ALGEBRAIC COMBINATORICS
Jérémie Guilhot & James Parkinson
Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures forC˜2
Volume 2, issue 5 (2019), p. 969-1031.
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Algebraic Combinatorics
Volume 2, issue 5 (2019), p. 969–1031 https://doi.org/10.5802/alco.75
Balanced representations, the asymptotic Plancherel formula, and Lusztig’s
conjectures for ˜ C 2
Jérémie Guilhot & James Parkinson
Abstract We prove Lusztig’s conjectures P1–P15 for the affine Weyl group of type ˜C2for all choices of positive weight function. Our approach to computing Lusztig’sa-function is based on the notion of a “balanced system of cell representations”. Once this system is established roughly half of the conjectures P1–P15 follow. Next we establish an “asymptotic Plancherel Theorem” for type ˜C2, from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig’s conjectures for all rank 1 and 2 affine Weyl groups for all choices of parameters.
The theory of Kazhdan–Lusztig cells plays a fundamental role in the representation theory of Coxeter groups and Hecke algebras. In their celebrated paper [14] Kazhdan and Lusztig introduced the theory in the equal parameter case, and in [16] Lusztig generalised the construction to the case of arbitrary parameters. A very specific feature in the equal parameter case is the geometric interpretation of Kazhdan–Lusztig theory, which implies certain “positivity properties” (such as the positivity of the structure constants with respect to the Kazhdan–Lusztig basis). This was proved in the finite and affine cases by Kazhdan and Lusztig in [15], and the case of arbitrary Coxeter groups was settled only very recently by Elias and Williamson in [5]. However, simple examples show that these positivity properties no longer hold for unequal parameters, hence the need to develop new methods to deal with the general case.
A major step in this direction was achieved by Lusztig in his book on Hecke algebras with unequal parameters [17, Chapter 14] where he introduced 15 conjectures P1–P15 which capture essential properties of cells for all choices of parameters. In the case of equal parameters these conjectures can be proved for finite and affine types using the above mentioned geometric interpretation (see [17]). For arbitrary parameters the existing state of knowledge is much less complete. A contemporary account of the theory outlining the state of the art can be found in [2].
Recently in [13] we developed an approach to proving P1–P15 and applied it to the case ˜G2 with arbitrary parameters. This provided the first irreducible affine Coxeter group, apart from the infinite dihedral group, where Lusztig’s conjectures have been established for arbitrary (unequal) parameters. Indeed, at the time of writing this paper, the only cases for which P1–P15 were known to hold (outside of the equal parameter case) were:
• the quasisplit case where a geometric interpretation is available [17, Chap- ter 16];
Manuscript received 28th March 2018, revised 15th November 2018 and 8th April 2019, accepted 15th March 2019.
Keywords. Kazhdan–Lusztig theory, Plancherel formula, affine Hecke algebras.
ISSN: 2589-5486 http://algebraic-combinatorics.org/
J. Guilhot & J. Parkinson
• finite dihedral type [8] and infinite dihedral type [17, Chapter 17] for arbitrary parameters;
• universal Coxeter groups for arbitrary parameters [23];
• finite typeBn in the “asymptotic” parameter case [3, 8];
• finite typeF4 for arbitrary parameters [8];
• affine type ˜G2 for arbitrary parameters [13].
We add that during the process of publishing this paper, Xie [25] announced a proof of P1–P15 for Coxeter groups whose Coxeter graph is either complete or right angled.
Our approach in [13] hinges on two main ideas:
(a) the notion of a balanced system of cell representationsfor the Hecke algebra, (b) theasymptotic Plancherel formula.
In the present paper we develop these ideas in type ˜C2. This three parameter case turns out to be considerably more complicated than the two parameter ˜G2 case, and this additional complexity requires us to take a somewhat more conceptual approach here.
We now briefly describe the ideas (a) and (b) above. Let (W, S) be a Coxeter system with weight functionL:W →N>0 and associated multi-parameter Hecke algebraH defined overZ[q,q−1]. LetΛbe the set of two-sided cells ofW with respect toL, and recall that there is a natural partial order 6LR on the setΛ. Let (Cw)w∈W denote the Kazhdan–Lusztig basis ofH.
One of the main challenges in proving Lusztig’s conjectures is to compute Lusztig’sa-function since, in principle, it requires us to have information on all the structure constants with respect to the Kazhdan–Lusztig basis. In [13] we showed that the existence of a balanced system of cell representations is sufficient to compute the a-function. Such a system is a family (πΓ)Γ∈Λ of representations of H, each equipped with a distinguished basis, satisfying various axioms including
(1) πΓ(Cw) = 0 for allw∈Γ0 with Γ06>LRΓ,
(2) the maximal degree of the coefficients that appear in the matrix πΓ(Cw) is bounded by a constantaπΓ,
(3) this bound is attained if and only ifw∈Γ.
This concept is inspired by the work of Geck [8] in the finite dimensional case.
Thus a main part of the present paper is devoted to establishing a balanced system of cell representations in type ˜C2 for each choice of parameters. For this purpose we use the explicit partition ofW into Kazhdan–Lusztig cells that was obtained by the first author in [12]. It turns out that the representations associated to finite cells naturally give rise to balanced representations and so most of our work is concerned with the infinite cells. In type ˜C2there are either 3 or 4 such two-sided cells depending on the choice of parameters. To each of these two-sided cells we associate a natural finite dimensional representation admitting an elegant combinatorial description in terms of alcove paths. Using this description we are able to give a combinatorial proof of the balancedness of these representations. In fact we study these representations as representations of the “generic” affine Hecke algebra of type ˜C2, thereby effectively analysing all possible choices of parameters simultaneously.
Once a balanced system of cell representations is established for each choice or parameters we are able to compute Lusztig’s a-function for type ˜C2, and combined with the explicit partition ofW into cells the conjectures P4, P8, P9, P10, P11, P12, and P14 readily follow.
The second main part of this paper is establishing an “asymptotic” Plancherel formula for type ˜C2, with our starting point being the explicit formulation of the Plancherel Theorem in type ˜C2 obtained by the second author in [20] (this is in turn a very special case of Opdam’s general Plancherel Theorem [19]). In particular we
Lusztig’s Conjectures forC˜2
show that in type ˜C2 there is a natural correspondence, in each parameter range, between two-sided cells appearing in the cell decomposition and the representations appearing in the Plancherel Theorem (these are thetempered representations ofH).
Moreover we define aq−1-valuationon the Plancherel measure, and show that in type C˜2 theq−1-valuation of the mass of a tempered representation is twice the value of Lusztig’sa-function on the associated cell. This observation allows us to introduce an asymptotic Plancherel measure, giving a descent of the Plancherel formula to Lusztig’s asymptotic algebraJ. In particular we obtain an inner product onJ, giving a satis- fying conceptual proof of P1 and P7. Moreover we are able to determine the setDof Duflo involutions, and conjectures P2, P3, P5, P6, and P13 follow naturally.
The remaining conjecture P15 is of a slightly different flavour. In [24] Xie has proved this conjecture under an assumption on Lusztig’s a-function. We are able to verify this assumption using our calculation of thea-function and the asymptotic Plancherel formula, hence proving P15 and completing the proof of all conjectures P1–P15.
We conclude this introduction with an outline of the structure of the paper. In Section 1 we recall the basics of Kazhdan–Lusztig theory, and we recall the axioms of a balanced system of cell representations from [13]. Section 2 provides background on affine Weyl groups, root systems, the affine Hecke algebra, and the combinatorics of alcove paths. In Section 3 we recall the partition of ˜C2 into cells for all choices of parameters from [12], and introduce some notions such as the generating set of a two-sided cell, cell factorisation and the ˜a-function. In Section 4 we define various representations of the affine Hecke algebra in preparation for the important Sections 5 and 6 where we establish the existence of the a balanced system of cell representations for each choice of parameters. The main work here is in Section 6, where we conduct a detailed combinatorial analysis of certain representations associated to the infinite two-sided cells. In Section 7 we establish connections between the Plancherel Theo- rem and the decomposition into cells, hence establishing the asymptotic Plancherel Theorem for type ˜C2. The proofs of P1–P15 are given progressively throughout the paper (see Corollaries 3.1, 6.2, 6.23, 7.9, 7.11, and Theorems 7.7 and 7.13).
1. Kazhdan–Lusztig theory and balanced cell representations In this section we recall the definition of the generic Hecke algebra and the setup of Kazhdan–Lusztig theory, including the Kazhdan–Lusztig basis, Kazhdan–Lusztig cells, and the Lusztig’s conjectures P1–P15. In this section (W, S) denotes an arbitrary Coxeter system (with |S|<∞) with length function `:W →N={0,1,2, . . .}. For I⊆S letWI be the standard parabolic subgroup generated by I.
1.1. Generic Hecke algebras and their specialisations. Let (qs)s∈S be a family of commuting invertible indeterminates with the property thatqs=qs0 when- eversands0are conjugate inW. LetRg=Z[(q±1s )s∈S]. Thegeneric Hecke algebra of type (W, S) is the Rg-algebra Hg with basis {Tw|w∈W} and multiplication given by (forw∈W ands∈S)
TwTs=
(Tws if`(ws) =`(w) + 1 Tws+ (qs−q−1s )Tw if`(ws) =`(w)−1.
(1)
We set qw :=qs1· · ·qsn where w=s1. . . sn ∈W is a reduced expression ofw. This can easily be seen to be independent of the choice of reduced expression (using Tits’
solution to the Word Problem).
Let L : W → N be a positive weight function on W. Thus L(w) > 0 for all w∈W different from the identity and L(ww0) =L(w) +L(w0) whenever `(ww0) =
`(w) +`(w0). Let qbe an invertible indeterminate and letR=Z[q,q−1] be the ring
Algebraic Combinatorics, Vol. 2 #5 (2019) 971
J. Guilhot & J. Parkinson
of Laurent polynomials in q. The Hecke algebra of type (W, S, L) is the R-algebra H=HLwith basis{Tw|w∈W}and multiplication given by (forw∈W ands∈S)
TwTs=
(Tws if`(ws) =`(w) + 1
Tws+ (qL(s)−q−L(s))Tw if`(ws) =`(w)−1.
(2)
We refer to (Tw)w∈W as the “standard basis” ofH. Of courseHis obtained fromHg
via the specialisationqs7→qL(s), with the multiplicative property of weight functions ensuring that this specialisation compatible with the fact that qs = qs0 whenever s and s0 are conjugate in W. For a given weight function L, we denote the above specialisation by ΘL:Hg→ H.
While Kazhdan–Lusztig theory is setup in terms of the specialised algebra H = HL, we will also need the generic algebraHg at times in this paper (particularly in Section 6). We sometimes writeQs=qs−q−1s , orQs=qL(s)−q−L(s)depending on context (particularly in matrices for typesetting purposes). IfS ={s0, . . . , sn}we will also often write, for example, 0121 as shorthand fors0s1s2s1, and thus in the Hecke algebraT0121=Ts0s1s2s1. In particular, note that 1 is shorthand fors1, and therefore to avoid confusion we denote the identity ofW bye.
1.2. The Kazhdan–Lusztig basis. Let L be a positive weight function and let H=HL. The involution ¯ onRwhich sendsqtoq−1can be extended to an involution onHby setting
X
w∈W
awTw= X
w∈W
awTw−1−1.
In [14], Kazhdan and Lusztig proved that there exists a unique basis{Cw|w∈W} ofHsuch that, for allw∈W,
Cw=Cw and Cw=Tw+X
y<w
Py,wTy where Py,w∈q−1Z[q−1].
This basis is called theKazhdan–Lusztig basis(KL basis for short) ofH. The polynomi- alsPy,ware called theKazhdan–Lusztig polynomials, and to complete the definition we setPw,w = 1 andPy,w= 0 whenevery6< w (here6denotes Bruhat order onW) and Pw,w= 1 for allw∈W. We note that the Kazhdan–Lusztig polynomials, and hence the elementsCw, depend on the the weight functionL(see the following example).
Example1.1.Let (W, S, L) be a Coxeter group and letJ ⊆Sbe such that the group WJ generated by J is finite. Let wJ be the longest element of WJ. The Kazhdan–
Lusztig element CwJ is equal to P
w∈WJqL(w)−L(wJ)Tw. Indeed, this element has the required triangularity with respect to the standard basis and it is stable under the bar involution. Further, if we set CwJ := P
w∈Wqwq−1wJTw ∈ Hg then we have ΘL(CwJ) =CwJ for all positive weight functionsLonW.
Now assume that S contains two elements s1, s2 such that (s1s2)4 =e. If we set a=L(s1) andb=L(s2) then we have
C212=
T212+q−b(T12+T21) + q−b−a−q−b+a
T2+q−2bT1 + q−2b−a−q−2b+a
Te ifb > a, T212+q−a(T21+T12) +q−2a(T1+T2) +q−3aTe ifa=b, T212+q−b(T12+T21) + q−a−b−q−a+b
T2+q−2bT1
+ q−a−q−a−2b
Te ifb < a.
Indeed, the expressions on the right-hand side are stable under the bar involution and since they have the required triangularity property, they have to be the Kazhdan–
Lusztig element associated to 212. Unlike the case wherew=wJ, there is no generic
Lusztig’s Conjectures forC˜2
element inHgthat specialises toC212∈ H(W, S, L) for all positive weight functionsL.
We also note that when b > a we have P2,212 = q−b−a−q−b+a, showing that the Kazhdan–Lusztig polynomials can have negative coefficients in the unequal parameter case.
Letx, y ∈W. We denote by hx,y,z ∈Rthe structure constants associated to the Kazhdan–Lusztig basis:
CxCy= X
z∈W
hx,y,zCz.
Definition1.2 ([17, Chapter 13]).TheLusztiga-functionis the functiona:W →N defined by
a(z) := min{n∈N|q−nhx,y,z∈Z[q−1] for allx, y∈W}.
WhenW is infinite it is, in general, unknown whether thea-function is well-defined.
However in the case of affine Weyl groups it is known thatais well-defined, and that a(z) 6 L(w0) where w0 is the longest element of the underlying finite Weyl group W0 (see [17]). Thea-function is a very important tool in the representation theory of Hecke algebras, and plays a crucial role in the work of Lusztig on the unipotent characters of reductive groups.
Definition1.3.Forx, y, z∈W letγx,y,z−1 denote the constant term ofq−a(z)hx,y,z. The coefficients γx,y,z−1 are the structure constants of the asymptotic algebra J introduced by Lusztig in [17, Chapter 18].
1.3. Kazhdan–Lusztig cells and associated representations. Define pre- orders6L,6R,6LRonW extending the following by transitivity:
x6Ly ⇐⇒ ∃ h∈ Hsuch thatCx appears in the KL expansion ofhCy, x6Ry ⇐⇒ ∃ h∈ Hsuch thatCx appears in the KL expansion ofhCy, x6LRy ⇐⇒ ∃ h, h0∈ Hsuch thatCxappears in the KL expansion ofhCyh0. We associate to these preorders equivalence relations ∼L, ∼R, and ∼LR by setting (for∗ ∈ {L,R,LR})
x∼∗yif and only if x6∗y andy6∗x.
The equivalence classes of ∼L, ∼R, and ∼LR are called left cells, right cells, and two-sided cells.
Example 1.4.For y, w ∈ W we write y w if and only if there exists x, z ∈ W such thatw=xyz and`(w) =`(x) +`(y) +`(y). In this case it is not hard to see, using the unitriangularity of the change of basis matrix from the standard basis to the Kazhdan–Lusztig basis, thatTxCyTz=Cw+P
z<wazCzand thereforew6LRy.
We denote byΛthe set of all two-sided cells (note that of courseΛdepends on the choice of weight function). Given any cell Γ (left, right, or two-sided) we set
Γ6∗:={w∈W |there existsx∈Γ such thatw6∗x}
and we define Γ>∗, Γ>∗ and Γ<∗ similarly.
To each right cell Υ of W there is a natural right H-module HΥ constructed as follows. TheR-modules
H6RΥ:=hCx|x∈Υ6Ri and H<RΥ:=hCx|x∈Υ<Ri are rightH-modules by definition and therefore the quotient
HΥ:=H6RΥ/H<RΥ
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is a right H-module with basis {Cw | w ∈ Υ} where Cw is the class ofCw in HΥ. Given a left cell (respectively a two-sided cell) we can follow a similar construction to produce leftH-modules (respectivelyH-bimodules).
1.4. Lusztig conjectures. Define ∆ :W →Nandnz∈Z\ {0}by the relation Pe,z =nzq−∆(z)+ strictly smaller powers ofq.
This is well defined becausePx,y∈q−1Z[q−1] for allx, y∈W. Let D={w∈W |∆(w) =a(w)}.
The elements ofDare calledDuflo elements(or, somewhat prematurely,Duflo invo- lutions; see P6 below).
In [17, Chapter 13], Lusztig has formulated the following 15 conjectures, now known as P1–P15.
P1. For anyz∈W we havea(z)6∆(z).
P2. Ifd∈ Dandx, y∈W satisfyγx,y,d6= 0, then y=x−1.
P3. Ifx∈W then there exists a uniqued∈ Dsuch thatγx,x−1,d6= 0.
P4. If z0 6LR z then a(z0) > a(z). In particular the a-function is constant on two-sided cells.
P5. Ifd∈ D,x∈W, andγx,x−1,d6= 0, thenγx,x−1,d=nd=±1.
P6. Ifd∈ Dthend2=e(the identity).
P7. For anyx, y, z∈W, we haveγx,y,z=γy,z,x.
P8. Let x, y, z ∈ W be such that γx,y,z 6= 0. Then x−1 ∼R y, y−1 ∼R z, and z−1∼R x.
P9. Ifz0 6L zanda(z0) =a(z), thenz0∼Lz.
P10. Ifz0 6R zanda(z0) =a(z), thenz0∼Rz.
P11. Ifz0 6LRzand a(z0) =a(z), thenz0 ∼LRz.
P12. IfI⊆S then thea-function ofWI is the restriction toWI of thea-function ofW.
P13. Each right cell Υ of W contains a unique element d ∈ D, and we have γx,x−1,d6= 0 for allx∈Υ.
P14. For eachz∈W we have z∼LRz−1.
P15. Ifx, x0, y, w∈W are such thata(w) =a(y) then X
y0∈W
hw,x0,y0⊗hx,y0,y= X
y0∈W
hy0,x0,y⊗hx,w,y0 in R⊗ZR.
1.5. Balanced system of cell representations. In [13] we introduced the no- tion of abalanced system of cell representations, inspired by the work of Geck [6, 8]
in the finite case. We recall this theory here.
IfSis anR-polynomial ring (including the possibilityS=R), we writeS60andS0 for the associatedZ[q−1]-polynomial andZ-polynomial subrings ofS, respectively. In particularR60=Z[q−1] andR0=Z. Let
sp|
q−1 =0:S60→S0 denote the specialisation atq−1= 0.
By a matrix representation of H we shall mean a triple (π,M,B) whereM is a right H-module over an R-polynomial ring S, and B is a basis ofM. We write (for h∈ Handu, v∈B)
π(h;B) and [π(h;B)]u,v
for the matrix ofπ(h) with respect to the basisB, and the (u, v)thentry ofπ(h;B).
Let deg(f(q)) denote the degree of the Laurent polynomial f(q) ∈ S (note that degree here refers to degree inq, not degree in the indeterminates of the polynomial
Lusztig’s Conjectures forC˜2
ringS). A matrix representation (π,M,B) is calledbounded if deg([π(Cw;B)]u,v) is bounded from above (for allu, v∈Band allw∈W). In this case we call the integer
aπ:= max{deg([π(Cw;B)]u,v)|u, v∈B, w∈W} (3)
thebound of the matrix representation and we define theleading matricesby cπ(w;B) := sp|
q−1 =0 q−aππ(Cw;B)
forw∈W . (4)
Definition 1.5.We say that H admits a balanced system of cell representations if for each two-sided cellΓ∈Λthere exists a matrix representation(πΓ,MΓ,BΓ)defined over an R-polynomial ring RΓ (where we could have RΓ =R) such that the following properties hold:
B1. If w /∈Γ>LR thenπΓ(Cw;BΓ) = 0.
B2. The matrix representation(πΓ,MΓ,BΓ)is bounded. LetaπΓdenote the bound.
B3. We havecπΓ(w;BΓ)6= 0 if and only if w∈Γ.
B4. The leading matricescπΓ(w;BΓ)(w∈Γ) are free over Z.
B5. For eachz∈Γthere existsx, y∈Γsuch thatγ˜x,y,z−16= 0, where˜γx,y,z−1 ∈Z is the coefficient ofqaπΓ inhx,y,z.
B6. If Γ06LRΓ thenaπ
Γ0 >aπΓ.
The natural numbers (aπΓ)Γ∈Λ are called the bounds of the balanced system of cell representations.
Remark 1.6.We make the following remarks:
(1) We note that B1 does not depend on the basis BΓ. A representation with property B1 is called acell representation for the two-sided cell Γ. It is clear that the representations associated to cells that we introduced in Section 1.3 are cell representations (see [13, Section 2.1]).
(2) If the basis BΓ of MΓ is clear from context we will sometimes writecπΓ(w) in place ofcπΓ(w;BΓ).
(3) By [13, Corollary 2.4] the axioms B1–B4 and B6 alone imply that theZ-span JΓ of the matricescπΓ(w;BΓ) withw∈Γ is aZ-algebra, and that
cπΓ(x;BΓ)cπΓ(y;BΓ) =X
z∈Γ
˜
γx,y,z−1cπΓ(z;BΓ) forx, y∈Γ
with ˜γx,y,z−1as defined in B5. Hence these integers are the structure constants of the algebraJΓ.
(4) We note that in (3) and (4) it is equivalent to replace Cw by Tw, because Cw=Tw+P
v<wPv,wTv withPv,w∈q−1Z[q−1]. However in B1 one cannot replaceCw byTw.
(5) Finally we note that we have slightly changed the numbering from [13], where B5 was denoted B40, and B6 was denoted B5.
In [13] we showed that the existence of a balanced system of cell representations is sufficient to compute Lusztig’sa-function. In particular, we have:
Theorem1.7 ([13, Theorem 2.5 and Corollary 2.6]).Suppose thatHadmits a balanced system of cell representations. Then a(w) = aπΓ for all w ∈Γ. Moreover, for each Γ∈Λ the Z-algebra JΓ spanned by the matrices{cπΓ(w;BΓ)|w∈Γ} is isomorphic to Lusztig’s asymptotic algebra associated to Γ, and˜γx,y,z=γx,y,z.
Note that the first part of this theorem implies that the boundsaπΓin Definition 1.5 are in fact unique. That is, if there exist two balanced systems of cell representations then their bounds coincide.
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2. Affine Weyl groups, affine Hecke algebras, and alcove paths We begin this section with some basic facts about root systems and Weyl groups. We then recall the combinatorial language of alcove paths from [21], and the concept of alcove paths confined to strips from [13]. We also discuss the combinatorics of the affine Hecke algebra (and extended affine Hecke algebra) of type ˜C2.
2.1. Root systems and Weyl groups. Let Φ be the non-reduced root system of typeBC2 in the vector spaceR2. Thus Φ consists of vectors
Φ = Φ+∪(−Φ+), where Φ+={α1, α2, α1+α2, α1+ 2α2,2α2,2(α1+α2)}, withkα1k=√
2,kα2k= 1, andhα1, α2i=−1. Let Φ0 and Φ1 be the subsystems Φ0=±{α1, α2, α1+α2, α1+ 2α2} and Φ1=±{α1,2α2, α1+ 2α2,2α1+ 2α2} of typesB2 andC2, respectively (see Figure 1).
Letα∨= 2α/hα, αi. The dual root system is
Φ∨=±{α∨1, α∨2/2, α∨1 +α∨2/2, α∨1 +α∨2, α∨2,2α∨1 +α∨2}.
The coroot lattice is theZ-latticeQspanned by Φ∨. Thus Q={mα∨1 +nα∨2/2|m, n∈Z}.
The fundamental coweightsω1 andω2 are defined byhωi, αji=δi,j, and thus ω1=α∨1 +α∨2/2 and ω2=α∨1 +α∨2.
In particular, note that ω1, ω2 ∈Q. LetQ+ be the cone Z>0ω1+Z>0ω2 (note that this notation is non-standard).
For eachα∈Φ let sα be the orthogonal reflection in the hyperplane Hα ={x∈ R2 | hx, αi = 0} orthogonal to α, and for i ∈ {1,2} let si = sαi. The Weyl group of Φ is the subgroupW0 of GL(R2) generated by the reflectionss1 ands2 (this is a Coxeter group of typeB2=C2). The Weyl groupW0 acts onQand theaffine Weyl group isW =QoW0 where we identify λ∈ Qwith the translationtλ(x) =x+λ.
The affine Weyl group is a Coxeter group with generating setS={s0, s1, s2}, where s0=tϕ∨sϕ, withϕ= 2α1+ 2α2 the highest root of Φ.
For each α∈ Φ and k∈Z letHα,k={x∈R2| hx, αi=k}, and let sα,k be the orthogonal reflection in the affine hyperplaneHα,k. Explicitly,sα,k(x) =x−(hx, αi − k)α∨. Each affine hyperplane Hα,k with α∈Φ+ andk∈Zdivides R2 into two half spaces, denoted
Hα,k+ ={x∈R2| hx, αi>k} and Hα,k− ={x∈R2| hx, αi6k}.
This “orientation” of the hyperplanes is called theperiodic orientation (see Figure 1).
Ifw∈W we define thelinear partθ(w)∈W0and thetranslation weightwt(w)∈Q by the equation
w=twt(w)θ(w).
LetF denote the union of the hyperplanesHα,k withα∈Φ andk∈Z. The closures of the open connected components of R2\ F are calledalcoves (these are the closed triangles in Figure 1). Thefundamental alcoveis given by
A0={x∈R2|06hx, αi61 for allα∈Φ+}.
The hyperplanes bounding A0 are called the walls ofA0. Explicitly these walls are Hαi,0withi= 1,2 andHϕ,1. We say that afaceofA0(that is, a codimension 1 facet) hastypesifori= 1,2 if it lies on the wallHαi,0and of types0if it lies on the wallHϕ,1. The affine Weyl group W acts simply transitively on the set of alcoves, and we use this action to identify the set of alcoves withW viaw↔wA0. Moreover, we use the action ofW to transfer the notions of walls, faces, and types of faces to arbitrary
Lusztig’s Conjectures forC˜2
alcoves. Alcoves A and A0 are called s-adjacent, written A ∼s A0, if A6=A0 andA and A0 share a common type s face. Thus under the identification of alcoves with elements ofW, the alcovesw andwsares-adjacent.
2 Affine Weyl groups, affine Hecke algebras, and alcove paths 6
2.1 Root systems and Weyl groups
LetΦbe the non-reduced root system of typeBC2in the vector spaceR2. ThusΦconsists of vectors Φ = Φ+∪(−Φ+), where Φ+={α1, α2, α1+α2, α1+ 2α2,2α2,2(α1+α2)}, withkα1k=√
2,kα2k= 1, andhα1, α2i=−1. LetΦ0 andΦ1 be the subsystems
Φ0=±{α1, α2, α1+α2, α1+ 2α2} and Φ1=±{α1,2α2, α1+ 2α2,2α1+ 2α2} of typesB2andC2, respectively.
Letα∨= 2α/hα, αi. The dual root system is
Φ∨=±{α∨1, α∨2/2, α∨1 +α∨2/2, α∨1 +α∨2, α∨2,2α∨1 +α∨2}. The corrot lattice is theZ-latticeQspanned byΦ∨. Thus
Q={mα∨1 +nα∨2/2|m, n∈Z}. The fundamental coweightsω1andω2 are defined byhωi, αji=δi,j, and thus
ω1=α∨1 +α∨2/2 and ω2=α∨1 +α∨2.
In particular, note thatω1, ω2∈Q. LetQ+be the coneZ≥0ω1+Z≥0ω2(note that this notation is non-standard).
For eachα∈Φletsαbe the orthogonal reflection in the hyperplaneHα={x∈R2 | hx, αi= 0}orthogonal toα, and fori∈ {1,2} letsi=sαi. TheWeyl group of Φis the subgroupW0 ofGL(R2)generated by the reflectionss1ands2
(this is a Coxeter group of typeB2=C2). The Weyl groupW0 acts onQand theaffine Weyl group isW =Q⋊W0
where we identifyλ∈Qwith the translationtλ(x) =x+λ. The affine Weyl group is a Coxeter group with generating setS={s0, s1, s2}, wheres0=tϕ∨sϕ, withϕ= 2α1+ 2α2the highest root ofΦ.
For each α∈Φ and k∈Z let Hα,k = {x ∈R2 | hx, αi= k}, and let sα,k be the orthogonal reflection in the affine hyperplaneHα,k. Explicitly,sα,k(x) =x−(hx, αi −k)α∨. Each affine hyperplaneHα,kwithα∈Φ+andk∈Zdivides R2into two half spaces, denoted
Hα,k+ ={x∈R2| hx, αi ≥k} and Hα,k− ={x∈R2| hx, αi ≤k}. This “orientation” of the hyperplanes is called theperiodic orientation(see Figure 1).
Ifw∈W we define thelinear partθ(w)∈W0and thetranslation weightwt(w)∈Qby the equation w=twt(w)θ(w).
LetF denote the union of the hyperplanesHα,kwithα∈Φandk∈Z. The closures of the open connected components ofR2\Fare calledalcoves(these are the closed triangles in Figure 1). Thefundamental alcove is given by
A0={x∈R2|0≤ hx, αi ≤1for allα∈Φ+}.
The hyperplanes boundingA0 are called thewalls ofA0. Explicitly these walls areHαi,0withi= 1,2 andHϕ,1. We say that aface ofA0(that is, a codimension1facet) hastype sifori= 1,2if it lies on the wallHαi,0and of types0if it lies on the wallHϕ,1.
The affine Weyl groupW acts simply transitively on the set of alcoves, and we use this action to identify the set of alcoves withW viaw↔wA0. Moreover, we use the action ofW to transfer the notions of walls, faces, and types of faces to arbitrary alcoves. AlcovesAandA′ are calleds-adjacent, writtenA∼s A′, if A6=A′ andAandA′ share a common typesface. Thus under the identification of alcoves with elements ofW, the alcoveswandwsares-adjacent.
α1=α∨1
ω2
α∨2/2 2α2
ω1
•
•
•
•
•
•
•
•
•
•
•
•
• s1
s2
s0
+
− +
−
+
−
− +
− + − +
+
− +
−
+
−
−
+ + − + −
e s1
s2
s0
Fig. 1: Root system of typeFigure 1. BCRoot system of type2, periodic orientation, and adjacency types (dotted, dashed, solidBC2, periodic orientation, and ad- =0,1,2) jacency types (dotted, dashed, solid = 0,1,2)
2.2. Alcove paths. For any sequencew~ = (si1, si2, . . . , si`) of elements ofSwe have e∼si1 si1 ∼si2 si1si2 ∼si3 · · · ∼si` si1si2· · ·si`.
In this way, sequences w~ of elements of S determine alcove paths (also called al- cove walks) of type w~ starting at the fundamental alcove e = A0. We will typi- cally abuse notation and refer to alcove paths of typew~ =si1si2· · ·si` rather than
~
w= (si1, si2, . . . , si`). Thus “the alcove path of typew~ =si1si2· · ·si`” is the sequence (v0, v1, . . . , v`) of alcoves, wherev0=eandvk =si1· · ·sik fork= 1, . . . , `.
Letw~ =si1si2· · ·si`be an expression forw∈W, and letv∈W. Apositively folded alcove path of typew~ starting atvis a sequencep= (v0, v1, . . . , v`) withv0, . . . , v`∈W such that
(1) v0=v,
(2) vk ∈ {vk−1, vk−1sik} for eachk= 1, . . . , `, and
(3) if vk−1 = vk then vk−1 is on the positive side of the hyperplane separating vk−1 andvk−1sik.
Theend ofpis end(p) =v`. Let wt(p) = wt(end(p)) andθ(p) =θ(end(p)). Let P(w, v) =~ {all positively folded alcove paths of type w~ starting atv}.
Less formally, apositively folded alcove path of typew~ starting atvis a sequence of steps from alcove to alcove inW, starting atv, and made up of the symbols (where thekth step hass=sik fork= 1, . . . , `):
2 Affine Weyl groups, affine Hecke algebras, and alcove paths 7
2.2 Alcove paths
For any sequencew~ = (si1, si2, . . . , siℓ)of elements ofS we have
e∼si1 si1∼si2 si1si2∼si3 · · · ∼siℓ si1si2· · ·siℓ.
In this way, sequences w~ of elements ofS determinealcove paths (also called alcove walks) of type w~ starting at the fundamental alcovee=A0. We will typically abuse notation and refer to alcove paths of typew~ =si1si2· · ·siℓ rather thanw~ = (si1, si2, . . . , siℓ). Thus “the alcove path of typew~ =si1si2· · ·siℓ” is the sequence(v0, v1, . . . , vℓ)of alcoves, wherev0=eandvk=si1· · ·sik fork= 1, . . . , ℓ.
Letw~=si1si2· · ·siℓ be an expression forw∈W, and letv∈W. Apositively folded alcove path of typew~ starting atv is a sequencep= (v0, v1, . . . , vℓ)withv0, . . . , vℓ∈W such that
1) v0 =v,
2) vk∈ {vk−1, vk−1sik}for eachk= 1, . . . , ℓ, and
3) ifvk−1=vk thenvk−1 is on the positive side of the hyperplane separatingvk−1 andvk−1sik. Theend ofpisend(p) =vℓ. Letwt(p) = wt(end(p))andθ(p) =θ(end(p)). Let
P(w, v) =~ {all positively folded alcove paths of typew~ starting atv}.
Less formally, a positively folded alcove path of typew~ starting at vis a sequence of steps from alcove to alcove in W, starting atv, and made up of the symbols (where thekth step hass=sik fork= 1, . . . , ℓ):
−
x xs
+
(positives-crossing)
−
xs x
+
(positives-fold)
+ x xs
−
(negatives-crossing)
Ifphas no folds we say thatpisstraight. Note that, by definition, there are no “negative” folds.
Ifpis a positively folded alcove path we define, for eachsj∈S,
fj(p) = #(positivesj-folds inp).
2.3 Alcove paths confined to strips
Letα′1=α1andα′2= 2α2 (these are the simple roots ofΦ1). Fori∈ {1,2}let Ui={x∈R2|0≤ hx, α′ii ≤1} be the region between the hyperplanesHα′
i,0 andHα′
i,1. It is also convenient to defineU3=U2.
Letw~=si1· · ·siℓ be an expression forw∈W. Leti∈ {1,2,3}. Ani-folded alcove path of typew~ starting atv∈ Uiis a sequencep= (v0, v1, . . . , vℓ)withv0, . . . , vℓ∈ Uisuch that
1) v0 =v, andvk∈ {vk−1, vk−1sik}for eachk= 1, . . . , ℓ, and 2) ifvk−1=vk then either:
(a) vk−1sik∈ U/ i, or
(b) vk−1 is on the positive side of the hyperplane separatingvk−1 andvk−1sik. We note that condition 2)(a) can only occur ifvk−1 andvk−1sik are separated by eitherHα′
i,0 orHα′ i,1. Theend of thei-folded alcove pathp= (v0, . . . , vℓ)isend(p) =vℓ. Let
Pi(w, v) =~ {alli-folded alcove paths of typew~ starting atv}.
Less formally,i-folded alcove paths are made up of the following symbols, wherex∈ Uiands∈S:
x− xs +
(positives-crossing)
xs− x +
(s-fold)
+ x xs−
(negatives-crossing) (a)When the alcovesxandxsboth belong toUi
+ xs x−
(s-bounce)
xs− x +
(s-bounce)
(b)Whenxslies outside ofUi We refer to the two symbols in (b) as “s-bounces” rather than folds, since they play a different role in the theory. It turns out that there is no need to distinguish between “positive” and “negative” s-bounces. We note that bounces only
If phas no folds we say that p is straight. Note that, by definition, there are no
“negative” folds.
Algebraic Combinatorics, Vol. 2 #5 (2019) 977
