Iwahori Hecke Algebras¶

The Iwahori Hecke algebra is defined in:

Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo Sect. I 10 1964 215–236 (1964).

In this original paper, the algebra occurs as the convolution ring of functions on a p-adic group that are compactly supported and invariant both left and right by the Iwahori subgroup. However Iwahori determined its structure in terms of generators and relations, and it turns out to be a deformation of the group algebra of the affine Weyl group.

Once the presentation is found, the Iwahori Hecke algebra can be defined for any Coxeter group. It depends on a parameter $q$ which in Iwahori’s paper is the cardinality of the residue field. But it could just as easily be an indeterminate.

Then the Iwahori Hecke algebra has the following description. Let $W$ be a Coxeter group, with generators (simple reflections) $s_1,\cdots,s_n$ . They satisfy the relations $s_i^2=1$ and the braid relations

$s_i s_j s_i s_j ... = s_j s_i s_j s_i ...$

where the number of terms on each side is the order of $s_i s_j$ .

The Iwahori Hecke algebra has a basis $T_1,\cdots,T_n$ subject to relations that resemble those of the $s_i$ . They satisfy the braid relations and the quadratic relation

$(T_i-q)(T_i+1)=0.$

This can be modified by letting $q_1$ and $q_2$ be two indeterminates and letting

$(T_i-q_1)(T_i-q_2) = 0.$

In this generality, Iwahori Hecke algebras have significance far beyond their origin in the representation theory of p-adic groups. For example, they appear in the geometry of Schubert varieties, where they are used in the definition of the Kazhdan-Lusztig polynomials. They appear in connection with quantum groups, and in Jones’s original paper on the Jones polynomial.

Here is how to create an Iwahori Hecke algebra:

sage: R.<q>=PolynomialRing(ZZ)
sage: H=IwahoriHeckeAlgebraT("B3",q); H
The Iwahori Hecke Algebra of Type B3 in q,-1 over Univariate
    Polynomial Ring in q over Integer Ring and prefix T
sage: T1,T2,T3 = H.algebra_generators()
sage: T1*T1
(q-1)*T1 + q

If the Cartan type is affine, the generators will be numbered starting with T0 instead of T1.

You may coerce a Weyl group element into the Iwahori Hecke algebra:

sage: W = WeylGroup("G2",prefix="s")
sage: [s1,s2]=W.simple_reflections()
sage: P.<q>=LaurentPolynomialRing(QQ)
sage: H = IwahoriHeckeAlgebraT(W,q)
sage: H(s1*s2)
T1*T2

Intertwining Operators¶

The Iwahori Hecke operator can be used to study representations of p-adic groups that have fixed vectors with respect to the Iwahori subgroup. References:

Rogawski, On modules over the Hecke algebra of a $p$ -adic group, Invent. Math., Inventiones Mathematicae, 79, 1985, 3, 443–465,

Reeder, On certain Iwahori invariants in the unramified principal series, Pacific J. Math., Pacific Journal of Mathematics, 153, 1992, 2, 313–342,
Haines, Kottwitz and Parsad, Iwahori-Hecke Algebras, http://arxiv.org/abs/math/0309168.
Bump and Nakasuji, Casselman’s basis of Iwahori vectors and the Bruhat order, http://arxiv.org/abs/1002.2996.
Casselman, The unramified principal series of $\mathfrak{p}$ -adic groups. I. The spherical function, Compositio Math., Compositio Mathematica, 40, 1980, 3, 387–406.

A key feature of the theory are the intertwining operators between different induced representations. We will show how to implement these intertwining operators in Sage.

For simplicity we will work over $G=SL(n,F)$ where $F=\mathbb{Q}_q$ , with $q$ a prime number, but everything works for an arbitrary split reductive group over a nonarchimedean local field. If $\mathbf{z}=(z_1,\cdots,z_n)\in\mathbb{C}^n$ , define the following quasicharacters of $T(F)$ , the diagonal subgroup in $G$ :

$\chi_{\mathbf{z}}\left(\begin{array}{ccc} t_1\\&\ddots\\&&t_n\end{array}\right)=\prod_{i=1}^n z_i^{\hbox{ord}(t_i)},\qquad \delta\left(\begin{array}{ccc} t_1\\&\ddots\\&&t_n\end{array}\right)=\prod_{i=1}^n |t_i|^{n+1-2i}$

We extend both to characters of the Borel subgroup $B(F)$ of upper triangular matrices by letting the subgroup $N(F)$ of upper triangular unipotent matrices be in the kernels. Let $V(\chi)$ with $\chi=\chi_{\mathbf{z}}$ is defined to be the space of locally constant functions $f$ on $G$ such that:

$f(b g) = (\delta^{1/2}\chi)(b)f(g),\qquad b\in B(F).$

We have a representation $\pi:G\to\hbox{End}(V(\chi))$ by right translation: $\pi(g)f(h)=f(h g)$ . These are the spherical principal series representations. Let $K=SL(n,\mathbb{Z}_q)$ . This is the standard maximal compact subgroup. It has a homomorphism $K\to SL(n,\mathbb{F}_q)$ by reduction modulo $q$ . The subgroup $J$ of elements of $K$ whose images are upper triangular is an open subgroup, the Iwahori subgroup.

Suppose that the $z_i$ are all distinct. Then for each $w\in W$ there exists an intertwining integral $M_w:V(\chi)\to V({^w\chi})$ where the Weyl group acts on characters of $T(F)$ by conjugation. This integral is defined by the formula

$M h(g)=\int_{N\cap w N_- w^{-1}} h(w^{-1}n g)\,d n$

where $N$ is the group of upper triangular unipotent matrices and $N_-$ is the group of lower triangular matrices. The integral is convergent provided $|z_i/z_{i+1}|<1$ and in general it may be defined by analytic continuation in the $z_i$ or another method. See Casselman loc. cit..

The problem is how to model the intertwining operators in Sage.

Let $W=N(F)/T(\mathbb{Z}_q)$ and $W'=N(F)/T(F)$ . These are isomorphic to the affine and finite Weyl groups, that is, the Weyl groups with Cartan types [`A`,n-1,1] and [`A`,n-1], respectively.

The Iwahori Hecke algebra $\mathcal{H}$ is the ring of compactly supported functions $h$ on $G$ such that $h(k g k')=h(g)$ when $k,k'\in J$ . It is a ring under convolution. $V(\chi)$ becomes a module over it by

$(h f)(g)=\int_G h(x) f(g x)\, d x$ .

The subspace $V(\chi)^J$ of $V(\chi)$ consisting of elements that are fixed by $J$ is invariant under $\mathcal{H}$ . This module is finite-dimensional: in fact, its order is $|W'|=n!$ , whereas $V(\chi)$ itself is infinite dimensional. However the finite-dimensional $\mathcal{H}$ -module $V(\chi)^J$ accurately reflects the structure of the $G$ -module $V(\chi)$ . For example $V(\chi)$ is irreducible as a $G$ -module if and only if $V(\chi)^J$ is irreducible as an $\mathcal{H}$ -module.

Normalizing the Haar measure so that $J$ has volume $1$ , let $T_w$ be the characteristic function of $J w J$ , and if $1 \le i \le r$ let $T_i$ denote $T_{\sigma_i}$ . The $T_w$ with $w \in W$ form a basis, and the $T_i$ form a set of algebra generators The $T_i$ satisfy the same braid relations as the $s_i$ , but the relation $\sigma_i^2 = 1$ is replaced by $T_i^2 = (q - 1) T_i + q$ .

The subalgebra elements of $\mathcal{H}$ consisting of functions that are supported in $K$ is the finite Iwahori Hecke algebra $\mathcal{H}'$ . Thus $\dim (H) = |W|$ but $\mathcal{H}$ is infinite-dimensional The subalgebra $H$ has generators $T_1, \cdots, T_r$ but omits $T_0$ .

We see that $W$ , $W'$ , $\mathcal{H}$ and $\mathcal{H'}$ are all objects that can be created in Sage.

We define a vector space isomorphism $\alpha = \alpha (\chi) : V (\chi)^J \to H'$ as follows. If $F \in V (\chi)^J$ then let $\alpha (F) = f$ where $f$ is the function $f (g) = F (g^{- 1})$ if $g \in K$ , 0 if $g \notin K$ . It may be checked that $\alpha (F) \in H'$ . This allows us to model the $|W'|$ -dimensional $\mathcal{H}$ -module $V(\chi)$ by the $|W'|$ -dimensional subalgebra $\mathcal{H}'$ . This idea, which appears in the paper of Rogawski cited above, is due to Joseph Bernstein.

Let $w\in W$ and define a map $\mathcal{M}_w:H\to H$ by requiring that

$\alpha({^w\chi})\circ M_w = \mathcal{M}_w\circ\alpha(\chi)$

as maps from $V(\chi)^J\to H$ . Since $M_w$ and $\alpha(\chi)$ , $\alpha({^w\chi})$ are all homomorphisms of left $\mathcal{H}'$ -modules, $\mathcal{M}_w$ is a homomorphism of left $\mathcal{H}'$ modules. Now let $\mu_{\mathbf{z}}(w)= \mathcal{M}_w(1)$ where $1$ denotes the identity element of $\mathcal{H}'$ .

Because $\mathcal{M}_w$ is a $\mathcal{H}'$ -module homomorphism, we have

$\mathcal{M}_w(h) = \mathcal{M}_w(h\cdot 1)= h \mathcal{M}_w(1) = h\cdot\mu_{\mathbf{z}}(w)$ .

In other words, the intertwining operator is modeled by multiplication by $\mu_{\mathbf{z}}(w)$ when we identify $V(\chi)^J$ and $V({^w\chi})^J$ with $H$ by means of the homomorphisms $\alpha$ . We are therefore left with the problem of computing $\mu_{\mathbf{z}}(w)$ . If $l(w_1w_2)=l(w_1)+l(w_2)$ then $M_{w_1w_2}=M_{w_1}M_{w_2}$ and it follows that

$\mu_{\mathbf{z}}(w_1w_2)=\mu_{\mathbf{z}}(w_2)\mu_{w_2\mathbf{z}}(w_1)$ .

(Here $w_2$ acts on the spectral parameters $\mathbf{z}$ in the obvious way.) Thus we are reduced to the determination of $\mu_{\mathbf{z}}(w)$ when $w=s_i$ is a simple reflection, and this is accomplished by Theorem 3.4 in Casselman, loc. cit.:

$\mu_{\mathbf{z}}(s_i)=\frac{1}{q}T_i+\left(1-\frac{1}{q}\right)\frac{\mathbf{z}^{\alpha_i}}{1-\mathbf{z}^{\alpha_i}}.$

Here the simple root $\alpha_i=(0,\cdots,0,1,-1,0,\cdots,0)$ in the ambient lattice, where the $1$ is in the $i$ -th position, so $\mathbf{z}^\alpha_i=z_i/z_{i+1}$ .

In addition to the intertwining operators, let us construct a basis $\psi_w$ of $V(\chi)^J$ indexed by elements of $W'$ . Let

$\psi_w(b k)=\left\{\begin{array}{ll}\delta^{1/2}\chi(b) & \text{if $k\in J u^{-1} J$ with $u \ge w$},\\0&\text{otherwise,}\end{array}\right.$

where $b\in B(F)$ and $k\in K$ . In order to make sense of this definition, bear in mind that every element of $G$ has a decomposition as a product $b k$ with $b\in B(F)$ and $k\in K$ , and every $k\in K$ belongs to a unique double coset $J u^{-1} J$ with $u\in W'$ .

The image of $\psi_w$ under the homomorphism $\alpha$ is just $\sum_{u\ge w} T_u$ , and by abuse of notation we will denote this element of $\mathcal{H}'$ also by $\psi_w$ . Now let us ask to evaluate $m(u,v)=(\Lambda_v\psi_u)(1)$ . It may be shown that $m(u,v)=0$ unless $u\le v$ , and that $m(u,u)=1$ . In other words, the “matrix” $m$ is upper triangular in the Bruhat order.

Bump and Nakasuji used Sage to investigate this question and arrived at the following conjecture. This is conjectured for more general groups than just the $SL(n)$ that we are considering, but only for the simply-laced groups (types A,D,E). It follows from the Deodhar conjecture that the set

$S(u,v)=\{\alpha\in\Phi^+|u\le v r_\alpha < v\}$

has cardinality $\ge l(v)-l(u)$ , where $l$ is the length function on $W'$ . (Remember that $r_\alpha$ is the reflection in the positive root $\alpha$ .) If the Kazhdan-Lusztig polynomial $P_{w_0v,w_0u}=1$ , then $|S(u,v)|=l(v)-l(u)$ .

Conjecture (Bump and Nakasuji) If the Kazhdan-Lusztig polynomial $P_{w_0v,w_0u}=1$ then

$m(u,v)=\prod_{\alpha\in S(u,v)}\frac{1-q^{-1}\mathbf{z}^\alpha}{1-\mathbf{z}^\alpha}$

The following Sage code is capable of verifying this statement for ['A',3], and of course can be adapted to any root system.

Intertwining Operator Program¶

This program can compute both the actual and conjectured values of $m(u,v)$ .

import copy

P1.<q>=PolynomialRing(QQ)
W = WeylGroup("A3",prefix="s")
KL = KazhdanLusztigPolynomial(W,q)

P.<q,z1,z2,z3,z4>=PolynomialRing(QQ)
F = Frac(P)
H = IwahoriHeckeAlgebraT(W,q,base_ring=F)
[s1,s2,s3]=W.simple_reflections()
w0 = W.long_element()
[T1,T2,T3]=H.algebra_generators()

def mu(z,s):
    """
    Produces an element of the Hecke algebra that mimics
    the intertwining integral corresponding to a simple
    reflection.
    """
    [z1,z2,z3,z4] = z
    if s == s1:
        return q^(-1)*T1+(1-1/q)*(z1/(z2-z1))
    elif s == s2:
        return q^(-1)*T2+(1-1/q)*(z2/(z3-z2))
    elif s == s3:
        return q^(-1)*T3+(1-1/q)*(z3/(z4-z3))

def reflect(s,z):
    """
    Applies the simple reflection s to the spectral data z.
    """
    return (Matrix(z)*s.matrix()).list()

def intertwiner(w,debug=False):
    """
    This is mu_z(w) in general. The function mu(z,s) implemented
    above is only for simple reflections.

    Run with debug=True to see what it is really doing.
    """
    decomp = w.reduced_word()
    decomp.reverse()
    prod = H(1)
    if debug:
        clist = ""
    for i in range(len(decomp)):
        alist = copy.deepcopy(decomp[:i])
        z = [z1,z2,z3,z4]
        for j in alist:
            z = reflect(W.simple_reflection(j),z)
        s = W.simple_reflection(decomp[i])
        if debug:
            clist +="mu(%s,%s)"%(z.__repr__(),s.__repr__())
            if i < len(decomp)-1:
                clist += "*"
        else:
            prod = prod*mu(z,s)
    if debug:
        return clist
    else:
        return prod

def psi(u):
    """
    u is a permutation. This produces the sum of the f_v for v>=u.
    """
    u = H(u).support_of_term()
    return sum(H(v) for v in W if u.bruhat_le(v))

def ev(f):
    """
    Evaluate the element of the space of Iwahori fixed vectors
    in the induced model at the identity.
    """
    return f.coefficient(W(1))

def m(u,v):
    """
    It is assumed that u <= v. Apply the v intertwiner to
    psi(u) and evaluate at 1.
    """
    u = W(u)
    v = W(v)
    return ev(psi(u)*intertwiner(H(v)))

def rfactor(alpha):
    """
    If alpha is a root, returns (1-q^(-1)*z^alpha)/(1-z^alpha).
    """
    z = [z1,z2,z3,z4]
    za = prod(z[i]**alpha[i] for i in range(4))
    return (1-q^(-1)*za)/(1-za)

def conjectured_m(u,v):
    """
    This is the value conjectured for m(u,v) by Bump and Nakasuji
    if u and v satisfy a hypothesis.
    """
    refdict = W.reflections()
    S = [refdict[r] for r in refdict.keys() if u.bruhat_le(v*r) and (v*r).bruhat_le(v)]
    return prod(rfactor(alpha) for alpha in S)

Iwahori Hecke Algebras¶

Intertwining Operators¶

Intertwining Operator Program¶

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Iwahori Hecke Algebras¶

Intertwining Operators¶

Intertwining Operator Program¶

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