Unique Functionals and Representations ofHecke Algebras
Benjamin Brubaker, Daniel Bump and Solomon Friedberg
October 5, 2012
To the memory of Jonathan David Rogawski
Abstract
In [24], Rogawski used the affine Hecke algebra to model the intertwining operators of unramified principal series representations of
p-adic groups. On the other hand, a representation of this Hecke algebra in which the standard generators act by Demazure-Lusztig operators was introduced by Lusztig [18] and applied by Kazhdan andLusztig [17] to prove the Deligne-Langlands conjecture. These operators appear in various other contexts. Ion [11] used them to express
matrix coefficients of principal series representations in terms of nonsymmetric Macdonald polynomials, while Brubaker, Bump and Licata[3] found essentially the same operators underlying recursive relationships for Whittaker functions. In this article, we will explain the roleof unique functionals and Hecke algebras in these contexts and revisit
the results of Ion [11] from the point of view of [3].

Keywords: Hecke algebra, unramified principal series, Demazure-Lusztig
operator, unique functional.
AMS subject classification: Primary 22E50, Secondary 22E35, 33D52.

1Introduction

One of the innovations in Rogawski [24] was the use of the Hecke algebra
to model the intertwining operators of unramified principal series representations of p-adic groups. His goal was the classification of the irreducible1representations of the Hecke algebra, or equivalently, the irreducible representations of a p-adic group having an Iwahori fixed vector. These hadalready been classified in the case of GLn by Zelevinsky [27]. It was known(Zelevinsky [28]) that there were analogies between this problem and the decomposition of Verma modules of a semisimple Lie algebra into irreducibles,
where deep connections with the theory of Hecke algebras had been foundby Kazhdan and Lusztig [15]. Rogawski sought to clarify the relationshipbetween p-adic representation theory and Kazhdan-Lusztig theory. As partof this effort, he expressed intertwining integrals between principal series
representations in terms of Kazhdan-Lusztig elements of the Hecke algebra.
Among the tools that had been brought to bear on the study of Vermamodules, Jantzen [14] had introduced a filtration of the Verma module basedon the Shapovalov bilinear form. With the analogy between p-adic groups
and Verma modules in mind, Rogawski gave an analog of the Jantzen filtration, and also reproved the results of Zelevinsky [27] for GLn using Hecke
algebra methods.
After Rogawski’s paper was written, Zelevinsky’s results were generalized
to arbitrary p-adic groups by Kazhdan and Lusztig [16, 17], who proved theLanglands-Deligne conjecture classifying the irreducible representations of ap-adic group G that have Iwahori fixed vectors. They made use of a representation of the affine Hecke algebra HJ on the ring of rational functions on
the maximal torus T̂ of the L-group Ĝ. In this action, the generators of the
Hecke algebra act by certain operators known as Demazure-Lusztig operators, which resemble the well-known Demazure operators that occur in thecohomology of line bundles over Schubert varieties. Given data consistingof a pair (s, u) of elements of Ĝ such that s is semisimple, u unipotent, andsus−1 = uq , where q is the cardinality of the residue field, a subquotient of
this representation can be found that gives an irreducible HJ -module. This
module may be identified with the space of Iwahori fixed vectors in an irreducible representation of G(F ). To prove that this gives every irreducible
representation of G(F ) uniquely, thus proving the Deligne-Lusztig conjecture,
Kazhdan and Lusztig made use of the “coincidence” that the same representation of the Hecke algebra by Demazure-Lusztig operators also occurs in the
K-theory of flag varieties, allowing statements about representations to betranslated into algebraic geometry, where suitable methods are available.
The representation of the Hecke algebra by Demazure-Lusztig operatorscomes up in yet another context, namely the study of special functions on ap-adic group realized as matrix coefficients involving Iwahori fixed vectors.
2

Our first task, after some preliminaries, will be to briefly retrace Rogawski’s
steps and to discuss the relationship between the Hecke algebra and the intertwining operators. The intertwining operators involve different principal
series representations, which must be taken together to obtain a representation of the Hecke algebra. We express this by saying that the principal series
representation is variable in this representation of the Hecke algebra.
As we will explain in Section 3, this representation may be converted into
something concrete by introducing a family of functionals on the principal
series representations. We will consider two particular such families: theWhittaker functionals, and the spherical functionals. The role of the uniquefunctional is that it converts the Iwahori fixed vectors in a variable principal
series representation into a family of regular functions on T̂ . The problem ofvariability of the principal series representation disappears, and the action of
the Hecke algebra generators is by some variant (depending on the functional)of the Demazure-Lusztig operator. The regularity of the functions obtained
this way is related to Bernstein’s method of showing that a unique functional
defined (typically by a suitable integral) on an open subspace extends toa meromorphic function for all Langlands parameters. (See, for example,
Banks [1] or Sakellaridis [25], Sect. 7 for Bernstein’s method.)
Both the spherical and the Whittaker functionals are defined and nonzero on a Zariski dense set of T̂ . By contrast, one may consider functionals
defined and non-zero on only a subset of T̂ that is not Zariski dense. An
example is the Shalika functional on representations of GLn , proved unique
by Jacquet and Rallis [13], with a Casselman-Shalika type formula established by Sakellaridis [25]. The Shalika functional, like the Bessel models forclassical groups, exhibits characteristics of both the spherical and Whittaker
functionals and can be studied using our methods. We hope to return to thisin a later paper.
Shortly before Rogawski’s paper [24], intertwining operators had beenused for another purpose, namely the computation of the spherical vector ineach of our two archetypal models. For the spherical model, Casselman [4]used intertwining operators to give a new proof of the Macdonald formulawhich expresses the values of the spherical function as specializations of Macdonald symmetric functions. This specialization is the Hall-Littlewood polynomial in the case of GLn . Later nonsymmetric Macdonald polynomials
were defined in Macdonald [20], Opdam [21] and Cherednik [6]. Generalizing
the Macdonald formula, Ion [11] showed that the Iwahori-fixed vectors in
the model are expressible in terms of nonsymmetric Macdonald polynomials,
3making use of recursions that they satisfy involving Demazure-Lusztig operators. See also Cherednik and Ostrik [7] for earlier hints of this connection.
For the Whittaker functional, Casselman and Shalika [5] used the intertwining operators to show that the values of the spherical Whittaker function
are the irreducible characters of Ĝ, multiplied by a factor which is a deformation of the denominator in the Weyl character formula. Regarding the moregeneral Iwahori-fixed vectors in the model, Reeder [22, 23] used the Hecke
algebra action to give recursions between these, usable for explicit computation. These relations can be understood in terms of Demazure-Lusztig
operators as proved by Brubaker, Bump and Licata [3].
In Section 2 we review the relation between the Iwahori Hecke algebra
and the intertwining operators for principal series. Then in Section 3 weshow that, given a unique functional on a Zariski dense subset of T̂ , to eachgenerator Ti of the Iwahori Hecke algebra we may attach a difference operator
on a suitable ring of regular functions that is similar to a Demazure operator.
This extends to an action of the Hecke algebra. In the case of the sphericalfunctional, this gives a new perspective on the work of Ion [11].
We thank David Kazhdan for helpful conversations regarding universal
principal series and Daniel Orr for discussions on nonsymmetric Macdonald polynomials. This work was partially supported by NSF grants DMS0844185, DMS-1001079, and DMS-1001326 and by NSA grant H98230-10-10183.

2

Hecke algebras and intertwiners of principal series

Let G be a reductive algebraic group defined and split over a nonarchimedean
local field F with ring o of integers and prime p. We may regard G as
defined over o in such a way that K = G(o) is a special maximal compact
subgroup. Let q = |o/p|. Let T be a split maximal torus contained in Borel
subgroup B, and let W0 = N (T )/T be the Weyl group. We will always
choose representatives for W0 from N (T ) ∩ K.
The connected L-group Ĝ is an algebraic group defined over C with amaximal torus T̂ that is in duality with T in the sense that elements of T̂ (C)are in bijection with the unramified characters of T (F ), that is, those thatare trivial on T (F ) ∩ K = T (o). Let J be the Iwahori subgroup, which is the4preimage in K of B(Fq ) under the canonical homomorphism G(o) −→ G(Fq ).
Let s1 , · · · , sr be the simple reflections in the Weyl group W0 . The affine
Weyl group Waff is obtained by adjoining one more “affine” simple reflections0 (Bourbaki [2], Section VI.2). It is the semidirect product of W0 by theroot lattice Q∨ of T̂ , which is the coroot lattice of T . The groups W0 and
Waff are Coxeter groups. The extended affine Weyl group Wext is a slightly
larger group that is the semidirect product of W0 by the weight lattice P ∨of T̂ , which is the coweight lattice of T , isomorphic to T (F )/T (o). In thispresentation, P ∨ is a normal subgroup. We will denote by λ 7−→ aλ a map
sending λ ∈ P ∨ to a representative of the corresponding coset in T (F )/T (o).
The group Wext is not a Coxeter group, but like Coxeter groups it has alength function. The finite subgroup Ω of elements of length 0 is isomorphic
to P ∨ /Q∨ . (For example, if G is semisimple of adjoint type, then Ω isisomorphic to the fundamental group of G.) The group Wext is the semidirect
product of Waff by Ω, with Waff being a normal subgroup. Conjugation by
an element of Ω permutes the si .
The (affine) Iwahori Hecke algebra HJ is the convolution algebra of compactly supported J-biinvariant functions on G(F ). Let H0 be the subringof function with support in K. Then H0 and HJ have the following explicit
description due to Iwahori and Matsumoto [12]. If r is the rank of G, the ring
H0 is generated by T1 , · · · , Tr where each Ti is the characteristic function of
Jsi J. These Ti then satisfy quadratic relations
Ti2 = (q − 1)Ti + q(1)

and the braid relations
Ti Tj Ti · · · = Tj Ti Tj · · ·where the number of terms on each side is the order of si sj . The affine
Hecke algebra Haff is obtained by adjoining an element T0 satisfying thesame quadratic and braid relations but allowing i = 0 corresponding to theaffine simple reflection s0 . The algebra HJ is slightly larger than Haff , and isisomorphic as a vector space to Haff ⊗ C[Ω]. The elements of Ω act on Haffby conjugation, and this action corresponds to permuting the Ti , just as in
Wext conjugation by elements of Ω permutes the si .
The algebra HJ also has a presentation analogous to the presentation of
Wext as the semidirect product of P ∨ by W0 . This presentation, sometimes
known as the Bernstein presentation was developed but not published by5

Bernstein and Zelevinsky. Possibly its first use in a published paper wasin Rogawski [24], and a full treatment was given by Lusztig [18]. In theBernstein presentation H0 is supplemented by a ring homomorphism θ :C[P ∨ ] −→ HJ . As a vector space HJ = H0 ⊗ C[P ∨ ]. To describe the ring
structure, it is sufficient to give one relation. Let 1 6 i 6 r and let λ ∈ P ∨ .
Thenθ(λ) − θ(si λ),θ(λ)Ti − Ti θ(si λ) = (q − 1)
1 − θ(−αi∨ )where αi∨ is the coroot corresponding to i.
Let z ∈ T̂ (C), and let τ = τz : T (F ) −→ C× be the corresponding
unramified character. We extend it to B(F ) by letting the unipotent radical
N (F ) be in the kernel. The principal series representation M (τ ) consists of
all locally constant maps f : G(F ) −→ C such that f (bg) = (δ 1/2 τ )(b) f (g)for b ∈ B(F ), g ∈ G(F ). The action of G(F ) is by right translation. The
module M (τ ) is irreducible if τ is in general position.
If (π, V ) is an irreducible representation having a J-fixed vector, thenV J is a finite-dimensional irreducible HJ module and its isomorphism classdetermines π. Any such (π, V ) with a J-fixed vector is a subquotient of M (τ )for some τ , and the category of smooth representations of finite length all of
whose composition factors have J-fixed vectors is equivalent to the categoryof finite-dimensional HJ -modules.
The Weyl group W0 acts on T and hence on unramified characters. We
will make this a right action, so τ w(a) = τ (waw−1 ) for a ∈ T (F ). If w ∈ W0 ,the modules M (τ ) and M (τ w) are isomorphic if irreducible, and in any casehave isomorphic semisimplifications. In order to see this, one may constructhomomorphisms Aw : M (τ ) −→ M (τ w) by means of intertwining integrals.
By definitionZ
Aw f (g) =f (wng) dn(2)
N ∩w−1 N− w

where N− is the unipotent radical of the negative Borel. The integral isconvergent if |τ (aλ )| < 1 for dominant weights λ. By the singular set wemean the union of hyperplanes in T̂ (C) that are the kernels of the corootsα∨ (regarded as characters of T̂ ). For arbitrary τ , the intertwining operator
may be defined by analytic continuation, except that the Aw can have polesin the singular set.
Now we return to the point in question: why may Hecke algebras be used
to model intertwining operators? The basic insight is that, to any ring R
6

regarded as a left R-module, a left R-module homomorphism λ : R −→ R isnecessarily of the form λ(x) = x · a for some a ∈ R. This is trivial to provewith a = λ(1).
The first way of applying this is to note that the space M (τ )J of Jinvariants is |W0 |-dimensional. It has several natural bases indexed by Weylgroup elements. A particular one is the basis Φτw = Φw (w ∈ W0 ) defined by 1/2
δ τ (b) if k ∈ BwJ,τΦw (bk) =0otherwisefor b ∈ B(F ) and k ∈ K.
We see that H0 and M (τ )J are both |W0 |-dimensional H0 -modules, andin fact they are isomorphic as left H0 -modules. A particular isomorphism%τ : M (τ )J −→ H0 is given by %τ (f ) = F wheref (g −1 ) if g ∈ K,
F (g) =
0otherwise .
It is not hard to check that this is an isomorphism of left H0 -modules.
Composing with this isomorphism, the intertwining integral Aw thus gives
rise to a homomorphism H0 −→ H0 . This can have poles (in the singular set)or zeros (if τ (aα∨ ) = q ±1 for some coroot α∨ ) but if τ is in general position it
is an isomorphism and so it agrees with right multiplication by a particularelement Fw of H0 , which was identified by Rogawski [24]. It is sufficient todescribe it when w = si is a simple reflection, and in this case1
Fsi = (Ti + 1) − Cαi (τ )qwhere Cαi (τ ) with τ = τz is the ubiquitous rational function∨

1 − q −1 zαi
Cαi (τ ) =.
∨1 − z αi(3)

In other words, the following diagram is commutative:%τM (τ )J −−−→

As
y i
%τ s

H0

·Fs
y i

i
M (τ si )J −−−→
H0

7

.

(4)

See Brubaker, Bump and Licata [3], Lemma 24 for a proof.
Instead of H0 one may also use HJ to model the intertwining integrals.
The factor Cαi (τ ), which depends on the spectral parameter, may then be
replaced by an element of θ(P ∨ ). This point of view is taken in Haines,
Kottwitz and Prasad [9]. In order to see how this should work, consider
that since the intertwining operator permutes the spaces M (τ )J one might
consider each Aw to be an endomorphism ofM
M (τ w)J .
(5)
w∈W0

The characters τ of B(F ) that are induced all have a common trivial restriction to the subgroup B0 = T (o)N (F ), so by Frobenius reciprocity theG(F )module M = indB0 (1) is the direct integral of the spaces (5), with τ runningover T̂ modulo the action of W0 . This is the universal principal series. It isG(F )almost true that the intertwining integrals are endomorphisms of indB0 (1);the difficulty is that the operators are polar in the singular set, so one must
restrict to the orthogonal complement in M of the spaces (5) with singular τ .
G(F )Alternatively, one may consider the compact induction Mc = c-indB0 (1).
Although this is no longer closed under the intertwining operators, at least
for f in Mc the intertwining integral (2) is always convergent, and Aw is
realized as a map Mc −→ M .
We now come to the point, which is that as an HJ -module Mc is a freemodule of rank one. This is to be expected from the Bruhat decomposition
because every coset in B0 \G(F )/J has a unique representative in the extended affine Weyl group, and this is also in bijection with J \G(F )/J . Thus
the extended affine Weyl group parametrizes both a basis of Mc and a basis
of HJ . For a proof that the module Mc is one dimensional, see Chriss andKhuri-Makdisi [8] or Haines, Kottwitz and Prasad [9], Lemma 1.6.1.
If Aw were a map Mc −→ Mc we could then transfer Aw to a map HJ −→HJ and conclude that it agreed with right multiplication by some element.
Due to the poles of the intertwining operators, this does not quite work.
What may be done is to consider a somewhat larger Hecke algebra. In theBernstein presentation HJ ∼= H0 ⊗ C [P ∨ ], we can enlarge C[P ∨ ] to any ring∨
R such that C[P ] ⊆ R ⊆ M, where M is the field of fractions of C[P ∨ ].
We take R to be the localization at the set of singular hyperplanes, that is,
the ring obtained by adjoining 1 − α∨ for all coroots α∨ . Let HJ0 = H0 ⊗ R.
Since the poles of the intertwining operators are contained in the singular8

locus, if M 0 denotes the submodule of M generated by the image of the Aw ,then we have a commutative diagramMc ∼= HJ −−−→


Aw y
M0

HJ0


y

−−−→ HJ0(6).

The top arrow is the natural inclusion and the bottom arrow extends the injection of Mc into HJ . The vertical arrows are HJ0 -module homomorphisms.
As before, if R is a ring, then a left R-module homomorphism R −→ R is
multiplication by some element, and therefore Aw may be realized as multiplication by some element of HJ0 .

3

Hecke algebra modules from unique functionals

There are two different kinds of actions of the affine Hecke algebra that weneed to consider. First, in any smooth representation, the Hecke algebra acts
by convolution. The Hecke algebra action on the Iwahori fixed vectors in an
induced representation is an action of this type.
Second, with the notation as in the prior section, the affine Hecke algebra
acts on the ring O(T̂ ) of rational functions on the maximal torus T̂ in the
L-group. This ring is isomorphic to the group algebra C[P ∨ ] of the coweightlattice P ∨ of T , which may be identified with the weight lattice of T̂ . The
generators Ti of the finite Hecke algebra H0 will act by so-called DemazureLusztig operators. This action was introduced by Lusztig [19] and has farreaching consequences.
We have seen in the previous section how the Hecke algebra can be used
to model intertwining integrals. In this section we will show how we maytranslate this action of the Hecke algebra into an action on rational functions.
We described two interpretations of intertwining operators via Hecke algebras
in the diagrams (4) and (6); the simpler point of view in (4) will be sufficientfor our purposes, as it was for Rogawski.
Let us consider, for every z in some Zariski-dense subset of T̂ , a linear
functional Lz on M (τ ). We shall suppose that Lz arises from a multiplicityfree representation in the following way. Let H be a subgroup of G(F )9G(F )and η a character of H such that the induced representation indH (η) is
multiplicity-free. Then by Frobenius reciprocity, a functional Lz (if it exists)
is characterized up to scalar multiple by the property that Lz (π(g)φ) =η(g)Lz (φ) for g ∈ H. As noted above, using the uniqueness of the functional
it is often possible to show by a method of Bernstein that Lz (φ) is a rational
function of z on this set. Alternatively, this rationality may be proved forone specific vector, together with recursions that imply the rationality forall φ.
As particular examples, Lz could be the spherical functionalZ
Sz (f ) =f (k) dkKor the Whittaker functionalZ
Wz (f ) =f (w0 ng) ψ(n) dn
N (F )

where ψ is a nondegenerate character of N (F ), the unipotent radical of the
Borel, and w0 is the long element of the Weyl group W . The sphericaland Whittaker functionals are both characterized up to scalar multiple by
the uniqueness property described above. Indeed, since the spherical Hecke
algebra of compactly supported K-binivariant functions is commutative, K
is a Gelfand subgroup of G(F ) so the functional Lz is determined (up toscalar) by the fact that Sz (π(k)φ) = Sz (φ) for all k ∈ K. For the Whittaker
functional, the corresponding uniqueness result was obtained by Gelfand andGraev, Piatetski-Shapiro and Shalika [26]. The Whittaker integral, like theintertwining integral, is convergent for z in an open set, and has analytic
continuation to all z.
It follows from these uniqueness results that for every pair z and w thefunctional Lzw ◦ Aw is a constant multiple of Lz . For the two examples
above, these constants were computed by Casselman [4] and Casselman andShalika [5], who found that for the spherical functional∨YY
1 − q −1 zαS=Cα (τ ) Sz
(7)
Szw ◦ Aw =z
1 − z α∨++α∈∆
w−1 α ∈ ∆−α∈∆
w−1 α ∈ ∆−10

with Cα (τ ) as in (3) and for the Whittaker functional∨Y
1 − q −1 z−α
Wzw ◦ Aw =Wz .
1 − zα∨+(8)α∈∆
w−1 α ∈ ∆−Here ∆+ and ∆− are the positive and negative roots, and if α is root thenα∨ is the associated coroot.
Our goal is to describe a Hecke algebra action on M (τ )J arising fromeach of these functionals and explain how this action gives a recursion forLz (π(g)Φw ) for any standard basis element Φw ∈ M (τ )J . To describe thefunction Lz (π(g)Φw ), it suffices to choose g from a set of representatives for
H\G(F )/J, where H = K when L = S and H = N when L = W. Thismeans that we may choose g = aλ , where λ ∈ P ∨ , and in the Whittakercase we may assume λ is dominant, since otherwise W(π(aλ )φ) = 0 for any
φ ∈ M (τ ) (cf. [3], Lemma 5).
For both the spherical and Whittaker functionals, there is a standard
basis vector Φw for which Lz (π(aλ )Φw ) has a particularly simple form. In
the Whittaker case, for any dominant weight λ we haveWz (π(aλ )Φτw0 ) = δ 1/2 (aλ )zw0 λ .
(See [3] Proposition 6.) In the spherical case, we have the following statement.
Proposition 1. Let λ ∈ P ∨ . Then Sz (πz (aλ )Φτ1 ) = c(λ)zλ where the constant c(λ) is independent of z.
Proof. We haveSz (πz (aλ )Φτ1 ) =ZΦ1 (kaλ ) dk = (δ1/2Zτ )(aλ )KΦ1 (a−1
λ kaλ ) dk.

K

The support of Φ1 is B(F )J. It is easy to see that if k ∈ K and a−1
λ kaλ ∈
−1
B(F )J then aλ kaλ ∈ B0 J where B0 is the kernel of τz : B(F ) −→ C. Hencethe integral is a constant independent of z.
For either functional, this choice of standard basis vector will be thestarting point for our recursion. It remains to define the Hecke action. From
Rogawski’s perspective, M (τ )J is understood abstractly as a Hecke algebra
module via the regular representation, which is determined by:Tw Φ1 = Φw ,and Tw Φy = Tw Ty Φ1
11

for all y, w ∈ W .

Remembering the quadratic relation given in (1), we see that for a simple
reflection s,(Φsw
if sw > w,
Ts Φw =
qΦsw + (q − 1)Φw if sw < w.
But (4) gives a relation between the intertwining operator As and Ts . So
combining these two ingredients and manipulating the terms we obtain thefollowing result.
Proposition 2. Let τ = τz and let w ∈ W0 . Let s = si be a simple reflection.
Then τ
if sw < w,
Φw + Φτsw
τsτ
(9)As Φw + Cα (τ )Φw =τ
−1τ
q (Φw + Φsw ) if sw > w.
Proof. This is an easy consequence of Casselman [4], Theorem 3.4; see Proposition 8 of [3].
Thus it is reasonable to expect that we may obtain an action of the
Hecke algebra on the ring O(T̂ ) of regular functions on T̂ via the regular
representation by applying the functional to both sides of (9).
The case where L = W is treated in detail in [3], so we will focus on thecase L = S, and return at the end to make some remarks about the difference
between the two cases. For L = S, we intend to start our recursion at Φ1
and move up in the Bruhat order. So let us rewrite (9) in the case sw > w:
qAs Φτws + qCα (τ ) − 1 Φτw = Φτsw .
(10)
We will find that applying Sz to both sides gives a recursive identity for thematrix coefficientFλ,w (z) := Sz (π(aλ )Φτw ),of the principal series representation. To state this more precisely, let us
introduce the Demazure-Lusztig operator Ti defined on an arbitrary function
F on the dual torus T̂ by
Ti F (z) := qC−αi (τ )F (zsi ) + qCαi (τ ) − 1 F (z),(11)
Here 1 ≤ i ≤ r corresponds to a simple reflection si of the finite Weyl group.
After some algebra, this is equivalent toTi F (z) = (zαi − 1)−1 F (z) − F (zsi ) − qF (z) + qzαi F (zsi ) ,12which is precisely the operator defined in Lusztig [19], equation (8.1). (Thiscan also be defined if i = 0, but for this discussion we are excluding thiscase.)
The following result is equivalent to a result of Ion [11], Proposition 5.8.
Theorem 1. The Demazure-Lusztig operators satisfy the quadratic and braid
relations, hence generate a ring isomorphic to the Hecke algebra H0 . If w ∈W0 and si w > w, thenFλ,si w = Ti Fλ,w .
(12)
The fact that the Demazure-Lusztig operators satisfy the quadratic and braid
relations is due to Lusztig [19], Section 8. However checking the braid relations directly depends on a tedious computation for rank 2 root systems, so it
may be of interest that we can avoid such computations using our methods.
Proof. Assume that sw > w. Apply π(aλ ) to both sides of (10) and thenapply Sz . We obtainFλ,sw (z) = qSzτ As (π(aλ )Φτws ) + (qCα (τ ) − 1) Fλ,w (z) .

(13)

Now we use (7), replacing z by zs and remembering that Cα (τzs ) = C−α (τ ).
τComparing with (11), the right-hand side of (13) is just Ti Fλ,w
as desired.
We turn to the fact that the Ti satisfy the generating relations of H0 . Letm be the order of si sj with 1 6 i, j 6 r, i 6= j. To show thatT i Tj T i · · · = T j Ti T j · · ·(m factors on both sides)

it is sufficient to show that they have the same effect on zλ , where λ ∈ P ∨ .
By Proposition 1, it is thus sufficient to show thatTi Tj Ti · · · Faλ ,1 (z) = Tj Ti Tj · · · Faλ ,1 (z) .
But applying (12), both sides equal Faλ ,w (z) where w = si sj si · · · = sj si sj · · ·is the longest element of the dihedral group generated by si and sj .
We next prove the quadratic relation. Assume now that si w < w. Then
applying (12) to Fλ,si w we have Fλ,si w = Ti−1 Fλ,w . Now we can compute thisby the first case of (9), and we findTi−1 F (z) = C−α F (zsi ) + (Cα (τ ) − 1) F (z) .
This means that Ti−1 = q −1 Ti + q −1 − 1, which is equivalent to the quadratic
relation Ti2 = (q − 1) Ti + q.
13This theorem guarantees that, given any reduced decomposition for the
Weyl group element w = si1 . . . sik , the operator
Tw := Ti1 · · · Tik
is well defined. As Ion notes in Theorem 3.1 of [10], the Demazure-Lusztig
operators applied to zλ for λ dominant give a recursive definition for a certain
limit of nonsymmetric Macdonald polynomials with weight w · λ ∈ P ∨ . Thus
our Iwahori-spherical functions are also limits of these polynomials for λdominant.
We caution the reader that our conventions for Demazure-Lusztig operators and nonsymmetric Macdonald polynomials differ slightly from those of
Ion. Instead they more closely parallel those of Cherednik [6]. In particular,
our Ti essentially match those in [6], equation (3.5), which are then used toconstruct nonsymmetric Macdonald polynomials.
Returning to the Whittaker case, similar arguments to those presented
above were given in [3]. The operators that result are not Demazure-Lusztig
operators, but are related in a way that is made precise in Section 5 of [3].
The difference results from the fact that the starting point for the Whittakerrecursion is the Iwahori fixed vector Φw0 , rather than Φ1 . Furthermore,
the constant of proportionality in (8) differs slightly from the spherical casegiven in (7). The resulting recursive operators for Whittaker functions are
Demazure-Lusztig operators conjugated by θ (ρ∨ ) and with q replaced by
q −1 . Here θ : C [P ∨ ] −→ HJ is as in the previous section and ρ∨ is half thesum of the positive coroots. In either the spherical or Whittaker cases, theresulting action of the finite Hecke algebra can be generalized to an actionof the (extended) affine Hecke algebra where the elements in θ(P ∨ ) act bytranslation. (See Theorem 28 of [3] for the Whittaker case.)

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School of Mathematics, University of Minnesota, Minneapolis, MN
55455
brubaker@math.umn.edu
Dept. of Mathematics, Stanford University, Stanford, CA 94305-2125
bump@math.stanford.edu
Dept. of Mathematics, Boston College, Chestnut Hill, MA 024673806
friedber@bc.edu

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