The structure of local regions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 13 November 2012

The structure of local regions

Recall that the Weyl group acts on

T=Hom(X,*)= { group homomorphismst:X* } by(wt) (Xλ)=t (Xw-1λ).

Any element tT is determined by the values t(Xω1), t(Xω2),, t(Xωn). For tT define the polar decomposition

t=trtc, tr,tcT such thattr(Xλ) >0and tc (Xλ) =1,

for all XλX. There is a unique γn and a unique vn/P such that

tr(Xλ)= eγ,λ andtc (Xλ)= e2πiv,λ for allλP. (4.1)

In this way we identify the sets Tr= { tT t=tr } and Tc= { tT t=tc } with 𝔥* and 𝔥*/P, respectively.

For this paragraph (our goal here is (4.3) below) assume that q is not a root of unity (we will treat the type A, root of unity case in detail in Section 7). The representation theory of H is “the same” for any q which is not a root of unity, i.e. provided q is not a root of unity, the classification and construction of simple H-modules can be stated uniformly in terms of the parameter q. Suppose tT is such that t=tr and γ𝔥* is such that

t=eγ, in the sense thatt(Xλ) =eγ,λ for allXλX.

For the purposes of representation theory (as in Theorem 3.5) t indexes a central character and so we should assume that γ is chosen nicely in its W-orbit. When

q=eandγ is dominant,i.e., γ,α0 for allαR+, (4.2)


Z(t)=Z(γ), P(t)=P(γ) ,and (t,J)= (γ,J) for a subsetJP(t),


Z(γ)= { αR+ γ,α=0 } ,P(γ)= { αR+ γ,α=1 } , (γ,J)= { wW R(w)Z(γ) =,R(w) P(γ)=J } . (4.3)

In this case the combinatorics of local regions is a new chapter in the combinatorics of the Shi arrangement defined in (1.16). Other aspects of the combinatorics of the Shi arrangement can be found in [ALi1999,33–35,37–39], and there are several additional places in the literature [Shi1987], [Xi1994, 1.11, 2.6], [Kos2000,KOP2000] which indicate that there is a deep (and not yet completely understood) connection between the structure and representation theory of the affine Hecke algebra and the combinatorics of the Shi arrangement.

4.4. Intervals in Bruhat order

Using the formulation in (4.3), Theorem 4.6 will give a complete description of the structure of (γ,J) as a subset of the Weyl group when q is not a root of unity. We will treat the type A, root of unity cases in Section 7.

The weak Bruhat order is the partial order on W given by

vwifR(v) R(w), (4.5)

where R(w) denotes the inversion set of wW as defined in (1.5). This definition of the weak Bruhat order is not the usual definition but is equivalent to the usual one by [Bjo1994, Proposition 2]. A set of positive roots K is closed if α,βK,α+βR+ implies that α+βK. The closure K of a subset KR+ is the smallest closed subset of R+ containing K. A set of positive roots KR+ is the inversion set of some permutation wW if and only if K is closed and Kc=R+\K is closed (see [Bjo1984, Proposition 2] or [KRa2002, Theorem 5.1]).

The following theorem is proved in [KRa2002, Section 5]. The proof of part (b) of the theorem relies crucially on a theorem of J. Losonczy [Los1999].

Theorem 4.6. Let γ𝔥* be dominant (i.e., γ,α0 for all αR+) and let JP(γ). Let (γ,J) be as given in (4.3).

  1. Then (γ,J) is nonempty if and only if J satisfies the condition ifβJ,αZ (γ)and β-αR+then β-αJ.
  2. The sub-root system R[γ]= { αR γ,α } , has Weyl group W[γ]= sα αR[γ] and if W[γ]= { σWR (σ)R[γ] = } then (γ,J)= W[γ]· [ τmax, τmin ] , where τmax,τminW[γ] are determined by R(τmax) R[γ]=J andR(τmin) R[γ]= (P(γ)\J) Z(γ)c , the complement is taken in the set of positive roots of R[γ], and [τmin,τmax] denotes the interval between wmin and wmax in the weak Bruhat order in W[γ].

4.7. Conjugation

Assume that γ is dominant (i.e., γ,α0 for all αR+) and JP(γ). Let (γ,J) be as given in (4.3). The conjugate of (γ,J) and of w(γ,J) are defined by

(γ,J)= ( -uγ,-u (P(γ)\J) ) and (t,J) 1-1 (t,J) w w=wu-1, (4.8)

where u is the minimal length coset representative of w0WγW/Wγ and w0 is the longest element of W. In Section 6.7 we shall show that these maps are generalizations of the classical conjugation operation on partitions.

Theorem 4.9. The conjugation maps defined in (4.8) are well defined involutions.


(a) Since γ is dominant, -uγ=-w0γ is dominant and thus -uγ,-uα=1 only if -uα>0. Thus the equation -uγ,-uα =1γ,α =1 gives that P(-uγ)=-uP(γ).

(b) Let vWγ such that w0=uv. (By [Bou1968, IV Section 1 Exercise 3], v is unique.) Then R+-w0Z(γ) =-uvZ(γ)=uZ (γ), and it follows that

Z(-uγ)=R+ { αR uγ,α =0 } =R+ ( uZ(γ)-uZ(γ) ) =uZ(γ).

(c) Let R-=-R+ be the set of negative roots in R. Let vWγ such that w0=uv. Then v is the longest element of Wγ and R(v)=Z(γ). Thus, since w0R-=R+,

R(u) = { αR αR+,w0v αR- } = { αRαR+ ,vαR+ } , = R+\R(v)=R+ \Z(γ).

(d) The weight -uγ=-uvγ=-w0γ is dominant and -u(P(γ)\J) P(-uγ) since -uP(γ)=P(-uγ). This shows that (γ,J) is well defined.

(e) Write w0=uv where v is the longest element of Wγ. Similarly, write w0=uv where u is the minimal length coset representative of w0Ww0γ and v is the longest element in Ww0γ. Conjugation by w0 is an involution on W which takes simple reflections to simple reflections and Ww0γ=w0 Wγw0. It follows that v=w0vw0. This gives

uu=(w0v) (w0v)=w0w0 vw0w0v=1,

and so the second map in (4.8) is an involution.

(f) Using (e) and (a),

-u ( P(-uγ)\ ( -u ( P(γ)\J ) ) ) = -u ( -uP(γ)\ ( -u (P(γ)\J) ) ) = P(γ)\ (P(γ)\J) =J,

and so the first map in (4.8) is an invoution.

(g) Let w𝒢(γ,J) and let w=wu-1. Since R(w)Z(γ)=,

u-1R (wu-1) Z(γ) = { βR uβR (wu-1), βZ(γ) } = { βR uβR+,wu-1 uβR-, βZ(γ) } = { βR βu-1R+, wβR-,β Z(γ) } = { βR βu-1R+, βR(w),β Z(γ) } (sinceZ(γ) R+) = { βR βu-1R+, βR(w)Z(γ) } = ,

and thus, by (b),

R(w)Z(-uγ) =R(wu-1)uZ (γ)=u ( u-1R(wu-1) Z(γ) ) =.

Since R(w)P(γ)=J,

-u-1R (wu-1) P(γ) = { βR -uβR (wu-1), βP(γ) } = { βR -uβR+, -wu-1 uβR-, βP(γ) } = { βR uβR-, wβR+,β P(γ) } = { βR βR(u), βR+\R(w), βP(γ) } (sinceP(γ)R+) = { βR βR+\Z(γ), βR+\R(w), βP(γ) } = { βR βR+\Z(γ), βP(γ)\J } (sinceR(w) P(γ)=J) = P(γ)\J,since Z(γ)and P(γ)are disjoint.

Thus, by (a),

R(w) P(-uγ) = R(wu-1) -uP(γ)=-u ( -u-1R (wu-1) P(γ) ) = -u(P(γ)\J),

and so the second map in (4.8) is well defined.

Remark 4.10. In type A, the conjugation involution coincides with the duality operation for representations of 𝔭-adic GL(n) defined by Zelevinsky [Zel1980]. Zelevinsky’s involution has been studied further in [KZe1996,LTV1999,MWa1986] and extended to general Lie type by Kato [Kat1993] and Aubert [Aub1995]. For H-modules in type A, this is the involution on modules induced by the Iwahori–Matsumoto involution of H and is detected on the level of characters: it sends an irreducible H-module L to the unique irreducible L* with dim ((L*)tgen)= dim ((L*)t-1gen) for each tT. I would like to thank J. Brundan for clarifying this remark and making it precise.

Hα1 Hα2 Hα1+2α2 Hα1+α2+δ Hα1+α2 Hα1+2α2+δ Hα1+2α2-δ Hα2+δ Hα2-δ Hα1-δ Hα1+δ Hα1+α2-δ γ s2s1s2γ s1s2γ s2γ Fig. 2.

Examples 4.11. (a) If γ is dominant and is generic (as an element of C) then Z(γ)=P(γ)= and (γ,)=W.

(b) Let ρ be defined by ρ,αi=1, for all 1in. Then

Z(ρ)=, P(ρ)= {α1,,αn}, and(ρ,J)= { wWD(w) =J } ,

where D(w)= { αiwsi <w } is the right descent set of wW. The sets (γ,J) which arise here are fundamental to the theory of descent algebras [GRe1989,Reu1993,Sol1976].

(c) This example is a generalization of (b). Suppose that (γ,J) is a local region such that γ is regular and integral (i.e., γ,α>0 for all αR+). Then

Z(γ)=,P (γ) {α1,,αn}, and(γ,J)= { wW D(w)P(γ)=J } .

(d) Let R be the root system of type C2 with simple roots α1=ε1α2 =ε2-ε1, where {ε1,ε2} is an orthonormal basis of 𝔥*=2. The positive roots are R+= { α1,α2,α1 +α2,α1+2 α2 } . Let γ2 be given by γ,α1=0 and γ,α2=1. Then γ is dominant (i.e., in C) and integral and

Z(γ)={α1} andP(γ)= {α2,α1+α2} . Hα1 Hα2 Hα1+2α2 Hα1+α2 J={α2,α1+α2} J={α2} J= C s2s1s2C s2s1C s2C Fig. 3.

Figure 3 displays the local regions (γ,J) as regions in 𝔥*, see the remarks after (2.18).

The solid line is the hyperplane corresponding to the root in Z(γ) and the dashed lines are the hyperplanes corresponding to the roots in P(γ).

(e) Let R be the root system of type C2 as in (d). Let γ2 be defined by

γ,α1 =0, γ,α2= 12.


Z(γ)={α1}, P(γ)= {α1+2α2}.

If J=P(γ) then the unique minimal element wmin of (γ,J) has R(wmin)= {α2,α1+2α2} J=J.

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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