of the  (see [6,10] and [20] for details). extended affine Weyl group W We are particularly interested in the following questions: • When is Xw˜ (b) = ∅? • If Xw˜ (b) is nonempty, what is dim Xw˜ (b)? These questions have been studied many times previously (see [7–9,14,21,22] and [24]), but an explicit answer to these questions is still unknown for general w ˜ and b. He [14] obtained a remarkable breakthrough in the study of affine Deligne–Lusztig varieties in affine flag varieties by showing that the emptiness/nonemptiness pattern and dimension formula for affine Deligne–Lusztig varieties can be deduced by class polynomials of affine Hecke algebras. Thus, we can reduce questions in arithmetic geometry and number theory to questions in representation theory and Lie theory. In this study, we obtain the following results. 2 . • We calculate class polynomials for the affine Hecke algebra of type (twisted) A • Using the class polynomials, we obtain the emptiness/nonemptiness pattern of Xw˜ (b) and its dimension formula for GL3 , U3 , and D× 3 (see §3.1 for details). • We verify a conjecture of Görtz–Haines–Kottwitz–Reuman for the general linear group, unitary group, and division algebra of semisimple rank 2 (see Theorem 3.13).

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• We obtain a closed formula for superbasic b on the number of the rational points in Xw˜ (b). • Based on information from the class polynomials and the reduction method, it is expected that the irreducible components of Xw˜ (b) of maximal dimension are controlled by the leading coefficient of the corresponding class polynomial. We then describe the leading coefficients of the corresponding class polynomials. The remainder of this paper is organized as follows. In §1, we recall some definitions, e.g., Coxeter systems, (affine) Hecke algebras, loop groups, class polynomials, and affine Deligne–Lusztig varieties. We also explain the algorithm for computing class polynomials as well as recalling the “Dimension = Degree” theorem. Section §2 contain the most technical details (see my PhD thesis [25] for more detailed calculations). First, we  and we then calculate the class classify all of the conjugacy (δ-conjugacy) classes of W polynomials. In §3, we apply class polynomials to the study of affine Deligne–Lusztig varieties. 1. Preliminary data 1.1. Coxeter systems and Hecke algebras To provide some context, let us first recall Hecke algebras of Coxeter groups. We follow [1], by letting W be a group with identity 1 and S be a set of generators of W such that S = S−1 and 1 ∈ / S. Every element of W is the product of a finite sequence of elements of S. We also assume that every element of S is of order 2. Definition 1.1. (W, S) is said to be a Coxeter system if it satisfies the following condition: For s, s ∈ S, let mss be the order of ss and let I0 be the set of pairs (s, s ) such that mss is finite. The generating set S and the relations ss mss = 1 for (s, s ) ∈ I0 form a presentation of the group W . Under this condition, we refer to W as a Coxeter group. Let (W, S) be a Coxeter system and w ∈ W . We recall that the length of w (with respect to S), denoted by S (w) or simply by (w) is the smallest integer r  0 such that w is the product of some (or equivalently, any) sequence of r elements of S. We retain the notations used in [17] §1.1. Let H be a group of automorphisms of the group W that preserves S. Set W  = W  H. Then, an element in W  is of the form wδ for some w ∈ W and δ ∈ H. We find that (wδ)(w δ  ) = wδ(w )δδ  ∈ W  with δ, δ  ∈ H. For w ∈ W and δ ∈ H, we set (wδ) = (w), where (w) is the length of w in the Coxeter group (W, S). Thus, H comprises length 0 elements in W  . For J ⊂ S, we denote WJ as the standard parabolic subgroup of W generated by sj for sj ∈ J and W J (resp. J W ) as the set of minimal coset representatives in W/WJ (resp. WJ \W ).

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Let δ ∈ H. For each δ-orbit in S, we pick a single element. Let g be the product of these elements (in any order) and put c = gδ ∈ W  . We refer to c as a Coxeter element of W  (see [23]). Let O be a δ-conjugacy class of W  and by definition, O is called Coxeter if it contains a Coxeter element of W  . Let (W, S) be a Coxeter system and R0 be a commutative ring with 1 (by an abuse of notation), and let q ∈ C∗ . Definition 1.2. The Hecke algebra H (with identity T1 ) associated with the Coxeter system (W, S) over R0 is the associative R0 -algebra, which is given by the following presentations: • Generators: Ts , s ∈ S; • Relations: Ts2 = (q − 1)Ts + qT1 and (Ts Tt )mst = (Tt Ts )mts for all s, t ∈ S. 1.2. Twisted loop groups Let k be the algebraic closure of a finite field Fq . Let F = Fq (()) and L = k(()) be the fields of Laurent series. Let G be a connected reductive group over F and splits over a tamely ramified extension of L. Let S ⊂ G be a maximal L-split torus defined over F , T = ZG (S) be its centralizer and N be the normalizer of T . Since k is algebraically closed, G is quasi-split over L. Furthermore, T is a maximal torus. Definition 1.3. The algebraic loop group LG associated with G is the ind-group scheme over k that represents the functor R −→ LG(R) = G(R(())) on the category of k-algebras. Let σ ∈ Gal(L/F ) be the Frobenius automorphism, which induces an automorphism on G(L), and we denote the induced automorphism by the same symbol. Let A be the apartment of G(L) corresponding to S and aC be a σ-invariant alcove S be the in A. Let I ⊂ G(L) be the Iwahori subgroup corresponding to aC over L and  set of simple reflections at the walls of aC . By definition, the finite Weyl group W associated with S is W = N (L)/T (L)  associated with S is and the Iwahori–Weyl group W  = N (L)/T (L)1 W

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∼ where T (L)1 is the unique parahoric subgroup of T (L). Since W = I\G(L)/I, we embed   in G(L) with W set-theoretically into G(L). Thus we identify representatives of W  elements in W . We have the Iwahori–Bruhat decomposition G(L) =

˜  I wI.

 w∈ ˜ W

¯ Let Γ be the absolute Galois group Gal(L/L) and P be the Γ-coinvariants of X∗ (T ). By [11] and by choosing a special vertex in A, we identify T (L)/T (L)1 with P . We also  −→ W −→ 1 and a semi-direct obtain a split short exact sequence 1 −→ P −→ W  , product W = P  W . The automorphism σ on G(L) induces an automorphism on W S. We choose a special vertex in A which we denote by δ. The map gives a bijection on  such that the previous split short exact sequence is preserved by δ. Thus, δ induces an automorphism on W and we denote it by the same symbol. Let Φ be the set of roots of (G, S) over L and Φa be the set of affine roots. Let S be the set of simple roots in Φ. We identify S with the set of simple reflections in W , and S. We denote Φ+ as the set of positive roots of Φ thus S is a δ-stable proper subset of  and ρ as half of the sum of all positive roots in Φ. Let G1 be the subgroup of G(L) generated by all parahoric subgroups. We take N1 = S) is a double Tits system with N (L) ∩ G1 , and thus by [2], the quadruple (G1 , I, N1 ,  affine Weyl group Wa = N1 /(N (L) ∩ I). We identify Wa with the Iwahori–Weyl group of the simply connected cover Gsc of the derived group Gder of G. Let Tsc be the maximal torus of Gsc given by T . Thus, we have Wa = X∗ (Tsc )Γ  W , which shows that a reduced root system Δ exists such that Wa = Q∨ (Δ)  W (Δ), where Q∨ (Δ) is the coroot lattice of Δ. We write Q for Q∨ (Δ) and identify Q with X∗ (Tsc )Γ and W (Δ) with W . S) are Coxeter systems. Thus, we already have Remark 1.4. The pairs (W, S) and (Wa ,   is not a length functions on W and Wa (see §1.1). However, the Iwahori–Weyl group W  , the length of w Coxeter group. For any element w ˜∈W ˜ (denoted as (w)) ˜ is the number  of “affine root hyperplanes” between w(a ˜ C ) and aC in A. Let Ω be the subgroup of W  comprising length 0 elements. The Iwahori–Weyl group W is a quasi-Coxeter group in  = Wa  Ω. the sense that W 1.3. Class polynomials By analogy with the definition of a Hecke algebra associated with a Coxeter system, we recall a Hecke algebra associated with a quasi-Coxeter system.  be the Hecke algebra associated with W  , i.e., H  is the associative Definition 1.5. Let H −1  A = Z[v, v ]-algebra with basis Tw˜ for w ˜ ∈ W and its multiplication is given by

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Tx˜ Ty˜ = Tx˜y˜,

507

if (˜ x) + (˜ y ) = (˜ xy˜);

(Ts − v)(Ts + v −1 ) = 0,

for s ∈  S.

 Note that the map Tw˜ → Tδ(w) ˜ defines an A-algebra automorphism of H, which we still denote as δ.  are said to be δ-conjugate if w  . For ˜ w ˜ ∈ W ˜ = x ˜wδ(˜ ˜ x)−1 for some x ˜∈W For any w, s i      and si ∈  w, ˜ w ˜ ∈W S, we write w ˜ −→δ w ˜ if w ˜ = si ws ˜ δ(i) and (w) ˜  (w). ˜ We write    such that w ˜− →δ w ˜ if there is a sequence w ˜=w ˜0 , w ˜1 , · · · , w ˜n = w ˜ of elements in W si   ˜ We write w for any k, w ˜k−1 −→δ w ˜k for some si ∈ S. ˜ ≈δ w ˜ if w ˜− →δ w ˜ and w ˜ − →δ w. ˜

˜ δw We write w ˜≈ ˜  if w ˜ ≈δ τ w ˜  δ(τ )−1 for some τ ∈ Ω.   ˜ w ˜ ∈ W are elementarily strongly δ-conjugate if (w) ˜ = (w ˜  ) and We say that w,  −1  exists such that w x ˜ ∈ W ˜ = x ˜wδ(˜ ˜ x) and (˜ xw) ˜ = (˜ x) + (w) ˜ or (wδ(  x)−1 ) = ˜ In addition, we say that w, ˜ w ˜  are strongly δ-conjugate if there is a sequence (˜ x) + (w).   such that for any i, w w ˜=w ˜0 , w ˜1 , · · · , w ˜n = w ˜ of elements in W ˜i−1 is elementarily   strongly δ-conjugate to w ˜i . We write w ˜∼ ˜ δw ˜ if w ˜ and w ˜ are strongly δ-conjugate. He and Nie [16] proved that minimal length elements wO of any δ-conjugacy class  satisfy some special properties, thereby generalizing the results of Geck and O of W Pfeiffer [5] on finite Weyl groups. These properties play a key role in the study of affine Deligne–Lusztig varieties and affine Hecke algebras. He [14] and [16] showed that for any δ-conjugacy class O, we can fix a minimal length representative wO and the image of ˜ H, ˜ H] ˜ δ where [H, ˜ H] ˜ δ is the A-submodule (we regard H ˜ as a left A-module) TwO in H/[  ˜ [h, h ]δ = hh − h δ(h)). ˜ generated by all δ-commutators (i.e., for any h, h ∈ H, of H ˜ H, ˜ H] ˜ δ . In Moreover, TwO is independent of the choice of wO and it forms a basis of H/[ [16, Theorem 6.7], the following was proved.  H,  H]  δ , where O runs Theorem 1.6 (He–Nie). The elements Tw˜O form an A-basis of H/[ . over all the δ-conjugacy classes of W  and a δ-conjugacy In the following, we denote Tw˜O as TO for simplicity. For any w ˜∈W class O, there exists a unique fw,O ∈ A such that ˜ Tw˜ ≡



fw,O ˜ TO

 H]  δ. mod [H,

O

fw,O is a polynomial in Z[v − v −1 ] with nonnegative coefficient, which is called the ˜ class polynomial attached to w ˜ and O, where this can be constructed inductively as follows.  , then we set ˜ is a minimal length element in a δ-conjugacy class of W If w  fw,O = ˜

1, 0,

if w ˜ ∈ O, if w ˜∈ / O.

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˜ If w ˜ is not a minimal length element in its δ-conjugacy class and for any w ˜ ∈ W    ˜ ) < (w), ˜ fw˜ ,O is constructed. By [16], w ˜1 ≈δ w ˜ and si ∈ S exist such that with (w (si w ˜1 sδ(i) ) < (w ˜1 ) = (w). ˜ In this case, (si w) ˜ < (w) ˜ and we define fw,O as ˜ fw,O = (v − v −1 )fsi w˜1 ,O + fsi w˜1 sδ(i) ,O . ˜ 1.4. The “Dimension = Degree” theorem We retain the notations given previously. Let Fl = G(L)/I be the f ppf quotient, then Fl is represented by an ind-scheme, which is ind-projective over k. We also have the Iwahori–Bruhat decomposition: Fl =

˜  I wI/I.

 w∈ ˜ W

 and b ∈ G(L), the affine Deligne–Lusztig variety associated Definition 1.7. For w ˜∈W with w ˜ and b is the locally closed sub-ind scheme Xw˜ (b)(k) in the affine flag variety Fl defined as Xw˜ (b)(k) = {gI ∈ G(L)/I | g −1 bσ(g) ∈ I wI}. ˜ Remark 1.8. The affine Deligne–Lusztig variety Xw˜ (b)(k) is a finite-dimensional k-scheme and it is locally of finite type over k.  a δ-straight element if and only if for any n ∈ N, we have We call an element w ˜∈W n−1  straight if it contains (wδ( ˜ w) ˜ ···δ (w)) ˜ = n(w). ˜ We call a δ-conjugacy class in W some straight element. We denote ·σ as the σ-conjugation action on G(L) and it is defined by that for any g, g  ∈ G(L), g ·σ g  = gg  σ(g)−1 . Let B(G) be the set of σ-conjugacy classes in G(L). We have the following Kottwitz’s classification of σ-conjugacy classes in G(L).  , we fix a minimal Theorem 1.9 (Kottwitz, He). For any straight δ-conjugacy class O of W length representative w ˜O . Then, G(L) =

 G(L) ·

σ

O

w ˜O ,

. where O runs over all the straight δ-conjugacy classes of W Let (P/Q)δ be the δ-coinvariants on P/Q and let  −→ W  /Wa ∼ κ:W = P/Q −→ (P/Q)δ be the natural projection. We call κ the Kottwitz map.

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Let PQ = P ⊗Z Q and PQ /W be the quotient of PQ by the natural action of W . We δ can identify PQ /W with PQ,+ = {χ ∈ PQ | α(χ)  0, for all α ∈ Φ+ }. Let PQ,+ be the set of δ-invariant points in PQ,+ . Since the image of W  δ in Aut(W ) is a finite group, then for each w ˜ = tχ w ∈  , there exists n ∈ N such that δ n = 1 and wδ(w)δ 2 (w) · · · δ n−1 (w) = 1. Then, W wδ( ˜ w)δ ˜ 2 (w) ˜ · · · δ n−1 (w) ˜ = tλ for some λ ∈ P . Let νw˜ = λ/n ∈ PQ and ν¯w˜ be the corresponding element in PQ,+ . Note that νw˜ is independent of the choice of n. Let ν¯w˜ be the unique element in PQ,+ that lies in the W -orbit of νw˜ . Since tλ = wt ˜ δ(λ) w ˜ −1 = twδ(λ) , δ δ  −→ P ν¯w˜ ∈ PQ,+ . We call the map W ˜ → ν¯w˜ the Newton map. We define Q,+ with w  −→ P δ × (P/Q)δ f :W Q,+

by

w ˜ −→ (¯ νw˜ , κ(w)). ˜

 . We denote the image of Following [18], f is constant on each δ-conjugacy class of W  the map by B(W , δ). Remark 1.10. (1) There is a bijection between the set of σ-conjugacy classes B(G) of . G(L) and the set of straight conjugacy classes of W  . Moreover, the (2) Any σ-conjugacy class of G(L) contains a representative in W δ  map f : W −→ PQ,+ × (P/Q)δ is in fact the restriction of a map defined on G(L) as b → (¯ νb , κ(b)) and we call ν¯b the Newton vector of b. The “Dimension = Degree” theorem is a main result of [14], which we quote as follows.  . Then Theorem 1.11 (He). Let b ∈ G(L) and w ˜∈W 1 ˜ + (O) + deg(fw,O dim(Xw˜ (b)) = max ((w) νb , 2ρ, ˜ )) − ¯ O 2  with f (O) = f (b) and (O) is the length of here O runs over δ-conjugacy classes of W any minimal length element in O. Remark 1.12. We use the convention that the dimension of an empty variety and the degree of a zero polynomial are both −∞. Theorem 1.11 relates the dimension of affine Deligne–Lusztig varieties to the degree of the class polynomials, which provides both a theoretic and practical method to determining the dimension of affine Deligne–Lusztig varieties. This shows that the dimension and emptiness/nonemptiness pattern of affine Deligne–Lusztig varieties Xw˜ (b) depend  , δ, w, only on the data (W ˜ f (b)), and thus independent of the choice of G. In addition, this implies that the emptiness/nonemptiness and dimension formulas of affine Deligne– Lusztig varieties rely only on the reduction method. This theorem inspired the present study of the class polynomials of affine Hecke algebras.

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2 2. Class polynomials for affine Hecke algebras of type (twisted) A 2.1. Explicit conjugacy classes  be the Iwahori–Weyl group, then W  = P  W = Wa  Ω, where P is the Let W coweight lattice and Wa is the affine Weyl group generated by the simple reflections ˜ = {s0 , s1 , s2 }. In this case, Ω = τ  with τ 3 = 1 and τ s0 τ −1 = s1 , τ s1 τ −1 = s2 , S τ s2 τ −1 = s0 . Let α1 , α2 be the simple roots corresponding to s1 , s2 , respectively. We  , in the split case, it is sufficient to consider elements note that by the symmetry on W  , we always write in Wa and Wa τ . In the following sections, for any w, ˜ w ˜ ∈ W Tw˜ ≡ Tw˜ 

 H]  (or [H,  H]  δ) mod [H,

as Tw˜ ≡ Tw˜  for short. 2.1.1. Split case-I  -conjugacy classes in Wa . First, we classify all of the W  -conjugate. Let O1 Lemma 2.1. Note that all of the elements in Wa with length 1 are W  -conjugacy class in Wa with minimal length 1. Then be the W O1 = {tkα1 s1 , tkα2 s2 , tk(α1 +α2 ) s1 s2 s1 | k ∈ Z}.  -conjugate. Let O2 Lemma 2.2. All of the elements in Wa with minimal length 2 are W  be the W -conjugacy class in Wa with minimal length 2. Then O2 = {tλ s1 s2 , tλ s2 s1 | λ ∈ Q}. Let Qsh = Q − {k(α1 + 2α2 ), k(2α1 + α2 ), k(α1 − α2 ) | k ∈ Z}. Lemma 2.3. For any λ ∈ P+ ∩ Qsh , i.e., λ = mα1 + nα2 , where m, n ∈ Z and 1  m   · tλ , then n  2m − 1 or 1  n  m  2n − 1. We set Oλ = W Oλ = {tmα1 +nα2 , t(n−m)α1 +nα2 , tmα1 +(m−n)α2 , t−nα1 −mα2 , t−nα1 +(m−n)α2 , t(n−m)α1 −mα2 }, with (Oλ ) = (tλ ). If λ = m(α1 + 2α2 ), then Oλ = {tm(α1 +2α2 ) , t−m(2α1 +α2 ) , tm(α1 −α2 ) }, or if λ = m(2α1 + α2 ), then Oλ = {tm(2α1 +α2 ) , t−m(α1 +2α2 ) , t−m(α1 −α2 ) }, with (Oλ ) = (tλ ).

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Lemma 2.4. For i ∈ N+ , set Ci = {tkα1 +iα2 s1 , t(−i)α1 +kα2 s2 , tkα1 +(k−i)α2 s1 s2 s1 | k ∈ Z} and Ci = {tkα1 −iα2 s1 , tiα1 +kα2 s2 , t(k−i)α1 +kα2 s1 s2 s1 | k ∈ Z}. Then, Ci and Ci are  -conjugacy classes of Wa with minimal length 3i if i is odd or with minimal length W 3i + 1 if i is even. The proofs of the four lemmas above, 2.1, 2.2, 2.3, and 2.4, are direct and the methods required are quite similar to each other. Thus, it is sufficient for us to give a proof of 2.2 as an example, and we omit the others. Proof of Lemma 2.2. Let O2 = {tλ s1 s2 , tλ s2 s1 | λ ∈ Q} and it is quite easy to show that  · O ⊂ O . Now, we prove that for any element w  -conjugate to W ˜ ∈ O2 , then w ˜ is W 2 2   s0 s1 . We use induction on length (w). ˜ Assume that for any w ˜ ∈ O2 and (w ˜  ) < (w), ˜   -conjugate to s0 s1 . We know that w then w ˜ is W ˜ can be written uniquely as xtμ y where x ∈ W, μ ∈ Q ∩ P+ , y ∈ I(μ) W , here I(μ) = {si ∈ S | μ, αi  = 0}. If w ˜ = xtμ y, x = si1 · · · sir = 1 (reduced expression) with sij ∈ S for 1  j  r, then set w ˜1 = si1 ws ˜ i1 = si2 · · · sir tμ ysi1 , and thus (w ˜1 ) = (μ) + (x) − 1 − (ysi1 )  (μ) + (x) − (y) = (w). ˜ If (w ˜1 ) < (w), ˜ then by induction we are complete. If (w ˜1 ) = (w), ˜ we set w ˜2 = μ si2 w ˜1 si2 = si3 · · · sir t ysi1 si2 and (w ˜2 ) = (μ) + (x) − 2 − (ysi1 si2 )  (μ) + (x) − 1 − (ysi1 ) = (w ˜1 ). By the same argument, we can reduce to the cases where w ˜ = tμ s1 s2 or tμ s2 s1 with mα1 +nα2 μ ∈ Q ∩ P+ . Now, if w ˜=t s1 s2 with 1  m  n < 2m − 1 or 1  n < m  2n. If (w) ˜ = 2, this is obvious. If (w) ˜ > 2, set w ˜1 = s0 ws ˜ 0 = t(1−n)α1 +(2−m)α2 s2 s1 ,  -conjugate to s0 s1 , and so is w. we have (w ˜1 ) = (w) ˜ − 2. By induction w ˜1 is W ˜ For kα1 +2kα2 w ˜ = t s1 s2 (k  1), check directly that w ˜1 = s1 ws ˜ 1 , then (w ˜1 ) = (w) ˜ − 2,  -conjugate to s0 s1 . For w so we can deduce that w ˜ is W ˜ = tkα1 +(2k−1)α2 s1 s2 , then  -conjugate to s0 s1 , and w ˜1 = s1 s0 ws ˜ 0 s1 and (w ˜1 ) = (w) ˜ − 2. By induction, w ˜1 is W mα1 +nα2 so is w. ˜ If w ˜ = t s2 s1 with 1  m  n  2m or 1  n < m < 2n − 1. If (w) ˜ = 2, this is obvious. If (w) ˜ > 2, set w ˜1 = s0 ws ˜ 0 = t(2−n)α1 +(1−m)α2 s1 s2 . In  -conjugate to s0 s1 , and so is w. addition, (w ˜1 ) = (w) ˜ − 2. By induction, w ˜1 is W ˜ For 2kα1 +kα1 w ˜=t s2 s1 (k  1), check directly that w ˜1 = s2 ws ˜ 2 , then (w ˜1 ) = (w) ˜ − 2, so  -conjugate to s0 s1 . For w we can deduce that w ˜ is W ˜ = t(2k−1)α1 +kα2 s2 s1 (k  1), then  -conjugate to s0 s1 , and so w ˜1 = s2 s0 ws ˜ 0 s2 and (w ˜1 ) = (w) ˜ − 2. By induction, w ˜1 is W is w. ˜ 2  -conjugacy classes in Wa are as follows: Theorem 2.5. The W {Id}, O1 , O2 , Oλ , Ci , and Ci where λ ∈ P+ ∩ Q, and i ∈ N+ .

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Proof. Following Lemmas 2.1, 2.2, 2.3, 2.4, and Wa = {Id}  O1  O2  (λ∈P+ ∩Q Oλ )  (i∈N+ (Ci  Ci )), we obtain the theorem directly. 2 2.1.2. Split case-II We set Iτ = {tmα1 +nα2 wτ | m ∈ N+ , n ∈ Z, 1 − m  n  2m − 1, w ∈ W }  {τ, s2 τ } {tkα1 +2kα2 τ, tkα1 +2kα2 s2 τ, tkα1 +2kα2 s2 s1 τ, tk(α1 −α2 ) τ, tk(α1 −α2 ) s1 τ, tk(α1 −α2 ) s2 τ | k ∈ N+ }. Since Wa τ = Iτ ∪ τ · (Iτ ) ∪ τ 2 · (Iτ ) and any element w ˜ ∈ Wa τ is ∼ ˜ to an element in -conjugacy Iτ , so it is sufficient to consider elements in Iτ . Now, we classify all of the W classes in Wa τ .  -conjugacy class in Wa τ with minimal length 0. Lemma 2.6. Let Oid,τ be the set of W Then, Oid,τ = {tmα1 +nα2 τ, tmα1 +nα2 s1 s2 τ | m, n ∈ Z}.  Lemma 2.7. For any λ ∈ P+ Q and λ = 2mα1 + mα2 (m ∈ N) i.e. λ = mα1 + nα2  · (tλ s2 s1 τ ). If n = 2m where 1  m  n  2m or 1  n < m  2n − 1, we set Oλ,τ = W or m = 2n − 1, where m, n ∈ N+ , then Oλ,τ = {tmα1 +nα2 s2 s1 τ, t(1−n)α1 +(m+1−n)α2 s2 s1 τ, t(n−m)α1 +(1−m)α2 s2 s1 τ }. If 2  m  n  2m − 1 or 1  n < m < 2n − 1, then Oλ,τ = {tmα1 +nα2 s2 s1 τ, t(1−n)α1 +(m+1−n)α2 s2 s1 τ, t(n−m)α1 +(1−m)α2 s2 s1 τ } {t(n−m)α1 +nα2 s2 s1 τ, t(1−n)α1 +(1−m)α2 s2 s1 τ, tmα1 +(m+1−n)α2 s2 s1 τ } and (Oλ,τ ) = (tλ s2 s1 τ ).  · (t( 2i +1)α1 +iα2 s1 s2 s1 τ ). Then, Oi,τ is the set of Lemma 2.8. For i ∈ Z, let Oi,τ = W  -conjugacy class in Wa τ with the W (Oi,τ ) = (t( 2 +1)α1 +iα2 s1 s2 s1 τ ). i

Moreover, Oi,τ = {tkα1 +iα2 s1 s2 s1 τ, t(k+i−1)α1 +kα2 s2 τ, t(1−i)α1 +kα2 s1 τ | k ∈ Z}.

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The approaches employed to prove the lemmas above 2.6, 2.7, and 2.8 are quite similar, so we only give a proof of 2.8 as an example. Proof of Lemma 2.8. (1) We set A = {tkα1 +iα2 s1 s2 s1 τ, t(k+i−1)α1 +kα2 s2 τ, t(1−i)α1 +kα2 s1 τ | k ∈ Z},  · A ⊂ A. Since then W A = {tkα1 +iα2 s1 s2 s1 τ | k ∈ Z}  {t(1−i)α1 +kα2 s1 τ | k ∈ Z}  {t(k+i−1)α1 +kα2 s2 τ | k ∈ Z} = {tkα1 +iα2 s1 s2 s1 τ | k ∈ Z}  τ · {tkα1 +iα2 s1 s2 s1 τ | k ∈ Z}  τ −1 · {tkα1 +iα2 s1 s2 s1 τ | k ∈ Z} = {t(k+i−1)α1 +kα2 s2 τ | k ∈ Z}  τ · {t(k+i−1)α1 +kα2 s2 τ | k ∈ Z}  τ −1 · {t(k+i−1)α1 +kα2 s2 τ | k ∈ Z} = {t(1−i)α1 +kα2 s1 τ | k ∈ Z}  τ · {t(1−i)α1 +kα2 s1 τ | k ∈ Z}  τ −1 · {t(1−i)α1 +kα2 s1 τ | k ∈ Z}, s0 ·(tkα1 +iα2 s1 s2 s1 τ ) = t(1−i)α1 +(1−k)α2 s1 τ , s1 ·(tkα1 +iα2 s1 s2 s1 τ ) = t(i−k)α1 +iα2 s1 s2 s1 τ , s2 · (tkα1 +iα2 s1 s2 s1 τ ) = t(k−1)α1 +(k−i)α2 s2 τ , s0 · (t(k+i−1)α1 +kα2 s2 τ ) = (1−k)α1 +(2−i−k)α2 (k+i−1)α1 +kα2 t s2 τ , s1 ·(t s2 τ ) = t(1−i)α1 +kα2 s1 τ , s2 ·(t(k+i−1)α1 +kα2 s2 τ ) = t(k+i)α1 +iα2 s1 s2 s1 τ , s0 · (t(1−i)α1 +kα2 s1 τ ) = t(1−k)α1 +iα2 s1 s2 s1 τ , s1 · (t(1−i)α1 +kα2 s1 τ ) = t(k+i−1)α1 +kα2 s2 τ , s2 · (t(1−i)α1 +kα2 s1 τ ) = t(1−i)α1 −(k+i)α2 s1 τ , and thus (1) is proved.  , by direct calculation, we have (2) For any w ˜∈W (t( 2 +1)α1 +iα2 s1 s2 s1 τ )  (wt ˜ ( 2 +1)α1 +iα2 s1 s2 s1 τ w ˜ −1 ). i

i

 -conjugate to t( 2i +1)α1 +iα2 s1 s2 s1 τ . We use induction on length. If For any w ˜ ∈ A, w ˜ is W  -conjugate to t( 2i +1)α1 +iα2 s1 s2 s1 τ . From for all w ˜  ∈ A with (w ˜  ) < (w), ˜ then w ˜  is W the proof of (1), it is sufficient for us to consider that w ˜ ∈ {tkα1 +iα2 s1 s2 s1 τ | k ∈ Z}. kα1 +iα2 Now, we take w ˜=t s1 s2 s1 τ . (a) For i > 0 and i is odd. If k  2i, then we take  -conjugate to w ˜1 = s0 · (w). ˜ Then (w ˜1 ) = (w) ˜ − 2, and thus by induction, w ˜ is W i ( 2i +1)α1 +iα2 t s1 s2 s1 τ . If  2  + 2  k  2i − 1, then we take w ˜1 = s2 s0 · (w). ˜ Then,  -conjugate to t( 2i +1)α1 +iα2 s1 s2 s1 τ . If (w ˜1 ) = (w) ˜ − 2, and thus by induction, w ˜ is W k =  2i  + 1, then w ˜ is already of minimal length. If k   2i , then we take w ˜1 = s1 · (w). ˜  -conjugate to t( 2i +1)α1 +iα2 s1 s2 s1 τ . Then (w ˜1 ) = (w) ˜ −2, and thus by induction, w ˜ is W (b) For i > 0 and i is even. If k  2i or  2i  + 2  k  2i − 1 or k   2i  − 1, then w ˜1 is  -conjugate to t( 2i +1)α1 +iα2 s1 s2 s1 τ . If i  0, the argument is as before, and thus w ˜ is W similar and it is omitted. By (1) and (2), the lemma is proved. 2 The following theorem follows directly from Lemmas 2.6, 2.7, and 2.8.

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 -conjugacy classes in Wa τ are: Oid,τ , Oλ,τ , and Oi,τ with i ∈ Z, Theorem 2.9. The W  λ ∈ P+ Q and λ = 2mα1 + mα2 (m ∈ N). 2.1.3. Quasi-split case  . In this section, we set · as the usual We classify the δ-conjugacy classes in W −1  W -conjugation (i.e., x · y = xyx ) and ·δ is the δ-conjugation (i.e., x · y = xyδ(x)−1 ). Let O0,δ = {tk(α1 +2α2 ) s2 s1 , tk(2α1 +α2 ) s1 s2 , tk(α1 −α2 ) | k ∈ Z} {(τ · tk(α1 +2α2 ) s2 s1 )τ 2 , (τ · tk(2α1 +α2 ) s1 s2 )τ 2 , (τ · tk(α1 −α2 ) )τ 2 | k ∈ Z} {(τ 2 · tk(α1 +2α2 ) s2 s1 )τ, (τ 2 · tk(2α1 +α2 ) s1 s2 )τ, (τ 2 · tk(α1 −α2 ) )τ | k ∈ Z}; O1,δ = {tλ s1 , tλ s2 | λ ∈ Q}  {(τ · tλ s1 )τ 2 , (τ · tλ s2 )τ 2 | λ ∈ Q} {(τ 2 · tλ s1 )τ, (τ 2 · tλ s2 )τ | λ ∈ Q}; O1,δ = {tkα1 +(2i+1)α2 s1 s2 s1 , t(2k+1)α1 +2iα2 s1 s2 s1 | k, i ∈ Z} {(τ · tkα1 +(2i+1)α2 s1 s2 s1 )τ 2 , (τ · t(2k+1)α1 +2iα2 s1 s2 s1 )τ 2 | k, i ∈ Z} {(τ 2 · tkα1 +(2i+1)α2 s1 s2 s1 )τ, (τ 2 · t(2k+1)α1 +2iα2 s1 s2 s1 )τ | k, i ∈ Z}; O3,δ = {t2kα1 +2iα2 s1 s2 s1 | k, i ∈ Z}  {(τ · t2kα1 +2iα2 s1 s2 s1 )τ 2 | k, i ∈ Z} {(τ 2 · t2kα1 +2iα2 s1 s2 s1 )τ | k, i ∈ Z}. For m ∈ N+ , we set  (m) =

1, 0,

O2m,δ = {t(k− 2 )α1 +(2k+(m))α2 s2 s1 , t(k+ m

 {t

(2k+(m))α1 +(k+ m+1 2 )α2

s1 s2 , t

m m

odd even.

m+1 2 )α1 +(2k+(m))α2

(2k+(m))α1 +(k− m 2 )α2

s2 s1 | k ∈ Z}

s1 s2 | k ∈ Z}

 {tkα1 +(m−k)α2 , t(k−m)α1 −kα2 | k ∈ Z}  {(τ · t(k− 2 )α1 +(2k+(m))α2 s2 s1 )τ 2 , (τ · t(k+ m

 {(τ · t(2k+(m))α1 +(k+

m+1 2 )α2

m+1 2 )α1 +(2k+(m))α2

s2 s1 )τ 2 | k ∈ Z}

s1 s2 )τ 2 , (τ · t(2k+(m))α1 +(k− 2 )α2 s1 s2 )τ 2 | k ∈ Z} m

 {(τ · tkα1 +(m−k)α2 )τ 2 , (τ · t(k−m)α1 −kα2 )τ 2 | k ∈ Z}  {(τ 2 · t(k− 2 )α1 +(2k+(m))α2 s2 s1 )τ, (τ 2 · t(k+ m

 {(τ 2 · t

(2k+(m))α1 +(k+ m+1 2 )α2

s1 s2 )τ, (τ 2 · t

m+1 2 )α1 +(2k+(m))α2

s2 s1 )τ | k ∈ Z}

(2k+(m))α1 +(k− m 2 )α2

s1 s2 )τ | k ∈ Z}

 {(τ 2 · tkα1 +(m−k)α2 )τ, (τ 2 · t(k−m)α1 −kα2 )τ | k ∈ Z}.

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 , then w Remark 2.10. Let w ˜∈W ˜ = wa τ  , where wa ∈ Wa , and τ  ∈ Ω. Thus, w ˜∼ ˜ δ wa .  In the following, to consider w ˜ ∈ W , it is sufficient to consider wa ∈ Wa . Theorem 2.11. O0,δ , O1,δ , O1,δ , O3,δ , O2m,δ , where m ∈ N+ form all the δ-conjugacy  . Moreover, classes of W (O0,δ ) = 0, (O3,δ ) = 3, (O1,δ ) = (O1,δ ) = 1, (O2m,δ ) = 2m. = Proof. By the definitions of O0,δ , O1,δ , O1,δ , O3,δ , O2m,δ for m ∈ N+ , we have W  O0,δ  O1,δ  O1,δ  O3,δ  m∈N+ O2m,δ . Therefore, the theorem is proved together with Lemma 2.12. 2  with minimal length 0; Lemma 2.12. (1) O0,δ is the δ-conjugacy class of W   (2) O1,δ and O1,δ are the δ-conjugacy classes of W with minimal length 1;  with minimal length 3; (3) O3,δ is the δ-conjugacy class of W  with minimal length 2m. (4) If m ∈ N+ , then O2m,δ is the δ-conjugacy class of W Proof. We prove (1) and the others are similar. We can easily check that for any  , then w w ˜ ∈ W ˜ ·δ O0,δ ⊂ O0,δ . Therefore, we need to show that for any element w ˜ ∈ O0,δ , w ˜ is δ-conjugate to id. By 2.10, it is sufficient to consider that w ˜ ∈ {tk(α1 +2α2 ) s2 s1 , tk(2α1 +α2 ) s1 s2 , tk(α1 −α2 ) | k ∈ Z} and we use induction on the length. If w ˜ = id, there is nothing to prove. If w ˜  ∈ {tk(α1 +2α2 ) s2 s1 , tk(2α1 +α2 ) s1 s2 , tk(α1 −α2 ) |   ˜ ) < (w), ˜ then w ˜ is δ-conjugate to id. If w ˜ = tk(α1 +2α2 ) s2 s1 or k ∈ Z}, and (w k(2α1 +α2 ) t s1 s2 , where k ∈ N+ , and we set w ˜1 = s0 ws ˜ 0 , then (w ˜1 ) = (w) ˜ − 2, thus k(α1 −α2 ) −k(α1 +2α2 ) w ˜1 is δ-conjugate to id, and so is w. ˜ If w ˜=t or t s2 s1 , where k ∈ N+ , and we set w ˜1 = s2 ws ˜ 1 , then (w ˜1 ) = (w) ˜ − 2, and thus w ˜1 is δ-conjugate to id, and so is w. ˜ If w ˜ = tk(α2 −α1 ) or t−k(2α1 +α2 ) s1 s2 , where k ∈ N+ , and we set w ˜1 = s1 ws ˜ 2 then (w ˜1 ) = (w) ˜ − 2, and thus w ˜1 is δ-conjugate to id, and so is w. ˜ Hence, (1) is proved. 2 Theorem 2.13. Let w, ˜ w ˜  ∈ Oi,δ , where i = 0, 1 or 2m, and m ∈ N+ . If (w) ˜ = (w ˜  ), then Tw˜ ≡ Tw˜  . Proof. We use induction on the length . First, we prove that w, ˜ w ˜  ∈ O1,δ . If (w) ˜ =   ˜ = (w ˜ ) = 2k + 1, where k ∈ N+ , and we (w ˜ ) = 1, which is obvious. Now, if (w) assume that for any w ˜1 , w ˜1 ∈ O1,δ and (w ˜1 ) = (w ˜1 ) < 2k + 1 then Tw˜1 ≡ Tw˜1 . It is sufficient to show that if w ˜1 = si wδ(s ˜ i ) and (w ˜1 ) = (w) ˜ − 2, then si w ˜ is a minimal length element in O2k,δ . If k = 3j + 3, then w ˜ = t(j+2)α1 −jα2 +i(α1 +2α2 ) s1 , where 0  i  j. If k = 3j + 4, then w ˜ = t(j+2)α1 −(j+1)α2 +i(α1 +2α2 ) s1 , where 0  (j+4)α1 −(j+1)α2 +i(α1 +2α2 ) ˜ =t s1 where 0  i  j + 1. i  j + 1. If k = 3j + 5, then w

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Thus, s0 w ˜ ∼ ws ˜ 0 and ws ˜ 0 is a minimal length element of O2k,δ . Similar arguments apply to the other cases. For w ˜ and w ˜  are contained in O0,δ . For k ∈ N+ , and if (1−k)(α1 +2α2 ) (w) ˜ = 6k − 4, then w ˜=t s2 s1 or w ˜ = t(1−k)(2α1 +α2 ) s1 s2 . If (w) ˜ = 6k − 2, k(α1 +2α2 ) k(2α1 +α2 ) then w ˜ = t s2 s1 or w ˜ = t s1 s2 . If (w) ˜ = 6k, then w ˜ = tk(α1 −α2 ) or w ˜ = tk(α2 −α1 ) . By symmetry, we know that Tt(1−k)(α1 +2α2 ) s2 s1 ≡ Tt(1−k)(2α1 +α2 ) s1 s2 , Ttk(α1 +2α2 ) s2 s1 ≡ Ttk(2α1 +α2 ) s1 s2 , or Ttk(α1 −α2 ) ≡ Ttk(α2 −α1 ) . Thus, the proof is complete for O0,δ . If w ˜ ∈ O2m,δ and (w) ˜ = 2m + 2k for some k ∈ N+ . If w ˜ = tλ s2 s1 , then we s0 s2 −1 have w ˜ −→δ s0 ws ˜ 0 or w ˜ −→δ s2 ws ˜ 1 . In either case, Tw˜ ≡ (v − v )Tsi w˜ + Tsi wδ(s where ˜ i) i = 0 or 2, si w ˜ ∈ O1,δ . A similar argument applies for w ˜  , where (w) ˜ = (w ˜  ). We have some i such that Tw˜  ≡ (v − v −1 )Tsi w˜  + Tsi w˜  δ(si ) and si w ˜  ∈ O1,δ . Then by the proof for O1,δ and induction, we have Tw˜ ≡ Tw˜  . 2 2.2. Explicit class polynomials  satisfy w The lemma [16, Lemma 5.1] states that if w, ˜ w ˜ ∈ W ˜∼ ˜w ˜  , then Tw˜ ≡ Tw˜  . Hence, we calculate the class polynomials attached to some typical affine Weyl group elements and list them in this section. Calculating the class polynomials attached to general elements can be reduced to these typical elements, so the methods are quite similar, and thus we omit the case-by-case argument and we simply list these class polynomials. 2.2.1. Split case-I We set I = {tmα1 +nα2 w | m ∈ N+ , n ∈ Z, 1 − m  n  2m − 1, w ∈ W }  {Id, s2 }  {tkα1 +2kα2 , tkα1 +2kα2 s2 , tkα1 +2kα2 s2 s1 , tk(α1 −α2 ) , tk(α1 −α2 ) s1 , tk(α1 −α2 ) s2 | k ∈ N+ }. Since Wa = I ∪ (τ · I) ∪ (τ 2 · I), it is sufficient to consider the elements in I. For α ∈ Q, we  -conjugacy set Qα = {λ ∈ Q ∩ P+ | λ < α} ∩ Qsh . For λ ∈ Q ∩ P+ , let C(tλ s1 ) be the W  λ class that contains t s1 , and let Otλ s1 be the set of all Ci with (Ci )  (C(tλ s1 )).  -conjugacy class that contains tλ s2 , and let O λ be the Similarly, let C (tλ s2 ) be the W set of all Ci with (Ci )  (C (tλ s2 )). Moreover, for λ ∈ Q ∩ P+ , we set

Dλ = {λ ∈ Q ∩ P+ | λ < λ − α1 , λ = λ − iα1 (i ∈ N+ )} ∩ Qsh , Dλ = {λ ∈ Q ∩ P+ | λ < λ − α2 , λ = λ − iα2 (i ∈ N+ )} ∩ Qsh . In addition, we set Eλ = {λ ∈ Q ∩ P+ | λ = λ − iα1 (i ∈ N+ )} ∩ Qsh , Eλ = {λ ∈ Q ∩ P+ | λ = λ − iα2 (i ∈ N+ )} ∩ Qsh .

t s2

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Proposition 2.14. If w ˜ ∈ Oλ , where λ ∈ P+ ∩ Q, then  fw,O = ˜

1, 0,

O = Oλ otherwise.

Proof. Since Oλ only has finitely many elements for each λ and any two elements in Oλ are ∼, ˜ then the proposition follows directly from the definition of class polynomials. 2 ˜ = tk(α1 +2α2 ) s2 s1 , with k ∈ N+ , then Proposition 2.15. (1) If w

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Q(k−1)α1 +(2k−2)α2  O ∈ O ∪ Ot(k−1)α 1 +(2k−2)α2 s2 tkα1 +(2k−1)α2 s1 O = O2 otherwise.

(2) If w ˜ = tk(α1 +2α2 ) s1 s2 , with k ∈ N+ , then

fw,O ˜

⎧ −1 2 ⎪ ⎪ (v − v ) , ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Qkα1 +2kα2  O ∈ O ∪ Otk(α 1 +2α2 ) s2 tkα1 +(2k−1)α2 s1 O = O2 otherwise.

Proposition 2.16. (1) If w ˜ = tk(2α1 +α2 ) s1 s2 , with k ∈ N+ , then

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Q(2k−2)α1 +(k−1)α2  O ∈ O ∪ Ot(2k−1)α 1 +kα2 s2 t(k−1)(2α1 +α2 ) s1 O = O2 otherwise.

(2) If w ˜ = tk(2α1 +α2 ) s2 s1 , with k ∈ N+ , then

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Qk(2α1 +α2 )  O ∈ O ∪ Ot(2k−1)α 1 +kα2 s2 tk(2α1 +α2 ) s1 O = O2 otherwise.

Next, we give a proof of 2.15. Symmetrically, we can obtain 2.16. Proof of Proposition 2.15. For (1), we have Ttk(α1 +2α2 ) s2 s1 ≡ (v − v −1 )(Ttkα1 +(2k−1)α2 s1 + Tt(k−1)α1 +(2k−2)α2 s1 ) + (v − v −1 )Tt(k−1)α1 +(2k−2)α2 s2 + Tt(k−1)(α1 +2α2 ) s2 s1

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······ ≡ (v − v −1 )[Ttα1 +α2 s1 +

k  (Tt(i−1)(α1 +2α2 ) s1 + Ttiα1 +(2i−1)α2 s1 )] i=2

+ (v − v −1 )

k−1 

Tti(α1 +2α2 ) s2 + TO2

i=1

······



≡ TO2 + (v − v −1 )2

TOλ

λ∈Q(k−1)α1 +(2k−2)α2



+ (v − v −1 )

TO .

  ∪O (k−1)(α +2α ) 1 2 s2 tkα1 +(2k−1)α2 s1 t

O∈O

Thus, (1) is proved. By Ttk(α1 +2α2 ) s1 s2 ≡ (v − v −1 )Ttk(α1 +2α2 ) s2 + Ttk(α1 +2α2 ) s2 s1 and (1), (2) is proved. 2 Similarly, we have the following proposition. Proposition 2.17. (1) For k ∈ N+ , we have tkα1 +(2k−1)α2 s1 s2 ∼t ˜ kα1 +(2k−1)α2 s2 s1 , and thus Ttkα1 +(2k−1)α2 s1 s2 ≡ Ttkα1 +(2k−1)α2 s2 s1 . If w ˜ = tkα1 +(2k−1)α2 s1 s2 or kα1 +(2k−1)α2 s2 s1 with k = 1, then w ˜ is already of minimal length in O2 . Therefore, t fw,O = 1 and 0 for other O. When k  2, then ˜ 2

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Q(k−1)(α1 +2α2 )  O ∈ O ∪ Ot(k−1)(α 1 +2α2 ) s2 t(k−1)(α1 +2α2 ) s1 O = O2 otherwise.

(2) For k ∈ N+ , we have t(2k−1)α1 +kα2 s1 s2 ∼t ˜ (2k−1)α1 +kα2 s2 s1 , thus Tt(2k−1)α1 +kα2 s1 s2 ≡ Tt(2k−1)α1 +kα2 s2 s1 . If w ˜ = t(2k−1)α1 +kα2 s1 s2 or t(2k−1)α1 +kα2 s2 s1 with k = 1, then w ˜ is already of minimal length in O2 . So fw,O ˜ 2 = 1 and 0 for other O. When k  2, then ⎧ (v − v −1 )2 , O = Oλ , λ ∈ Q(2k−2)α1 +(k−1)α2 ⎪ ⎪ ⎪  ⎨ (v − v −1 ), O ∈ O ∪ Ot(k−1)(2α 1 +α2 ) s2 t(k−1)(2α1 +α2 ) s1 fw,O = ˜ ⎪ 1, O = O 2 ⎪ ⎪ ⎩ 0, otherwise.

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Proposition 2.18. (1) If w ˜ = tλ s1 s2 with λ = mα1 +nα2 (m, n ∈ Z, 2  m  n < 2m−1), then ⎧ ⎪ (v − v −1 )2 , O = Oλ , λ ∈ Dλ ⎪ ⎪  ⎨ (v − v −1 ), O ∈ O ∪ Otλ−α 1 s2 tλ−α1 −α2 s1 fw,O = ˜ ⎪ 1, O = O 2 ⎪ ⎪ ⎩ 0, otherwise; (2) If w ˜ = tλ s2 s1 with λ = mα1 + nα2 (m, n ∈ Z, 2  m < n < 2m − 1), then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ with λ ∈ Dλ  O ∈ O ∪ Otλ−α 1 −α2 s2 tλ−α2 s1 O = O2 otherwise.

Proposition 2.19. (1) If w ˜ = tλ s2 s1 with λ = mα1 +nα2 (m, n ∈ Z, 2  n  m < 2n −1), then ⎧ ⎪ (v − v −1 )2 , O = Oλ , λ ∈ Dλ ⎪ ⎪  ⎨ (v − v −1 ), O ∈ O ∪ Otλ−α 1 −α2 s2 tλ−α2 s1 fw,O = ˜ ⎪ 1, O = O2 ⎪ ⎪ ⎩ 0, otherwise; (2) If w ˜ = tλ s1 s2 with λ = mα1 + nα2 (m, n ∈ Z, 2  n < m < 2n − 1), then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Dλ  O ∈ O ∪ Otλ−α 1 s2 tλ−α1 −α2 s1 O = O2 otherwise.

To avoid repetition, we only prove 2.18. Proof of Proposition 2.18. For (1), let λ = mα1 + nα2 Ttλ s1 s2 ≡ (v − v −1 )Ts0 tλ s1 s2 + Ts0 tλ s1 s2 s0 = (v − v −1 )Tt(1−n)α1 +(1−m)α2 s1 + Tt(1−n)α1 +(2−m)α2 s2 s1 ≡ (v − v −1 )Tτ −1 ·t(1−n)α1 +(1−m)α2 s1 + Tτ −1 ·t(1−n)α1 +(2−m)α2 s2 s1 = (v − v −1 )Tt(n−m+1)α1 +nα2 s1 s2 s1 + Tt(n−m+1)α1 +nα2 s2 s1 ≡ (v − v −1 )Ts1 t(n−m+1)α1 +nα2 s1 s2 s1 s1 + (v − v −1 )Ts0 t(n−m+1)α1 +nα2 s2 s1 + Ts0 t(n−m+1)α1 +nα2 s2 s1 s0 ≡ (v − v −1 )Tt(m−1)α1 +nα2 s2 + (v − v −1 )Tτ −1 ·t(1−n)α1 +(m−n)α2 s2

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+ Tτ −1 ·t(2−n)α1 +(m−n)α2 s1 s2 = (v − v −1 )(Tt(m−1)α1 +nα2 s2 + Tt(m−1)α1 +(n−1)α2 s1 ) + Tt(m−1)α1 +(n−1)α2 s1 s2 ······

(Inductively)

≡ (v − v −1 )[(Tt(m−1)α1 +nα2 s2 + Tt(m−1)α1 +(n−1)α2 s1 ) + (Tt(m−2)α1 +(n−1)α2 s2 + Tt(m−2)α1 +(n−2)α2 s1 ) + · · · · · · + (Tt(n−m+1)α1 +2(n−m+1)α2 s2 + Tt(n−m+1)α1 +(2(n−m+1)−1)α2 s1 )] + Tt(n−m+1)α1 +(2(n−m+1)−1)α2 s1 s2 ······ ≡ TO2

(2.17)  + (v − v −1 )2 TOλ + (v − v −1 ) λ ∈Dλ



TO .

  O∈O λ−α −α ∪O λ−α 1 2 s1 1 s2 t t

For (2), we combine (1) to obtain Ttλ s2 s1 ≡ (v − v −1 )Ttλ−α2 s1 + Ttλ−α2 s1 s2 ······



≡ TO2 + (v − v −1 )2



TOλ + (v − v −1 )

 λ ∈Dλ

TO .

  O∈O λ−α ∪O λ−α −α 2 s1 1 2 s2 t t

Corollary 2.20. (1) If w ˜ = t(2k+1)α1 +kα2 s1 s2 , k ∈ N, then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Q2kα1 +kα2  O ∈ O ∪ Ot2kα 1 +kα2 s2 t2kα1 +kα2 s1 O = O2 otherwise.

(2) If w ˜ = t(2k+1)α1 +kα2 s2 s1 , k ∈ N, then

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Q2kα1 +kα2 ∪ E(2k+1)α1 +(k+1)α2  O ∈ O ∪ Ot2kα 1 +kα2 s2 t(2k+1)α1 +(k+1)α2 s1 O = O2 otherwise.

(3) If w ˜ = tmα1 +nα2 s1 s2 , m, n ∈ N and m − 2n > 1, then

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

 O = Oλ , λ ∈ Dmα 1 +(m−n)α2  O ∈ O ∪ Ot(m−1)α mα +(m−n−1)α 1 2 s1 1 +(m−n−1)α2 s2 t O = O2 otherwise.

2

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(4) If w ˜ = tmα1 +nα2 s2 s1 , m, n ∈ N and m − 2n > 1, then

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Dmα1 +(m−n)α2 ∪ Emα1 +(m−n)α2  O ∈ O ∪ Ot(m−1)α 1 +(m−n)α2 s2 tmα1 +(m−n)α2 s1 O = O2 otherwise.

(5) If w ˜ = tmα1 −nα2 s1 s2 , m, n ∈ N+ and m − n > 1, then

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

 O = Oλ , λ ∈ Dmα 1 +(m+n)α2

 O ∈ O ∪ Ot(m−1)α 1 +(m+n−1)α2 s2 tmα1 +(m+n−1)α2 s1 O = O2 otherwise.

(6) If w ˜ = tmα1 −nα2 s2 s1 , m, n ∈ N and m − n > 1, then

fw,O ˜

⎧ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ , λ ∈ Dmα1 +(m+n)α2 ∪ Emα1 +(m+n)α2  O ∈ O ∪ Ot(m−1)α 1 +(m+n)α2 s2 tmα1 +(m+n)α2 s1 O = O2 otherwise.

Proof. For any w, ˜ which lies in the Corollary, if we take w ˜  = s2 ws ˜ 2 , then either Tw˜ ≡ −1  (v − v )Ts2 w˜ + Tw˜  or Tw˜ ≡ Tw˜  . At this time, w ˜ appears in the previous propositions. Thus, by combining the previous propositions, we obtain the Corollary. 2 Proposition 2.21. (1) If w ˜ ∈ O1 and (w) ˜ = 4k − 1(k ∈ N+ ), then ⎧ −1 3 −1 ⎪ ⎪ (k − i)(v − v ) + (v − v ), ⎪ ⎪ ⎪ (k − i)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ fw,O = ˜

(k − i)(v − v −1 )2 , ⎪ ⎪ ⎪ k(v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oi(α1 +α2 ) (1  i  k − 1)  O = Oλ , λ ∈ Ei(α1 +α2 ) ∪ Ei(α 1 +α2 ) (3  i  k − 1) O = Ci or Ci (1  i  k − 1) O = O2 O = O1 otherwise.

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(2) If w ˜ ∈ O1 and (w) ˜ = 4k − 3(k ∈ N+ ), then ⎧ ⎪ (v − v −1 ), ⎪ ⎪ ⎪ ⎪ (k − 1 − i)(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ (k − 1 − i)(v − v −1 )3 , ⎪ ⎪ ⎨ fw,O = ˜

⎪ (k − 1 − i)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ (k − 1)(v − v −1 ), ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎪ ⎩ 0,

O = O(k−1)(α1 +α2 ) O = Oi(α1 +α2 ) (1  i  k − 2)  O = Oλ , λ ∈ Ei(α1 +α2 ) ∪ Ei(α 1 +α2 ) (3  i  k − 2) O = Ci or Ci (1  i  k − 2) O = O2 O = O1 otherwise.

Proof. We prove (1) and the proof of (2) is similar. Since any element w ˜ ∈ O1 with (w) ˜ = 4k − 1 is ∼ ˜ to an element in {w ˜ = tkα1 s1 | k ∈ N+ }, then it is sufficient for us to consider w ˜ = tkα1 s1 (k ∈ N+ ). Tw˜ ≡ (v − v −1 )Ts2 w˜ + Ts2 ws ˜ 2 = (v − v −1 )Ttk(α1 +α2 ) s2 s1 + Ttk(α1 +α2 ) s1 s2 s1 ≡ (v − v −1 )Ttk(α1 +α2 ) s2 s1 + (v − v −1 )Ts0 tk(α1 +α2 ) s1 s2 s1 + Ts0 tk(α1 +α2 ) s1 s2 s1 s0 = (v − v −1 )Ttk(α1 +α2 ) s2 s1 + (v − v −1 )Tt(1−k)(α1 +α2 ) + Tt(2−k)(α1 +α2 ) s1 s2 s1 ≡ (v − v −1 )Ttk(α1 +α2 ) s2 s1 + (v − v −1 )Tτ ·t(1−k)(α1 +α2 ) + Tτ ·t(2−k)(α1 +α2 ) s1 s2 s1 = (v − v −1 )Ttk(α1 +α2 ) s2 s1 + (v − v −1 )Tt(k−1)α1 + Tt(k−1)α1 s1 ≡ (v − v −1 )Ttk(α1 +α2 ) s2 s1 + (v − v −1 )Ts2 t(k−1)α1 s2 + Tt(k−1)α1 s1 = (v − v −1 )Ttk(α1 +α2 ) s2 s1 + (v − v −1 )TO(k−1)(α1 +α2 ) + Tt(k−1)α1 s1 ······ ≡ (v − v −1 )

k 

Tti(α1 +α2 ) s2 s1 + (v − v −1 )

i=1

k−1 

TOi(α1 +α2 ) + TO1 .

i=1

Together with Proposition 2.19 (1), we obtain Ttkα1 s1 ≡

k−1 

[(k − i)(v − v −1 )3 + (v − v −1 )]TOi(α1 +α2 )

i=1

+

k−1 



 i=3 λ∈Ei(α1 +α2 ) ∪Ei(α

+

k−1  i=1

(1) is proved. 2

(k − i)(v − v −1 )3 TOλ 1 +α2 )

(k − i)(v − v −1 )2 (TCi + TCi ) + k(v − v −1 )TO2 + TO1 .

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Proposition 2.22. (1) If w ˜ ∈ Ci (i ∈ N+ ) and (Ci )  (w) ˜  6i + 1, then

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

O = Ojα1 +iα2 ( 2i  + 1  j  O = Ci otherwise.

(w)−1 ˜ 2

− i)

(2) If w ˜ ∈ Ci (i ∈ N+ ) and (w) ˜ = 6i − 1 + 4k (k  1), then

fw,O ˜

⎧ (v − v −1 ), ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎪ k(v − v ), ⎪ −1 2 ⎪ ⎪ ⎪ k(v − v ) , ⎪ ⎪ ⎪ k(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎨ (k − j)(v − v −1 )2 , = ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )2 + 1, ⎪ ⎪ ⎩ 0,

O = O2iα1 +iα2 O = O2  O ∈ O ∪ Ot2iα \Ci 1 +iα2 s2 t2iα1 +iα2 s1 O = Oλ , λ ∈ Q2iα1 +iα2 \E2iα1 +iα2 O = Oλ , λ ∈ E2iα1 +iα2 O ∈ C(t(2i+j+1)α1 +(i+j)α2 s1 ) ∪ C (t(2i+j)α1 +(i+j)α2 s2 ) (1  j  k − 1) O = Oλ λ ∈ E(2i+j)α1 +(i+j)α2  ∪ E(2i+j)α (1  j  k − 1) 1 +(i+j)α2 O = O(2i+j)α1 +(i+j)α2 (1  j  k − 1) O = Ci otherwise.

(3) If w ˜ ∈ Ci (i ∈ N+ ) and (w) ˜ = 6i + 1 + 4k (k  1), then ⎧ −1 ⎪ ⎪ (v − v ), ⎪ ⎪ ⎪ k(v − v −1 ), ⎪ ⎪ ⎪ ⎪ k(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ k(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ −1 2 ⎪ ⎨ (k − j)(v − v ) , fw,O = ˜

⎪ ⎪ ⎪ (k − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )2 + 1, ⎪ ⎪ ⎪ ⎪ (v − v −1 ), ⎪ ⎪ ⎩ 0,

O = O2iα1 +iα2 O = O2  O ∈ O ∪ Ot2iα \Ci 1 +iα2 s2 t2iα1 +iα2 s1 O = Oλ , λ ∈ Q2iα1 +iα2 \E2iα1 +iα2 O = Oλ , λ ∈ E2iα1 +iα2 O ∈ C(t(2i+j+1)α1 +(i+j)α2 s1 )∪ C (t(2i+j)α1 +(i+j)α2 s2 ) (1  j  k − 1) O = Oλ λ ∈ E(2i+j)α1 +(i+j)α2 ∪  E(2i+j)α (1  j  k − 1) 1 +(i+j)α2 O = O(2i+j)α1 +(i+j)α2 (1  j  k − 1) O = Ci O = O(2i+k)α1 +(i+k)α2 otherwise.

Proposition 2.23. (1) If w ˜ ∈ Ci (i ∈ N+ ) and (Ci )  (w) ˜  6i + 1, then

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

O = Oiα1 +( 2i +j)α2 (1  j  O = Ci otherwise.

 (w)− (C ˜ i) ) 2

524

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(2) If w ˜ ∈ Ci (i ∈ N+ ) and (w) ˜ = 6i − 1 + 4k (k  1), then ⎧ (v − v −1 ), ⎪ ⎪ ⎪ ⎪ k(v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ k(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )2 , ⎪ ⎪ ⎨ fw,O = ˜

⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )2 + 1, ⎪ ⎪ ⎩ 0,

O = Oiα1 +2iα2 O = O2  O ∈ O ∪ Oti(α \Ci 1 +2α2 ) s2 ti(α1 +2α2 ) s1  O = Oλ , λ ∈ Qiα1 +2iα2 \Eiα 1 +2iα2  O = Oλ , λ ∈ Eiα1 +2iα2 O ∈ C(t(i+j)α1 +(2i+j)α2 s1 ) ∪ C (t(i+j)α1 +(2i+j+1)α2 s2 ) (1  j  k − 1) O = Oλ ,  λ ∈ E(i+j)α1 +(2i+j)α2 ∪ E(i+j)α 1 +(2i+j)α2 (1  j  k − 1) O = O(i+j)α1 +(2i+j)α2 (1  j  k − 1) O = Ci otherwise.

(3) If w ˜ ∈ Ci (i ∈ N+ ) and (w) ˜ = 6i + 1 + 4k (k  1), then ⎧ ⎪ O = Oiα1 +2iα2 (v − v −1 ), ⎪ ⎪ ⎪ −1 ⎪ k(v − v ), O = O2 ⎪ ⎪ ⎪  −1 2 ⎪ ⎪ O ∈ O ∪ Oti(α \Ci ⎪ k(v − v ) , 1 +2α2 ) s2 ti(α1 +2α2 ) s1 ⎪ ⎪  ⎪ O = Oλ , λ ∈ Qiα1 +2iα2 \Eiα ⎪ k(v − v −1 )3 , ⎪ 1 +2iα2 ⎪ ⎪ −1 3 −1  ⎪ k(v − v ) + (v − v ), O = O , λ ∈ E λ ⎪ iα1 +2iα2 ⎪ ⎪ ⎪ (k − j)(v − v −1 )2 , O ∈ C(t(i+j)α1 +(2i+j)α2 s1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ∪ C (t(i+j)α1 +(2i+j+1)α2 s2 ) ⎨ fw,O = (1  j  k − 1) ˜ ⎪ ⎪ −1 3 ⎪ ) , O = Oλ , (k − j)(v − v ⎪ ⎪ ⎪  ⎪ λ ∈ E(i+j)α1 +(2i+j)α2 ∪ E(i+j)α ⎪ ⎪ 1 +(2i+j)α2 ⎪ ⎪ ⎪ (1  j  k − 1) ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 + (v − v −1 ), O = O(i+j)α1 +(2i+j)α2 (1  j  k − 1) ⎪ ⎪ ⎪ ⎪ O = Ci k(v − v −1 )2 + 1, ⎪ ⎪ ⎪ −1 ⎪ ⎪ (v − v ), O = O(i+k)α1 +(2i+k)α2 ⎪ ⎪ ⎩ 0, otherwise. The proofs of 2.22 and 2.23 are similar. Thus, we choose to prove 2.22 and omit the other. Proof of Proposition 2.22. Since (3) is a simple consequence of (2), we prove (1) and (2). Any element w ˜ ∈ Ci with i ∈ N+ is ∼ ˜ to an element in {tkα1 +(k−i)α2 s1 s2 s1 , tkα1 +iα2 s1 | i i k   2  + 1} ⊂ Ci . For  2  + 1  k  2i, tkα1 +iα2 s1 = s2 tkα1 +(k−i)α2 s1 s2 s1 s2 and (tkα1 +iα2 s1 ) = (tkα1 +(k−i)α2 s1 s2 s1 ). Thus, it is sufficient for us to consider w ˜ ∈ {tkα1 +iα2 s1 | k   2i  + 1}  {tkα1 +(k−i)α2 s1 s2 s1 | k  2i + 1}.

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(1) For w ˜ ∈ Ci and (Ci )  (w) ˜  6i − 1, we only consider w ˜ = tkα1 +iα2 s1 with  2i  + 1  k  2i. Tw˜ ≡ (v − v −1 )TO(k−1)α1 +iα2 + Tt(k−1)α1 +iα2 s1 ··· ≡ (v − v −1 )

k−1 

TOjα1 +iα2 + TCi .

j= 2i +1

If (w) ˜ = 6i + 1, we can take w ˜ = t(2i+1)α1 +(i+1)α2 s1 s2 s1 , then 

Tt(2i+1)α1 +(i+1)α2 s1 s2 s1 ≡ (v − v −1 )

TOλ + TCi .

λ∈E2iα1 +iα2 ∪{2iα1 +iα2 }

(2) We consider the case where w ˜ = t(2i+k)α1 +iα2 s1 , with k  1 Tw˜ ≡ (v − v −1 )Tt(2i+k)α1 +(k+i)α2 s2 s1 + Tt(2i+k)α1 +(k+i)α2 s1 s2 s1 ≡ (v − v −1 )Tt(2i+k)α1 +(k+i)α2 s2 s1 + (v − v −1 )TO(2i+k−1)α1 +(k+i−1)α2 + Tt(2i+k−1)α1 +iα2 s1 ······ ≡ (v − v −1 )

k 

Tt(2i+j)α1 +(j+i)α2 s2 s1 + (v − v −1 )

j=1

+ (v − v −1 )

k−1 

TO(2i+j)α1 +(j+i)α2

j=0 2i−1 

TOjα1 +iα2 + TCi .

j= 2i +1

Together with Proposition 2.19(1), then (2) is proved. 2 2.2.2. Split case-II We set Iτ = {tmα1 +nα2 wτ | m ∈ N+ , n ∈ Z, 1 − m  n  2m − 1, w ∈ W }  {τ, s2 τ } {tkα1 +2kα2 τ, tkα1 +2kα2 s2 τ, tkα1 +2kα2 s2 s1 τ, tk(α1 −α2 ) τ, tk(α1 −α2 ) s1 τ, tk(α1 −α2 ) s2 τ | k ∈ N+ }. Since Wa τ = Iτ ∪ τ · (Iτ ) ∪ τ 2 · (Iτ ) and any element w ˜ ∈ Wa τ is ∼ ˜ to an element in Iτ , then it is sufficient to consider the elements in Iτ .

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Proposition 2.24. For any w ˜ ∈ Oλ,τ , where λ ∈ P+ then  fw,O = ˜

1, 0,



Q and λ = 2mα1 + mα2 (m ∈ N),

O = Oλ,τ . otherwise.

Proof. According to the definition of class polynomials, this proposition is obvious. 2 For λ = mα1 + nα2 ∈ P+ ∩ Qsh , we set n Emα1 +nα2 ,τ = {λ = kα1 + nα2 |   + 1  k  m − 1}, 2 m+1  = {λ = mα1 + kα2 |   + 1  k  n}. Emα 1 +nα2 ,τ 2 Proposition 2.25. (1) Let λ = mα1 + nα2 ∈ P+ ∩ Qsh , and w ˜ = tλ s1 s2 s1 τ . Then,

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

O = Oλ ,τ , with λ ∈ Emα1 +nα2 ,τ O = On,τ otherwise

(2) Let λ = mα1 + nα2 ∈ P+ ∩ Qsh with n = 2m, and w ˜ = tλ s1 τ . Then,

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

 O = Oλ ,τ , with λ ∈ Emα 1 +nα2 ,τ O = O1−m,τ otherwise

Note: w ˜ = t2kα1 +kα2 s1 τ for k ∈ N+ is a minimal length element of O1−2k,τ . Proof. (1) If λ = kα1 + (2k − 1)α2 or λ = (k + 1)α1 + 2kα2 , where k ∈ N+ , then  -conjugacy class. Thus it is sufficient w ˜ = tλ s1 s2 s1 τ is already of minimal length in its W to consider those λ = mα1 + nα2 ∈ P+ ∩ Qsh with n  2m − 3, and in this case, we have Ttmα1 +nα2 s1 s2 s1 τ ≡ Ts0 tmα1 +nα2 s1 s2 s1 τ s0 = Tt(1−n)α1 +(1−m)α2 s1 τ ≡ (v − v −1 )Ts2 t(1−n)α1 +(1−m)α2 s1 τ + Ts2 t(1−n)α1 +(1−m)α2 s1 τ s2 ≡ (v − v −1 )Tτ −1 ·t(1−n)α1 +(m−n)α2 s2 s1 τ + Tτ −1 ·t(1−n)α1 +(m−n−1)α2 s1 τ ≡ (v − v −1 )Tt(m−1)α1 +nα2 s2 s1 τ + Tt(m−1)α1 +nα2 s1 s2 s1 τ ······ ≡ (v − v −1 )

m−1  i= n 2 +1

TOiα1 +nα2 ,τ + TOn,τ .

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 -conjugacy (2) If λ = (2k−1)α1 +kα2 , then w ˜ = tλ s1 τ is a minimal length element in its W class. Thus, it is sufficient to consider λ = mα1 + nα2 ∈ P+ ∩ Qsh with m  2n − 2, so by a similar argument we have Ttmα1 +nα2 s1 τ ≡ Ts0 tmα1 +nα2 s1 τ s0 = Tt(1−n)α1 +(1−m)α2 s1 s2 s1 τ ······ ≡ (v − v −1 )

n 

TOmα1 +iα2 ,τ + TO1−m,τ .

i= m+1 2 +1

Thus, (2) is proved. 2 Proposition 2.26. (1) If w ˜ = tkα1 +2kα2 τ with k ∈ N, then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

 O = Oλ,τ , where λ ∈ kj=2 Ejα 1 +(2j−1)α2 ,τ O = Oi,τ , where −k + 1  i  2k O = Oid,τ otherwise

(2) If w ˜ = tkα1 +(2k−1)α2 s1 s2 τ with k ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

 O = Oλ,τ , where λ ∈ k−1 j=2 Ejα1 +(2j−1)α2 ,τ O = Oi,τ , where −k + 2  i  2k − 1 O = Oid,τ otherwise

(3) If w ˜ = tkα1 +(2k−1)α2 τ with k ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

 O = Oλ,τ , where λ ∈ kj=2 Ejα 1 +(2j−1)α2 ,τ O = Oi,τ , where − k + 1  i  2k − 1 O = Oid,τ otherwise

Remark 2.27. For k ∈ N+ , since t(k+1)α1 +2kα2 s1 s2 τ = s2 (τ −1 · tkα1 +2kα2 τ )s2 , then we have tkα1 +2kα2 τ ∼t ˜ (k+1)α1 +2kα2 s1 s2 τ and thus Ttkα1 +2kα2 τ ≡ Tt(k+1)α1 +2kα2 s1 s2 τ . Proposition 2.28. (1) If w ˜ = t2kα1 +kα2 s1 s2 τ , where k ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ kj=2 E(2j−1)α1 +jα2 ,τ O = Oi,τ , where 2 − 2k  i  k O = Oid,τ otherwise

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(2) If w ˜ = t2kα1 +kα2 τ , where k ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ kj=2 E(2j−1)α1 +jα2 ,τ O = Oi,τ , where 1 − 2k  i  k O = Oid,τ otherwise

(3) If w ˜ = t(2k−1)α1 +kα2 s1 s2 τ , where k ∈ N2 , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ kj=2 E(2j−1)α1 +jα2 ,τ O = Oi,τ , where 3 − 2k  i  k O = Oid,τ otherwise

The proof of Proposition 2.26 is symmetric to that of Proposition 2.28, and thus we only prove one. Proof of Proposition 2.28. We prove (1), but the others are similar. If k = 1, then Tt2α1 +α2 s1 s2 τ ≡ (v − v −1 )(TO0,τ + TO1,τ ) + TOid,τ . If for k  2, then Tt2kα1 +kα2 s1 s2 τ ≡ Ts0 t2kα1 +kα2 s1 s2 τ s0 = Tt(1−k)α1 +(1−2k)α2 τ ≡ (v − v −1 )Tt(1−k)α1 +kα2 s2 τ + Tt(2−k)α1 +kα2 s1 s2 τ ≡ (v − v −1 )Tτ −1 ·t(1−k)α1 +kα2 s2 τ + Tτ −1 ·t(2−k)α1 +kα2 s1 s2 τ ≡ (v − v −1 )Tt(2k−1)α1 +(k−1)α2 s1 τ + (v − v −1 )Ts2 t(2k−1)α1 +(k−1)α2 s1 s2 τ + Ts2 t(2k−1)α1 +(k−1)α2 s1 s2 τ s2 ······ ≡ (v − v −1 )(Tt(2k−1)α1 +(k−1)α2 s1 τ + Tt(2k−1)α1 +kα2 s1 s2 s1 τ + Tt(2k−2)α1 +(k−1)α2 s1 τ ) + Tt(2k−2)α1 +(k−1)α2 s1 s2 τ ······ 0 

≡ (v − v −1 )

TOi,τ + (v − v −1 )

Tt(2i−1)α1 +iα2 s1 s2 s1 τ + TOid,τ

i=1

i=−2(k−1)

≡ (v − v −1 )2

k 

 λ∈ k j=2 E(2j−1)α1 +jα2 ,τ

TOλ,τ + (v − v −1 )

k 

TOi,τ + TOid,τ .

2

i=−2(k−1)

Remark 2.29. For k ∈ N2 , since t(2k−1)α1 +kα2 τ = s1 (τ · (t2kα1 +kα2 s1 s2 τ ))s1 and (t2kα1 +kα2 s1 s2 τ ) = (t(2k−1)α1 +kα2 τ ), and thus Tt2kα1 +kα2 s1 s2 τ ≡ Tt(2k−1)α1 +kα2 τ .

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Proposition 2.30. (1) Let w ˜ = tmα1 +nα2 τ , where m, n ∈ N+ , mα1 + nα2 ∈ P+ ∩ Qsh , m = 2n − 1 and n = 2m − 1. Then,

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = (v − v −1 ), ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎩ 0,

 n+1 

 O = Oλ,τ , where λ ∈ j=22 Ejα  1 +(2j−1)α2 ,τ m  j= n+1 +1 Ejα1 +nα2 ,τ 2

O = Oi,τ , where 1 − m  i  n O = Oid,τ otherwise

(2) Let w ˜ = tmα1 +nα2 s1 s2 τ , where m, n ∈ N+ , λ ∈ P+ ∩ Qsh , m = 2n − 1 and n = 2m − 1, n = 2m − 2. Then,

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = (v − v −1 ), ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎩ 0,

 m+1 

O = Oλ,τ , where λ ∈ j=22 nj= m+1 +1 Emα1 +jα2 ,τ

E(2j−1)α1 +jα2 ,τ 

2

O = Oi,τ , where 2 − m  i  n O = Oid,τ otherwise

Proof. We prove (1) next. If n is odd, then Ttmα1 +nα2 τ ≡ (v − v −1 )Tt(1−n)α1 +(1−m)α2 s1 s2 s1 τ + Tt(1−n)α1 +(1−m)α2 s1 s2 τ ≡ (v − v −1 )Tτ −1 ·(s2 t(1−n)α1 +(1−m)α2 s1 s2 s1 τ s2 ) + Tτ −1 ·(s2 t(1−n)α1 +(1−m)α2 s1 s2 τ s2 ) = (v − v −1 )Ttmα1 +nα2 s1 τ + Tt(m−1)α1 +nα2 τ ······ m 

≡ (v − v −1 )

Ttjα1 +nα2 s1 τ + T

j= n+1 2 +1 − n+1 2



≡ (v − v −1 )



TOj,τ + (v − v −1 )2

n 



TOj,τ + (v − v −1 )2

j=1− n+1 2

= (v − v

n+1 2 λ∈ j=2



−1 2

)

n+1 2 λ∈ j=2

+ (v − v −1 )

TOλ,τ

E j= n+1 +1 jα1 +nα2 ,τ 2

λ∈ m

j=1−m

+ (v − v −1 )

n+1 α +nα 1 2τ 2

t

 Ejα

n  i=1−m

1 +(2j−1)α2 ,τ

1 +(2j−1)α2 ,τ

TOλ,τ

m

TOi,τ + TOid,τ .

 Ejα

TOλ,τ + TOid,τ

j= n+1 +1 2

 Ejα 1 +nα2 ,τ

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If n is even, then by a similar argument, m 

Ttmα1 +nα2 τ ≡ (v − v −1 )

Ttiα1 +nα2 s1 τ + Tt n2 α1 +nα2 τ

i= n 2 +1



≡ (v − v −1 )2

TOλ,τ

n 2 E m  λ∈ j=2 jα1 +(2j−1)α2 ,τ

j= n +1 Ejα1 +nα2 ,τ 2

+ (v − v −1 )

n 

TOi,τ + TOid,τ .

i=1−m

The proposition is proved by combining these together. 2 Remark 2.31. If w ˜ = tmα1 +nα2 τ , where n  1, m  2n + 1 or n  0, m + n  1, and we set w ˜1 = s2 ws ˜ 2 = t(m+1)α1 +(m−n)α2 s1 s2 τ , then (w ˜1 ) = (w), ˜ w ˜∼ ˜w ˜1 and hence Tw˜ ≡ Tw˜1 . We know that w ˜1 is contained in the situation considered above. Corollary 2.32. (1) If w ˜ = t(2n+1)α1 +nα2 τ , where n ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ n+1 j=2 E(2j−1)α1 +jα2 ,τ O = Oi,τ , where − 2n  i  n + 1 O = Oid,τ otherwise.

(2) If w ˜ = t(2n+2)α1 +nα2 τ , where n ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ n+2 j=2 E(2j−1)α1 +jα2 ,τ O = Oi,τ , where − 1 − 2n  i  n + 2 O = Oid,τ otherwise.

(3) If w ˜ = tmα1 +nα2 τ , where n  1, m  2n + 3 or n  0, m + n  1, then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = (v − v −1 ), ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎩ 0,

 m+2 

O = Oλ,τ , where λ ∈ j=22 m−n E j= m+2 +1 (m+1)α1 +jα2 ,τ

E(2j−1)α1 +jα2 ,τ 

2

O = Oi,τ , where 1 − m  i  m − n O = Oid,τ otherwise.

Remark 2.33. If w ˜ = tmα1 +nα2 s1 s2 τ , where n  1, m  2n + 1 or n  0, m + n  2, then Ttmα1 +nα2 s1 s2 τ ≡ (v − v −1 )Ttmα1 +(m−n)α2 s1 s2 s1 τ + Tt(m−1)α1 +(m−n−1)α2 τ .

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Corollary 2.34. (1) If w ˜ = t(2k+1)α1 +kα2 s1 s2 τ , where k ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ k+1 j=2 E(2j−1)α1 +jα2 ,τ O = Oi,τ , where 1 − 2k  i  k + 1 O = Oid,τ otherwise.

(2) If w ˜ = t(2k+2)α1 +kα2 s1 s2 τ , where k ∈ N+ , then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ k+1 j=2 E(2j−1)α1 +jα2 ,τ  E(2k+2)α1 +(k+2)α2 ,τ O = Oi,τ , where − 2k  i  k + 2 O = Oid,τ otherwise.

(3) If w ˜ = t(k+2)α1 −kα2 s1 s2 τ , where k ∈ N, then

fw,O ˜

⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎨ (v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

 O = Oλ,τ , where λ ∈ k+1 j=2 Ejα1 +(2j−1)α2 ,τ O = Oi,τ , where − k  i  2k + 2 O = Oid,τ otherwise.

(4) If w ˜ = tmα1 +nα2 s1 s2 τ , where n ∈ N+ , m  2n + 3 or n ∈ Z0 , m + n  3 then ⎧ ⎪ (v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ fw,O = ˜

(v − v ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎩ 0,

−1

),

 m−n 

 O = Oλ,τ , where λ ∈ j=22 Ejα  1 +(2j−1)α2 ,τ m−1  j= m−n +1 Ejα1 +(m−n−1)α2 ,τ  Emα1 +(m−n)α2 ,τ 2

O = Oi,τ , where 2 − m  i  m − n O = Oid,τ otherwise.

Proofs of Corollaries 2.32 and 2.34. We choose to prove 2.32(1) and 2.34(4). For 2.32(1), the proof follows directly from Tt(2n+1)α1 +nα2 τ ≡ Tt(2n+2)α1 +(n+1)α2 s1 s2 τ ≡ (v − v −1 )2



TOλ,τ + (v − v −1 )

λ∈ n+1 j=2 E(2j−1)α1 +jα2 ,τ

In addition, for 2.34(4), we obtain the class polynomials from

n+1  i=−2n

TOi,τ + TOid,τ .

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Ttmα1 +nα2 s1 s2 τ ≡ (v − v −1 )Ttmα1 +(m−n)α2 s1 s2 s1 τ + Tt(m−1)α1 +(m−n−1)α2 τ ≡ (v − v −1 )2 

×

TOλ,τ

 m−n   λ∈ j=22 Ejα

m−1m−n E

Emα1 +(m−n)α2 ,τ 1 +(2j−1)α2 ,τ j= +1 jα1 +(m−n−1)α2 ,τ 2

+ (v − v −1 )

m−n 

TOi,τ + TOid,τ .

2

i=2−m

Proposition 2.35. For i ∈ N+ and w ˜ ∈ Oi,τ . (1) If (Oi,τ )  (w) ˜  6i − 5, then

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ E

( 2i +1+

O = Oi,τ otherwise

(w)−(O ˜ i,τ ) )α1 +iα2 ,τ 2

(2) If (w) ˜ = 6i − 3 + 4k, where k ∈ N, then

fw,O ˜

⎧ k(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k − 1 − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (k − j)(v − v −1 )3 + (v − v −1 ), = ⎪ k(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ k(v − v −1 )2 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ k(v − v ), ⎪ ⎪ −1 ⎪ (v − v ), ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ i−1 j=2 E(2j−1)α1 +jα2 ,τ O = Oλ,τ , where λ ∈ E(2i−1)α1 +iα2 ,τ O = Oλ,τ , where 1  j  k − 1, λ ∈ E(2i−1+j)α1 +(i+j)α2 ,τ O = Oλ,τ , where 2  j  k − 2,  λ ∈ E(2i+j)α 1 +(i+j)α2 ,τ O = O(2i−1+j)α1 +(i+j)α2 ,τ , where 1  j  k O = Ol,τ , where 2 − 2i  l  i − 1 O = Oi,τ O = O2−2i−j,τ or Oi+j,τ , where 1  j  k − 1 O = Oid,τ O = O(2i−1)α1 +iα2 ,τ otherwise

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(3) If (w) ˜ = 6i − 1 + 4k, where k ∈ N, then

fw,O ˜

⎧ ⎪ (k + 1)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ (k + 1 − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 3 ⎪ ⎪ ⎪ (k − j)(v − v ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (k + 1 − j)(v − v −1 )3 + (v − v −1 ), = ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 )2 + 1, ⎪ ⎪ ⎪ ⎪ (k + 1 − j)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ (v − v −1 ), ⎪ ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ i−1 j=2 E(2j−1)α1 +jα2 ,τ O = Oλ,τ , where λ ∈ E(2i−1)α1 +iα2 ,τ O = Oλ,τ , where 1  j  k, λ ∈ E(2i−1+j)α1 +(i+j)α2 ,τ O = Oλ,τ , where 2  j  k − 1,  λ ∈ E(2i+j)α 1 +(i+j)α2 ,τ O = O(2i−1+j)α1 +(i+j)α2 ,τ , where 1  j  k O = Ol,τ , where 2 − 2i  l  i − 1 O = Oi,τ O = O2−2i−j,τ or Oi+j,τ , where 1  j  k O = Oid,τ O = O(2i−1)α1 +iα2 ,τ otherwise

Proposition 2.36. For i ∈ N and w ˜ ∈ O−i,τ . (1) If (O−i,τ )  (w) ˜  6i + 3, then

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

O = Oλ,τ , where λ ∈ E 

(i+1)α1 +( i+1 2 +

O = O−i,τ otherwise.

(w)−(O ˜ −i,τ ) )α2 ,τ 2

(2) If (w) ˜ = 6i + 5 + 4k, where k ∈ N, then ⎧ ⎪ k(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ k(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (k − j)(v − v −1 )3 + (v − v −1 ), fw,O = ˜

⎪ ⎪ ⎪ k(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 )2 + 1, ⎪ ⎪ ⎪ ⎪ (k − j)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k(v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ (v − v −1 ), ⎪ ⎪ ⎩ 0,

 O = Oλ,τ , where λ ∈ ij=2 Ejα 1 +(2j−1)α2 ,τ  O = Oλ,τ , where λ ∈ E(i+1)α +(2i+1)α 1 2 ,τ O = Oλ,τ , where 1  j  k − 1,  λ ∈ E(i+1+j)α 1 +(2i+1+j)α2 ,τ O = Oλ,τ , where 2  j  k − 1, λ ∈ E(i+1+j)α1 +(2i+2+j)α2 ,τ O = O(i+1+j)α1 +(2i+2+j)α2 ,τ , where 1  j  k O = Ol,τ , where 1 − i  l  2i + 2 O = O−i,τ O = O−i−j,τ or O2i+2+j,τ , where 1  j  k − 1 O = Oid,τ O = O(i+1)α1 +(2i+2)α2 ,τ otherwise.

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(3) If (w) ˜ = 6i + 7 + 4k, where k ∈ N, then

fw,O ˜

⎧ (k + 1)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 )3 + (v − v −1 ), ⎪ ⎪ ⎪ ⎪ (k + 1 − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1 − j)(v − v −1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (k + 1 − j)(v − v −1 )3 + (v − v −1 ), = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 )2 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1 − j)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (k + 1)(v − v −1 ), ⎪ ⎪ ⎪ ⎪ ⎪ (v − v −1 ), ⎪ ⎪ ⎩ 0,

O = Oλ,τ ,  where λ ∈ ij=2 Ejα 1 +(2j−1)α2 ,τ  O = Oλ,τ , where λ ∈ E(i+1)α 1 +(2i+1)α2 ,τ O = Oλ,τ , where 1  j  k,  λ ∈ E(i+1+j)α 1 +(2i+1+j)α2 ,τ O = Oλ,τ , where 2  j  k, λ ∈ E(i+1+j)α1 +(2i+2+j)α2 ,τ O = O(i+1+j)α1 +(2i+2+j)α2 ,τ , where 1  j  k O = Ol,τ , where 1 − i  l  2i + 2 O = O−i,τ O = O−i−j,τ or O2i+2+j,τ , where 1  j  k O = Oid,τ O = O(i+1)α1 +(2i+2)α2 ,τ otherwise.

Proof of Proposition 2.35. Since (3) is easily deduced from (2), then it is sufficient for us to prove (1) and (2). We set A = {tkα1 +iα2 s1 s2 s1 τ | k   2i  + 1}  {t(2i−1+k)α1 +(i+k)α2 s2 τ | k ∈ N}. Then, we can check directly that any element in Oi,τ is ∼ ˜ to a unique element in A. Hence, it is sufficient to consider the elements in A. For (1), it is sufficient to consider that w ˜ = t( 2 +1+ i

(w)−(O ˜ i,τ ) )α1 +iα2 2



Tw˜ ≡ (v − v −1 )

s1 s2 s1 τ , and thus

+ TOid,τ .

E

(w)−(O ˜ i,τ ) )α1 +iα2 ,τ ( i +1+ 2 2

For (2), it is sufficient to consider that w ˜ = t(2i−1+k)α1 +(i+k)α2 s2 τ , where k ∈ N. If k  3, we calculate the class polynomials one by one, and if k  4, then Tt(2i−1+k)α1 +(i+k)α2 s2 τ ≡ (v − v −1 )Tt(2i−1+k)α1 +(i+k)α2 s2 s1 τ + (v − v −1 )Tt(2i−1+k)α1 +(i+k−1)α2 s1 s2 τ + Tt(2i−2+k)α1 +(i+k−1)α2 s2 τ ······ ≡ (v − v

−1

)

k 

TO(2i−1+j)α1 +(i+j)α2 ,τ

j=3

+ (v − v

−1

)

k−1  j=2

Tt(2i+j)α1 +(i+j)α2 s1 s2 τ + Tt(2i+1)α1 +(i+2)α2 s2 τ

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Furthermore, we obtain the class polynomials from the right-hand side, which is actually 

k(v − v −1 )3

TOλ,τ + (k(v − v −1 )3 + (v − v −1 ))

λ∈ i−1 j=2 E(2j−1)α1 +jα2 ,τ



×

TOλ,τ

λ∈E(2i−1)α1 +iα2 ,τ

+

k−1  j=1

+



(k − j)(v − v −1 )3

k−2 



(k − 1 − j)(v − v −1 )3

 λ∈E(2i+j)α

j=2

+

TOλ,τ

λ∈E(2j−1+j)α1 +(i+j)α2 ,τ

k 

TOλ,τ

1 +(i+j)α2 ,τ

((k − j)(v − v −1 )3 + (v − v −1 ))TO(2i−1+j)α1 +(i+j)α2 ,τ

j=1

+

i−1 

k(v − v −1 )2 TOj,τ + (k(v − v −1 )2 + 1)TOi,τ

j=2−2i

+ (v − v −1 )TO(2i−1)α1 +iα2 ,τ +

k−1 

(k − j)(v − v −1 )2 (TO2−2i−j,τ + TOi+j,τ ) + k(v − v −1 )TOid,τ .

2

j=1

2.2.3. Quasi-split case  , then w Let w ˜∈W ˜ = wa τ  , where wa ∈ Wa and τ  ∈ Ω. Thus, w ˜ ∼w ˜ a . In the following,  for w ˜ ∈ W , it is sufficient to consider wa ∈ Wa . Proposition 2.37. If w ˜ ∈ O1,δ and (w) ˜ = 2k + 1, where k ∈ N, then

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

O = O2j,δ (1  j  k) O = O1,δ otherwise.

Proof. This follows directly from the proof of Lemma 2.12 and Theorem 2.13. Proposition 2.38. If w ˜ ∈ O0,δ and (w) ˜ = 2k, where k ∈ N, then

fw,O ˜

⎧ −1 2 ⎪ ⎪ (k − j)(v − v ) , ⎪ ⎨ k(v − v −1 ), = ⎪ 1, ⎪ ⎪ ⎩ 0,

O = O2j,δ (1  j  k − 1) O = O1,δ O = O0,δ otherwise.

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536

Proof. For w ˜ ∈ O0,δ and (w) ˜ = 2k where k ∈ N. If k = 0, this is obvious. For k > 0, si with i = 0, 1, or 2 exist such that Tw˜ ≡ (v − v −1 )Tsi w˜ + Tsi wδ(s . ˜ i) In this case, si w ˜ ∈ O1,δ with (si w) ˜ = 2k − 1 and (si wδ(s ˜ i )) = 2k − 2. By using Proposition 2.37 and calculating inductively, we have Tw˜ ≡

k−1 

(k − j)(v − v −1 )2 TO2j,δ + k(v − v −1 )TO1,δ + TO0,δ .

2

j=1

Proposition 2.39. Let m ∈ N+ . If w ˜ ∈ O2m,δ and (w) ˜ = 2m + 2k, where k ∈ N, then,

fw,O ˜

⎧ (k − j)(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ −1 2 ⎪ ⎨ k(v − v ) + 1, = k(v − v −1 )2 , ⎪ ⎪ ⎪ k(v − v −1 ), ⎪ ⎪ ⎩ 0,

O = O2m+2j,δ (1  j  k − 1) O = O2m,δ O = O2m−2j,δ (1  j  m − 1) O = O1,δ otherwise.

Proof. The proof is quite similar to that of Proposition 2.38 and thus we omit it. 2 If w, ˜ w ˜  ∈ O1,δ lie in the critical strip and (w) ˜ = (w ˜  ), then Tw˜ ≡ Tw˜  . Moreover, if  w ˜ ∈ O1,δ with (w) ˜ = 2k + 1, where k ∈ N, then

fw,O ˜

⎧ −1 ⎪ ⎨ (v − v ), = 1, ⎪ ⎩ 0,

O = O2j,δ , 1  j  k O = O1,δ otherwise.

If w ˜ ∈ O1,δ lies in the shrunken Weyl chambers, we do not have a uniform formula for these class polynomials. For example, if w ˜ = t2α1 +α2 s1 s2 s1 , then

fw,O ˜

⎧ ⎪ (v − v −1 )3 + 2(v − v −1 ), O = O2,δ ⎪ ⎪ ⎨ (v − v −1 )2 , O = O1,δ = ⎪ 1, O = O1,δ ⎪ ⎪ ⎩ 0, otherwise.

If (w) ˜ = 7 and w ˜ = tα1 −α2 s1 s2 s1 , then

fw,O ˜

⎧ (v − v −1 ), ⎪ ⎪ ⎪ ⎪ (v − v −1 )3 + 2(v − v −1 ), ⎪ ⎨ = (v − v −1 )2 , ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎩ 0,

O = O6,δ O = O2,δ O = O1,δ O = O1,δ otherwise.

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If (w) ˜ = 7 and w ˜ = t−α1 −α2 s1 s2 s1 , then ⎧ (v − v −1 )3 + (v − v −1 ), O = O4,δ ⎪ ⎪ ⎪ ⎪ −1 3 −1 ⎪ ⎪ ⎨ (v − v ) + 2(v − v ), O = O2,δ fw,O = ˜

2(v − v −1 )2 , ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎩ 0,

O = O1,δ O = O1,δ otherwise.

Inductively, if fw,O = 0, we deduce that deg fw,O ˜ ˜ 1,δ = 2 and deg fw,O ˜ 2m,δ = 3 or 1 for certain m. Moreover, fw,O ˜ 1,δ = 1. A similar argument applies for w ˜ ∈ O3,δ . 3. Applications

3.1. Affine Deligne–Lusztig varieties of basic elements . First, we assume that b ∈ PGL3 (L) and w ˜∈W Theorem 3.1. 1. If b = 1, then the affine Deligne–Lusztig variety Xw˜ (b) = ∅ if and only if w ˜ satisfies one of the following conditions: (a) w ˜ = id (b) w ˜ ∈ O1 or w ˜ ∈ O2 (c) w ˜ ∈ Ci or Ci , where i ∈ N+ and (w) ˜  6i + 3. 2. If b = τ , then the affine Deligne–Lusztig variety Xw˜ (b) = ∅ if and only if w ˜ satisfies one of the following conditions: (a) w ˜ ∈ Oid,τ (b) w ˜ ∈ Oi,τ , where i ∈ N+ and (w) ˜  6i − 1 (c) w ˜ ∈ O1−i,τ , where (w) ˜  6i + 1. Proof. Following the “Dimension = Degree” Theorem 1.11, we return to §2 to check these nonzero class polynomials. 2 When the affine Deligne–Lusztig variety Xw˜ (b) = ∅, we do have a neat dimension formula.

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Theorem 3.2. 1. If b = 1 and Xw˜ (b) = ∅, then ⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1, (w) ˜ dim Xw˜ (b) = 2 + 1, ⎪ ⎪ (w)+3 ˜ ⎪ ⎪ , ⎪ 2 ⎪ ⎩

w ˜ = Id ˜ =1 w ˜ ∈ O1 and (w) w ˜ ∈ O2 w ˜ ∈ O1 with (w) ˜ > 1, or  w ˜ ∈ Ci or Ci for i ∈ N+ .

2. If b = τ and Xw˜ (b) = ∅, then  dim Xw˜ (b) =

(w) ˜ 2 , (w)+1 ˜ , 2

w ˜ ∈ Oid,τ w ˜ ∈ Oi,τ where i ∈ Z.

Proof. By the “Dimension = Degree” theorem, if b = 1, then dim Xw˜ (1) =

1 max{(w) ˜ + deg fw,Id , (w) ˜ + 1 + deg fw,O ˜ + 2 + deg fw,O ˜ ˜ 1 , (w) ˜ 2 }. 2

We check that the degree of these class polynomials in §2 and the theorem is proved. 2 Remark 3.3. Symmetrically, we obtain a similar description of the emptiness/nonemptiness pattern and dimension formula of Xw˜ (b) for w, ˜ b = τ 2. In the following, the proofs of the theorems for the emptiness/nonemptiness pattern and dimension formula are similar to those for Theorems 3.1 and 3.2, so they are omitted. Now, we assume that b ∈ U3 (L). min Theorem 3.4. If b is basic, then Xw˜ (b) = ∅ if and only if w ˜∈ / m∈N+ Omin 2m,δ , where O2m,δ is the set of minimal length elements of O2m,δ .

Theorem 3.5. If b = 1 and if Xw˜ (b) = ∅, then ⎧ 0, ⎪ ⎪ ⎪1 ⎪ ⎪ ˜ + 1), ⎪ ⎨ 2 ((w) dim Xw˜ (b) = 12 ((w) ˜ + 2), ⎪ ⎪ ⎪ 1 ((w) ⎪ ˜ + 3), ⎪ ⎪ ⎩2

˜ =0 w ˜ ∈ O0,δ and (w) w ˜ ∈ O1,δ and lies in critical strips, or w ˜ ∈ O1,δ w ˜ ∈ O0,δ with (w) ˜ > 0, or w ˜ ∈ O2m,δ where m ∈ N+ w ˜ ∈ O1,δ corresponds to shrunken Weyl chambers, or w ˜ ∈ O3,δ .

3.2. Affine Deligne–Lusztig varieties of nonbasic elements In this section, all proofs of the following theorems are direct and they are similar to those in 3.1, thus we omit them.

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 , the affine By the “Dimension = Degree” theorem, for any b ∈ G(L) and w ˜ ∈ W Deligne–Lusztig variety Xw˜ (b) = ∅ if and only if the corresponding class polynomial is nonzero. Thus, for nonbasic b, we check the corresponding class polynomial in Section §2 and we know the emptiness/nonemptiness pattern. The emptiness/nonemptiness pattern of Xw˜ (b) with basic b is neat, however, when b is nonbasic, the pattern turns to be annoying, and it cannot be as simple as that of basic b. To avoid tedious repetition, we simply list the following criteria of the emptiness/nonemptiness pattern. Theorem 3.6. Let G(L) = PGL3 (L). 1. If b corresponds to O1 or O2 , then the emptiness/nonemptiness pattern of Xw˜ (b) is the same as that of Xw˜ (1). 2. If b corresponds to Oλ0 , where λ0 ∈ P+ ∩ Qsh , then Xw˜ (b) = ∅ if and only if w ˜ satisfies one of the following conditions: (a) w ˜ ∈ Oλ 0 (b) w ˜ ∈ O2 such that it satisfies one of the following conditions: i. w ˜ = tk(α1 +2α2 ) s2 s1 , tkα1 +(2k−1)α2 s1 s2 , or tkα1 +(2k−1)α2 s2 s1 , and λ0 ∈ Q(k−1)(α1 +2α2 ) ii. w ˜ = tk(α1 +2α2 ) s1 s2 , and λ0 ∈ Qk(α1 +2α2 ) iii. w ˜ = tk(2α1 +α2 ) s1 s2 , t(2k−1)α1 +kα2 s1 s2 , or t(2k−1)α1 +kα2 s2 s1 , and λ0 ∈ Q(k−1)(2α1 +α2 ) iv. w ˜ = tk(2α1 +α2 ) s2 s1 or t(2k+1)α1 +kα2 ) s1 s2 , and λ0 ∈ Qk(2α1 +α2 ) v. w ˜ = tλ s1 s2 , where λ = mα1 + nα2 with m, n ∈ Z, 2  m  n < 2m − 1 or 2  n < m < 2n − 1, and λ0 ∈ Dλ vi. w ˜ = tλ s2 s1 , where λ = mα1 + nα2 with m, n ∈ Z, 2  m < n < 2m − 1 or 2  n  m < 2n − 1, and λ0 ∈ Dλ vii. w ˜ = t(2k+1)α1 +kα2 s2 s1 and λ0 ∈ Qk(2α1 +α2 ) ∪ E(2k+1)α1 +(k+1)α2  viii. w ˜ = tmα1 +nα2 s1 s2 , where m, n ∈ N, m − 2n > 1, and λ0 ∈ Dmα 1 +(m−n)α2 mα1 +nα2 ix. w ˜=t s2 s1 , where m, n ∈ N, m −2n > 1, and λ0 ∈ Dmα1 +(m−n)α2 ∪ Emα1 +(m−n)α2  x. w ˜ = tmα1 −nα2 s1 s2 , where m, n ∈ N, m − n > 1, and λ0 ∈ Dmα 1 +(m+n)α2 mα1 −nα2 xi. w ˜=t s2 s1 , where m, n ∈ N, m − n > 1, and λ0 ∈ Dmα1 +(m+n)α2 ∪ Emα1 +(m+n)α2 (c) w ˜ ∈ O1 , and it satisfies one of the following: i. (w) ˜ = 4k − 1, and λ0 = i(α1 + α2 ) (1  i  k − 1) or λ0 ∈ Ei(α1 +α2 ) ∪  Ei(α1 +α2 ) (3  i  k − 1) ii. (w) ˜ = 4k − 3, and λ0 = i(α1 + α2 ) (1  i  k − 1) or λ0 ∈ Ei(α1 +α2 ) ∪  Ei(α1 +α2 ) (3  i  k − 2) (d) w ˜ ∈ Ci , where i ∈ N+ , and it satisfies one of the following: ˜ i. (w) ˜  6i − 1 and λ0 = jα1 + iα2 , where  2i  + 1  j  (w)−1 −i 2  ii. (w) ˜ = 6i −1 +4k and λ0 ∈ Qi(2α1 +α2 ) ∪E(2i+j)α1 +(i+j)α2 ∪E(2i+j)α1 +(i+j)α2 (1  j  k − 1) or λ0 = (2i + j)α1 + (i + j)α2 (1  j  k − 1)

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 iii. (w) ˜ = 6i +1 +4k and λ0 ∈ Qi(2α1 +α2 ) ∪E(2i+j)α1 +(i+j)α2 ∪E(2i+j)α 1 +(i+j)α2 (1  j  k − 1) or λ0 = (2i + j)α1 + (i + j)α2 (1  j  k) (e) w ˜ ∈ Ci , where i ∈ N+ , and it satisfies one of the following:  (w)+ (C ˜ i) i. (w) ˜  6i + 1 and λ0 = iα1 + ( 2i  + j)α2 , where 1  j  2  ii. (w) ˜  6i −1 +4k and λ0 ∈ Qi(α1 +2α2 ) ∪E(i+j)α1 +(2i+j)α2 ∪E(i+j)α1 +(2i+j)α2 (1  j  k − 1) or λ0 = (i + j)α1 + (2i + j)α2 (1  j  k − 1)  iii. (w) ˜  6i +1 +4k and λ0 ∈ Qi(α1 +2α2 ) ∪E(i+j)α1 +(2i+j)α2 ∪E(i+j)α 1 +(2i+j)α2 (1  j  k − 1) or λ0 = (i + j)α1 + (2i + j)α2 (1  j  k) 3. If b corresponds to Oi0 (α1 +2α2 ) (symmetrically, for b corresponds to Oi0 (2α1 +α2 ) ), where i0 ∈ N+ , then Xw˜ (b) = ∅ if and only if w ˜ satisfies one of the following conditions. (a) w ˜ ∈ Oi0 (α1 +2α2 ) (b) w ˜ ∈ O1 , (w) ˜ = 4k − 1 and 2i0  k − 1, or (w) ˜ = 4k − 3 and 2i0  k − 2 (c) w ˜ ∈ Ci , where i ∈ N+ , and if (w) ˜  6i + 1, then i = 2i0 ; or if (w) ˜ = 6i − 1 + 4k or 6i + 1 + 4k, then i  2i0  i + k − 1 (d) w ˜ ∈ Ci , where i ∈ N+ , and if (w) ˜ = 6i −1 +4k or 6i +1 +4k, then 2i0  2i +k−1 (e) w ˜ ∈ O2 , and it satisfies one the following conditions. i. w ˜ = tk(α1 +2α2 ) s2 s1 , tk(α1 +2α2 ) s1 s2 , and 2i0  2k − 1 ii. w ˜ = tk(2α1 +α2 ) s1 s2 , t(2k−1)α1 +kα2 s1 s2 , or t(2k−1)α1 +kα2 s2 s1 , and 2i0  k−1 iii. w ˜ = tk(2α1 +α2 ) s2 s1 , t(2k+1)α1 +kα2 ) s1 s2 , and 2i0  k iv. w ˜ = t(2k+1)α1 +kα2 s2 s1 , and 2i0  k + 1 v. w ˜ = tmα1 +nα2 s1 s2 (2  m  n < 2m − 1 or 2  n < m < 2n − 1), or w ˜ = tmα1 +nα2 s2 s1 (2  m < n < 2m − 1 or 2  n  m < 2n − 1), and 2i0  n − 1 vi. w ˜ = tmα1 +nα2 s1 s2 , where m, n ∈ N, m − 2n > 1, and 2i0  m − n − 1 vii. w ˜ = tmα1 +nα2 s2 s1 , where m, n ∈ N, m − 2n > 1, and 2i0  m − n viii. w ˜ = tmα1 −nα2 s1 s2 , where m, n ∈ N, m − n > 1, and 2i0  m + n − 1 ix. w ˜ = tmα1 −nα2 s2 s1 , where m, n ∈ N, m − n > 1, and 2i0  m + n 4. If b corresponds to Ci0 (symmetrically, for b corresponds to Ci0 ), where i0 ∈ N+ , then Xw˜ (b) = ∅ if and only if w ˜ satisfies one of the following conditions. (a) w ˜ ∈ O1 , (w) ˜ = 4k − 1 and i0  k − 1, or (w) ˜ = 4k − 3 and i0  k − 2 (b) w ˜ ∈ Ci , where i ∈ N+ , and if (w) ˜  6i + 1, then i = i0 ; or if (w) ˜ = 6i − 1 + 4k or 6i + 1 + 4k, then i0  i + k − 1 (c) w ˜ ∈ Ci , where i ∈ N+ , and if (w) ˜ = 6i −1 +4k or 6i +1 +4k, then i0  2i +k −1 (d) w ˜ ∈ O2 , and it satisfies one the following conditions. i. w ˜ = tk(α1 +2α2 ) s2 s1 , tk(α1 +2α2 ) s1 s2 , and i0  2k − 1 ii. w ˜ = tk(2α1 +α2 ) s1 s2 , t(2k−1)α1 +kα2 s1 s2 , or t(2k−1)α1 +kα2 s2 s1 , and i0  k −1 iii. w ˜ = tk(2α1 +α2 ) s2 s1 , t(2k+1)α1 +kα2 ) s1 s2 , and i0  k iv. w ˜ = t(2k+1)α1 +kα2 s2 s1 , and i0  k + 1

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v. w ˜ = tmα1 +nα2 s1 s2 (2  m  n < 2m − 1 or 2  n < m < 2n − 1), or w ˜ = tmα1 +nα2 s2 s1 (2  m < n < 2m − 1 or 2  n  m < 2n − 1), and i0  n − 1 vi. w ˜ = tmα1 +nα2 s1 s2 , where m, n ∈ N, m − 2n > 1, and i0  m − n − 1 vii. w ˜ = tmα1 +nα2 s2 s1 , where m, n ∈ N, m − 2n > 1, and i0  m − n viii. w ˜ = tmα1 −nα2 s1 s2 , where m, n ∈ N, m − n > 1, and i0  m + n − 1 ix. w ˜ = tmα1 −nα2 s2 s1 , where m, n ∈ N, m − n > 1, and i0  m + n Theorem 3.7. 1. If b ∈ PGL3 (L) corresponds to Oλ0 , where λ0 ∈ P+ ∩ Qsh and Xw˜ (b) = ∅, then ⎧1 ˜ + (Oλ0 )) − λ0 , 2ρ, ⎪ 2 ((w) ⎪ ⎪ ⎪ 1 ⎪ ˜ + (Oλ0 ) + 1) − λ0 , 2ρ, ⎪ ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dim Xw˜ (b) =

1 ⎪ ˜ + (Oλ0 ) + 2) − λ0 , 2ρ, ⎪ 2 ((w) ⎪ ⎪ ⎪ 1 ⎪ ˜ + (Oλ0 ) + 3) − λ0 , 2ρ, ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

w ˜ ∈ Oλ0 w ˜ ∈ O1 and ˜ (α1 + α2 ) λ0 = (w)−1 4 or w ˜ ∈ Ci or Ci i ∈ N+ where (w) ˜  6i + 1 w ˜ ∈ O2 w ˜ ∈ O1 and ˜ λ0 = (w)−1 (α1 + α2 ) 4 or w ˜ ∈ Ci or Ci i ∈ N+ where (w) ˜ > 6i + 1

2. If b ∈ PGL3 (L) corresponds to Oi0 (α1 +2α2 ) , where i0 ∈ N+ and Xw˜ (b) = ∅, then ⎧1 ˜ + 6i0 + 1) − t0 , ⎪ ⎨ 2 ((w) 1 dim Xw˜ (b) = 2 ((w) ˜ + 6i0 + 2) − t0 , ⎪ ⎩1 ˜ + 6i0 + 3) − t0 , 2 ((w)

w ˜ ∈ Ci or Ci with (w) ˜  6i + 1, i ∈ N+ w ˜ ∈ O2 w ˜ ∈ O1 or w ˜ ∈ Ci , Ci with (w) ˜ > 6i + 1,

where t0 = i0 α1 + 2α2 , 2ρ. Symmetrically, if b corresponds to Oi0 (2α1 +α2 ) , then a similar dimension formula is obtained. 3. If b ∈ PGL3 (L) corresponds to Ci0 , where i0 ∈ N+ and Xw˜ (b) = ∅, then ⎧1 ˜ + (Ci0 )) − t0 , ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎨ 1 ˜ + (Ci0 ) + 1) − t0 , dim Xw˜ (b) = 2 ((w) ⎪ ⎪ ⎪ ˜ + (Ci0 ) + 2) − t0 , ⎪ 12 ((w) ⎪ ⎩

w ˜ ∈ Ci or Ci with (w) ˜  6i + 1, i ∈ N+ w ˜ ∈ O2 w ˜ ∈ O1 or w ˜ ∈ Ci , Ci with (w) ˜ > 6i + 1, i ∈ N+ ,

where t0 = i0 α1 + 2α2 , ρ. Symmetrically, when b corresponds to Ci0 , we have a similar dimension formula.

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Theorem 3.8. 1. Let b ∈ PGL3 (L). If b corresponds to Oλ0 ,τ , where λ0 ∈ P+ ∩Qsh , λ0 = (2i−1)α1 +iα2 for i ∈ N+ , and Xw˜ (b) = ∅, then ⎧ ⎪ (w) ˜ − ¯ νOλ0 ,τ , 2ρ, ⎪ ⎪1 ⎪ ⎪ ˜ + (Oλ0 ,τ ) + 1) − ¯ νOλ0 ,τ , 2ρ, ⎪ 2 ((w) ⎪ ⎪ ⎨ dim Xw˜ (τ˙ ) =

1 ˜ + (Oλ0 ,τ ) + 2) − ¯ νOλ0 ,τ , 2ρ, ⎪ 2 ((w) ⎪ ⎪ ⎪ 1 ⎪ ⎪ ˜ + (Oλ0 ,τ ) + 3) − ¯ νOλ0 ,τ , 2ρ, ⎪ 2 ((w) ⎪ ⎩

w ˜ ∈ Oλ0 ,τ w ˜ ∈ Oi,τ or O1−i,τ and (w) ˜  6i − 3 w ˜ ∈ Oid,τ w ˜ ∈ Oi,τ or O1−i,τ and (w) ˜ > 6i − 3.

2. Let b ∈ PGL3 (L). If b corresponds to Oi0 (α1 +2α2 ),τ , where i0 ∈ N+ and Xw˜ (b) = ∅, then ⎧1 ˜ + (O2i0 ,τ )) − l0 , ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎨ 1 ˜ + (O2i0 ,τ ) + 1) − l0 , dim Xw˜ (b) = 2 ((w) ⎪ ⎪ 1 ⎪ ((w) ˜ + (O2i0 ,τ ) + 2) − l0 , ⎪ ⎪ ⎩2

w ˜ ∈ Oi,τ , i ∈ N+ and (w) ˜  6i − 3 w ˜ ∈ Oid,τ w ˜ ∈ Oi,τ and (w) ˜ > 6i − 3 or w ˜ ∈ O1−i,τ ,

where l0 = (i0 − 13 )(α1 + 2α2 ), 2ρ. 3. Let b ∈ PGL3 (L). If b corresponds to O(2i0 −1)α1 +i0 α2 ),τ , where i0 ∈ N+ and Xw˜ (b) = ∅, then ⎧1 ˜ + (O2(1−i0 ),τ )) − l1 , ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎨ 1 ˜ + (O2(1−i0 ),τ ) + 1) − l1 , dim Xw˜ (b) = 2 ((w) ⎪ ⎪ ⎪ ˜ + (O2(1−i0 ),τ ) + 2) − l1 , ⎪ 12 ((w) ⎪ ⎩

w ˜ ∈ O1−i,τ , i ∈ N+ and (w) ˜  6i − 1 w ˜ ∈ Oid,τ w ˜ ∈ O1−i,τ and (w) ˜ > 6i − 1 or w ˜ ∈ Oi,τ ,

where l1 = (i0 − 23 )(2α1 + α2 ), 2ρ. 4. Let b ∈ PGL3 (L). If b corresponds to Oi0 ,τ , where i0 ∈ N+ . If Xw˜ (b) = ∅, then ⎧1 ˜ + (Oi0 ,τ )) − l2 , ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎨ 1 ˜ + (Oi0 ,τ ) + 1) − l2 , dim Xw˜ (b) = 2 ((w) ⎪ ⎪ 1 ⎪ ((w) ˜ + (Oi0 ,τ ) + 2) − l2 , ⎪ ⎪ ⎩2 where l2 = ( i20 − 13 )(α1 + 2α2 ), 2ρ.

w ˜ ∈ Oi,τ i ∈ N+ and (w) ˜  6i − 3 w ˜ ∈ Oid,τ w ˜ ∈ Oi,τ and (w) ˜ > 6i − 3 or w ˜ ∈ O1−i,τ ,

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Similarly, if b corresponds to O1−i0 ,τ and Xw˜ (b) = ∅, then ⎧1 ˜ + (O1−i0 ,τ )) − l3 , ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎨ 1 ˜ + (O1−i0 ,τ ) + 1) − l3 , dim Xw˜ (b) = 2 ((w) ⎪ ⎪ 1 ⎪ ((w) ˜ + (O1−i0 ,τ ) + 2) − l3 , ⎪ ⎪ ⎩2

w ˜ ∈ O1−i,τ i ∈ N+ and (w) ˜  6i − 1 w ˜ ∈ Oid,τ w ˜ ∈ O1−i,τ and (w) ˜ > 6i − 1 or w ˜ ∈ Oi,τ ,

where l3 = ( i20 − 16 )(2α1 + α2 ), 2ρ. Theorem 3.9. Let b ∈ U3 (L). If b corresponds to O2m0 ,δ , where m0 ∈ N+ , then ν¯O2m0 ,δ = m0 νO2m0 ,δ , 2ρ = 2m0 . If Xw˜ (b) = ∅, then 2 (α1 + α2 ) and ¯ ⎧ 0, ⎪ ⎪ ⎪ ⎪ 1 ⎪ ˜ + 1) − m0 , ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dim Xw˜ (b) = 12 ((w) ˜ + 2) − m0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ˜ + 3) − m0 , ⎪ ⎪ 2 ((w) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

˜ = 2m0 w ˜ ∈ O2m0 ,δ and (w) w ˜ ∈ O1,δ , or w ˜ ∈ O1,δ and lies in critical strips, or ˜ = 2m0 + 1 w ˜ ∈ O1,δ or O3,δ with (w) w ˜ ∈ O0,δ or w ˜ ∈ O2m0 ,δ and (w) ˜ > 2m0 , or w ˜ ∈ O2m,δ where m = m0 (w) ˜ > 2m0 + 1, w ˜ ∈ O3,δ or w ˜ ∈ O1,δ and corresponds to shrunken Weyl chambers.

3.3. Affine Deligne–Lusztig varieties: GL3 and D× 3 cases In general, there is a canonical projection π : GLn (L) −→ PGLn (L). Let I1 be the Iwahori subgroup of GLn (L) described in Section 2, and let I2 be the corresponding Iwa1 (or W 2 , respectively) be the Iwahori–Weyl group. Let hori subgroup of PGLn (L). Let W  w ˜ ∈ W1 and b ∈ GLn (L), such that κ(w) ˜ = κ(b). Then the affine Deligne–Lusztig variety Xw˜ (b) in the affine flag variety of GLn (L) is isomorphic to the affine Deligne–Lusztig variety Xπ(w) ˜ (π(b)) in the affine flag variety of PGLn (L) (see [9] for more details). Fur= fπ(w),π(O) . Since we already know the emptiness/nonemptiness thermore, we have fw,O ˜ ˜ pattern and the dimension formulas of the affine Deligne–Lusztig variety for PGL3 (L), then we have the same pattern and dimension formulas for GL3 (L). In a different manner, if the automorphism σ : GL3 (L) −→ GL3 (L) is replaced by   σ = Ad(τ ) ◦ σ : GL3 (L) −→ GL3 (L), then GL(L)σ3 is the group of units of the division algebra D3 (i.e., D× ˜ (τ ) for 3 ). As described in [9], the affine Deligne–Lusztig variety Xwτ PGL3 is isomorphic to the affine Deligne–Lusztig variety Xw˜ (1) for the group D× 3 . Thus, we have the same pattern and dimension formulas for D× . 3

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3.4. Rational points for some affine Deligne–Lusztig varieties In general, if Xw˜ (b) = ∅, then it has infinitely many irreducible components. However, if b ∈ G(L) is superbasic, then Xw˜ (b) contains only finite irreducible components and the number of rational points is finite. , Proposition 3.10 (He). Let G = PGLn be split over F . If x is a superbasic element in W  then for any w ˜∈W Xw˜ (x)(Fq ) = nq

(w) ˜ 2

√ fw,O ˜ |v= q ,

. where x ∈ O and it is the conjugacy class of W By applying Proposition 3.10 to the group G = PGL3 , we can obtain an explicit formula.  we ˜∈W Corollary 3.11. Let G = PGL3 be split over F . Let τ ∈ G(L), then for any w have ⎧ (w) ˜ 3q 2 , ⎪ ⎪ ⎪ (w)−1 ˜ ⎨ (w)−6i+3 3 ˜ 4 q 2 (q − 1), Xw˜ (τ )(Fq ) = (w)−1 ˜ (w)−6i+1 ˜ ⎪ q 2 (q − 1), ⎪ 4 ⎪ 3 ⎩ 0,

w ˜ ∈ Oid,τ w ˜ ∈ Oi,τ , i ∈ N+ and (w) ˜  6i − 1 w ˜ ∈ O1−i,τ and (w) ˜  6i + 1 otherwise.

Proof. The proof follows directly from Proposition 3.10 and Propositions 2.35, 2.36. 2 3.5. The Görtz–Haines–Kottwitz–Reuman conjecture Before we recall a conjecture of Görtz–Haines–Kottwitz–Reuman, we give some notations first. Let G(L) be as given in Section 2. For any b ∈ G(L), we denote Jb as the σ-centralizer of b (i.e., Jb = {g ∈ G(L) | gbσ(g)−1 = b}). By definition, the defect of b (def (b)) is the F -rank of G minus the F -rank of Jb (i.e., def (b) = rkF G − rkF Jb ). We employ Lemma 4.6 from [12] to calculate the defect of b. The Görtz–Haines–Kottwitz–Reuman conjecture is stated as follows. Conjecture 3.12. Let b ∈ G(L) and b be a basic element in G(L) such that κ(b) = κ(b ).  with sufficiently large length, Xw˜ (b) = ∅ if and only if Xw˜ (b ) = ∅. In Then for w ˜∈W this case, 1 dim Xw˜ (b) = dim Xw˜ (b ) − νb , ρ + (def (b ) − def (b)). 2

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Theorem 3.13. The Conjecture 3.12 is true for the following groups: GL3 (L),

PGL3 (L),

D× 3.

U3 (L),

Proof. By §3.3, it is sufficient for us to check for the groups PGL3 (L) and U3 (L). By §3.1, we know the emptiness/nonemptiness pattern for basic b . For nonbasic b, using the  with suffiDimension = Degree theorem and class polynomials in Section 3, if w ˜∈W ciently large length, then we check case-by-case that Xw˜ (b) = ∅ if and only if Xw˜ (b ) = ∅. If they are nonempty, we need to compare their dimensions and we also check case-bycase. For example, let G(L) = PGL3 (L) and b = 1. If b corresponds to Oλ0 , where  with sufficiently large length, and Xw˜ (b) = ∅, we have λ0 ∈ P+ ∩ Qsh . For those w ˜∈W dim Xw˜ (b) − dim Xw˜ (1) =

1 (Oλ0 ) − λ0 , 2ρ. 2

Since λ0 ∈ P+ , (Oλ0 ) = (tλ0 ) = λ0 , 2ρ. By direct calculation, we have def (1) = def (b). Thus, 1 dim Xw˜ (b) = dim Xw˜ (1) − νb , ρ + (def (1) − def (b)). 2 If b = τ and b corresponds to Oλ0 ,τ , where λ0 ∈ P+ ∩ Qsh and λ0 = (2i − 1)α1 + iα2 for all i ∈ N+ . Thus, we have dim Xw˜ (b) − dim Xw˜ (τ ) =

1 ((Oλ0 ,τ ) + 2) − νOλ0 ,τ , 2ρ. 2

In this case, we have νb = νOλ0 ,τ = λ0 − 13 (α1 + 2α2 ), (λ0 ) = (Oλ0 ,τ ) + 2, and def (τ ) − def (b) = 2, thus 1 dim Xw˜ (b) = dim Xw˜ (τ ) − νb , ρ + (def (τ ) − def (b)). 2 For other cases, the methods are quite similar and thus we omit them. 2 3.6. Affine Deligne–Lusztig varieties in the affine Grassmannian First, we recall the affine Deligne–Lusztig variety in the affine Grassmannian. We retain the notations from Section §1. Let G be the smooth affine group scheme associated with the special vertex of the Bruhat–Tits building of G. We denote L+ G(R) = G(R[[]]) as the infinite dimensional affine group scheme. The twisted affine Grassmannian is defined by the f pqc (fidèlement plate quasi-compacte) quotient Gr = LG/L+ G. We have the Cartan decomposition G(L) =

L

+

μ∈P+

G(k)μ L+ G(k),

Gr(k) =

L

+

μ∈P+

G(k)μ L+ G(k)/L+ G(k).

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Definition 3.14. Let b ∈ G(L) and μ ∈ P+ , then the affine Deligne–Lusztig variety Xμ (b) in the affine Grassmannian Gr is defined by Xμ (b)(k) = {g ∈ G(L) | gbσ(g)−1 ∈ L+ G(k)μ L+ G(k)/L+ G(k)}.  contains Let w0 be the longest element in W . We note that any W × W -coset of W λ a unique maximal element and this element is of the form w0 t for some λ ∈ P+ . An element in this double coset is of the form xtλ y for x ∈ W and y ∈I(λ) W , where I(λ) W = {si ∈ S | λ, αi  = 0}. Let w ˜ = w0 tλ , and λ ∈ P+ . For the special element λ w0 t , we have the following Theorem 3.15 (He). Let λ ∈ P+ , x ∈ W and y ∈I(λ) W . For any b ∈ G(L), dim Xxtλ y (b)  dim Xw0 tλ (b) − (w0 ) + (x). Furthermore, for the special element w0 tλ , there is a complete solution for the emptiness/nonemptiness pattern and dimension formula (see [15] Theorem 6.1). 3.7. Leading coefficient of fw0 tλ ,O Let p : Fl −→ Gr be the projection. Each fiber of p is isomorphic to L+ G/I, which is of dimension (w0 ). Since p−1 Xλ (b) = w∈W ˜ (b) and Theorem 3.15, ˜ tλ W Xw then for any w ˜ ∈ W tλ W , we have dim Xw˜ (b)  dim Xw0 tλ (b). Based on information about the class polynomials and the reduction method, we expect that the irreducible components of Xw˜ (b) of maximal dimension can be controlled by the leading coefficient of the corresponding class polynomial fw0 tλ ,O . More precisely, on one hand, for  (w w, ˜ w ˜ ∈ W ˜  = swδ(s) ˜ and (w) ˜ = (w ˜  ) + 2), and Xw(b) = ∅, then Xw(b) = X 1  X2 , ˜ ˜ where X1  Xsw˜ (b) and X2  Xswδ(s) (b) (see [14, Proposition 4.2]). On the other ˜ hand, Tw˜ ≡ (v − v −1 )Tsw˜ + Tswδ(s) . And then, we apply the reduction method to ˜ Xw(b) time and time again until all of them are irreducible. Meanwhile, each reduc˜ tion step splits Tw˜ into more terms, which add the degrees or the coefficients of the corresponding class polynomials. The more reduction steps we have the larger coefficients of top degrees we will get, and which indicate that the more irreducible compo can be written as xtμ y (see nents we may obtain. Recall that every element w ˜ ∈ W §3.6). If x = si1 si2 · · · sir , then w ˜ → si2 · · · sir tμ yδ(si1 ) → · · · → tμ yδ(x) such that μ μ (t yδ(x))  (xt y). And the affine Deligne–Lusztig variety Xw0 tμ (b) may involve the most reduction steps. Instead of considering the irreducible components of Xw˜ (b) of maximal dimension, we consider the leading coefficient of the corresponding class polynomial. In addition, we denote fw,O ˜ b as the class polynomial that corresponds to Xw ˜ (b), which is indicated by the “Dimension = Degree” theorem.

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 , we denote N0 as the leading coefficient of fw,O Proposition 3.16. Given w ˜ ∈W ˜ 2 . We denote L(fw,O ˜ = w0 t λ , ˜ ) to be the leading coefficient of fw,O ˜ . Moreover, we assume that w where λ ∈ P+ . (1) If λ = k0 (α1 + α2 ) with k0 ∈ N and it is sufficiently large, then ⎧ ⎪ ⎨ N0 − 2i, N0 − i, L(fw,O ˜ b) = ⎪ ⎩

b ←→ Oiα1 +2iα2 or O2iα1 +iα2 ;  b ←→ Oλ , λ ∈ {i(α1 + α2 )}  Ei(α1 +α2 )  Ei(α , or 1 +α2 )  b ←→ Ci or Ci ,

where i ∈ N and k0 related to i is quite large. (2) If λ = i0 (α1 + 2α2 ) + k0 (α1 + α2 ) with k0 ∈ N and it is sufficiently large, then ⎧ ⎪ N0 , ⎪ ⎪ ⎨ N − j, 0 L(fw,O ˜ b) = ⎪ ⎪ ⎪ ⎩

b ←→ Oi0 α1 +2i0 α2 , or Ci , Ci (i  i0 ); b ←→ Oλ , λ ∈ {(i0 + j)α1 + (2i0 + j)α2 }  E(i0 +j)α1 +(2i0 +j)α2  E(i , or 0 +j)α1 +(2i0 +j)α2  b ←→ Ci0 +j or Ci0 +j .

(3) If λ = i0 (2α1 + α2 ) + k0 (α1 + α2 ) with k0 ∈ N and it is sufficiently large, then ⎧ ⎪ N0 , ⎪ ⎪ ⎨ N − j, 0 L(fw,O ˜ b) = ⎪ ⎪ ⎪ ⎩

b ←→ O2i0 α1 +i0 α2 , or Ci , Ci (i  i0 ); b ←→ Oλ , λ ∈ {(2i0 + j)α1 + (i0 + j)α2 }  E(2i0 +j)α1 +(i0 +j)α2  E(2i , or 0 +j)α1 +(i0 +j)α2 b ←→ Ci0 +j or Ci0 +j .

Proof. We check (1) and the others are similar. If b corresponds to Oiα1 +2iα2 (or −1 O2iα1 +iα2 ), where i ∈ N+ , then Ob = C2i (or Ob = C2i ). In this case, fw,O ) ˜ 2 = N0 (v−v −1 2 ) . If b corresponds to Oλ0 , where λ0 = i(α1 +α2 ) and fw,C ˜ 2i = fw,C ˜ 2i = (N0 −2i)(v −v  −1 3 (or λ0 ∈ Ei(α1 +α2 ) Ei(α ), then Ob = Oλ0 , and fw,O ) +(v−v −1 ) ˜ λ0 = (N0 −i)(v−v 1 +α2 ) −1 3 ) ). If b corresponds to Ci orCi , then Ob = Ci or Ob = Ci , (or fw,O ˜ λ0 = (N0 − i)(v − v −1 2 and fw,C ) . Thus (1) is proved. 2 ˜ i = fw,C ˜ i = (N0 − i)(v − v Remark 3.17. We still do not know how to compute the connected (irreducible) components of affine Deligne–Lusztig varieties in affine flag variety. However, [3] made some progress on connected components of affine Deligne–Lusztig varieties in affine Grassmannian, and we hope that their method can be generated to the affine flag setting. Acknowledgments This study is based on my HKUST PhD thesis. I would like to express my deepest gratitude to my thesis supervisor Professor Xuhua He, who supported my PhD study generously and kindly, gave me careful guidance, and shared his brilliant ideas with me. I also thank the Department of Mathematics at The Hong Kong University of Science and Technology for providing me with postgraduate studentships. This paper was written

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when I am a postdoc in Beijing International Center for Mathematical Research, and I thank the institute for their hospitality as well as Prof. Ruochuan Liu for helpful discussions. I thank the referee for careful reading and helpful suggestions in improving the exposition of this article. In particular, following the referee’s suggestions, we add Theorem 3.6 and rewrite §3.7. References [1] N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Elements of Mathematics, Springer-Verlag, Berlin, 2002. [2] F. Bruhat, J. Tits, Groupes réductifs sur un corps local, II: schémas en groupes, existence d’une donnée radicielle valuée, Publ. Math. 60 (1984) 5–184. [3] M. Chen, M. Kisin, E. Viehmann, Connected components of affine Deligne–Lusztig varieties in mixed characteristic, Compos. Math. 151 (2015) 1697–1762. [4] P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976) 103–161. [5] M. Geck, G. Pfeiffer, On the irreducible characters of Hecke algebras, Adv. Math. 102 (1) (1993) 79–94. [6] U. Görtz, T. Haines, R. Kottwitz, D. Reuman, Dimensions of some affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006) 467–511. [7] U. Görtz, T. Haines, R. Kottwitz, D. Reuman, Affine Deligne–Lusztig varieties in affine flag varieties, Compos. Math. 146 (5) (2010) 1339–1382. [8] U. Görtz, X. He, Dimensions of affine Deligne–Lusztig varieties in affine flag varieties, Doc. Math. 15 (2010) 1009–1028. [9] U. Görtz, X. He, S. Nie, P -alcoves and nonemptiness of affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015) 647–665. [10] T. Haines, Introduction to Shimura varieties with bad reduction of parahoric type, Clay Math. Proc. 4 (2005) 583–642. [11] T. Haines, M. Rapoport, On parahoric subgroups, appendix to [19]. [12] P. Hamacher, The dimension of affine Deligne–Lusztig varieties in the affine Grassmannian of unramified groups, arXiv:1312.0486. [13] X. He, Minimal length elements of extended affine Weyl group, I, arXiv:1004.4040. [14] X. He, Geometric and homological properties of affine Deligne–Lusztig varieties, Ann. of Math. 179 (2014) 367–404. [15] X. He, Note on affine Deligne–Lusztig varieties, arXiv:1309.0075. [16] X. He, S. Nie, Minimal length elements of extended affine Weyl group, Compos. Math. 150 (11) (2014) 1903–1927. [17] X. He, Z. Yang, Elements with finite Coxeter part in an affine Weyl group, J. Algebra 372 (2012) 204–210. [18] R. Kottwitz, Isocrystals with additional structure II, Compos. Math. 109 (3) (1997) 255–339. [19] G. Pappas, M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008) 118–198. [20] M. Rapoport, A guide to the reduction modulo p of Shimura varieties, Astérisque (298) (2005) 271–318. [21] D. Reuman, Determining whether certain affine Deligne–Lusztig sets are nonempty, Thesis Chicago, 2002, arXiv:math/0211434. [22] D. Reuman, Formulas for the dimensions of some affine Deligne–Lusztig varieties, Michigan Math. J. 52 (2) (2004) 435–451. [23] T.A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974) 159–198. [24] E. Viehmann, The dimension of affine Deligne–Lusztig varieties, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006) 513–526. [25] Z. Yang, Class polynomials for some affine Hecke algebras, PhD thesis, HKUST, 2014, http://lbezone.ust.hk/bib/b1334294.