Parabolic Kazhdan-Lusztig R-polynomials for Hermitian symmetric pairs ^{1}

Francesco Brenti^{2} Dipartimento di Matematica Universitá di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00133 Roma, Italy brenti@mat.uniroma2.it

1 Introduction

In their fundamental paper [13] Kazhdan and Lusztig defined, for any Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W (see, e.g., [11, Chap.7] or [2, Chap.5]). These polynomials play an important role in several areas of mathematics, including the algebraic geometry and topology of Schubert varieties and representation theory (see, e.g., [2, Chap.5], and the references cited there). In order to prove the existence of these polynomials Kazhdan and Lusztig introduced another family of polynomials, usually called the R-polynomials, whose knowledge is equivalent to that of the Kazhdan-Lusztig polynomials.

In 1987 Deodhar ([7]) introduced parabolic analogues of all these polynomials. These parabolic Kazhdan-Lusztig and R-polynomials reduce to the ordinary ones for the trivial parabolic subgroup of W and are also related to them in other ways (see, e.g., Proposition 2.4 below). Besides these connections the parabolic polynomials also play a direct role in several areas including the theories of generalized Verma modules ([6]), tilting modules ([15], [16]), quantized Schur algebras ([19]), Macdonald polynomials ([10], [9]), Schubert varieties in partial flag manifolds ([12]), and in the representation theory of the Lie algebra gl_{n} ([14]).

The purpose of this work is to study the parabolic Kazhdan-Lusztig R-polynomials for Hermitian symmetric pairs. More precisely, we give explicit combinatorial product formulas for these polynomials. These imply, in particular, that all the roots of these polynomials are (either zero or) roots of unity. Some of these results are used in [5] to obtain explicit combinatorial formulas for the parabolic Kazhdan-Lusztig polynomials.

12000 Mathematics Subject Classification: Primary 05E99; Secondary 20F55.

2Partially supported by EU grant HPRN-CT-2001-00272. Part of this research was carried out while the author was a member of the Mittag-Leffler Institut in Djürsholm, Sweden, whose hospitality and financial support are gratefully acknowledged.

The organization of the paper is as follows. In the next section we recall definitions, notation and results that are used in the rest of this work. In §3 we study the parabolic R-polynomials of Hermitian symmetric pairs. We obtain two main combinatorial formulas for them, one in terms of signed permutations and one in terms of lattice paths. As an application of our results, we derive combinatorial closed product formulas for certain sums and alternating sums of ordinary Kazhdan-Lusztig R-polynomials.

2 Preliminaries

In this section we collect some definitions, notation and results that are used in the rest of this work. We let P =^{def} {1, 2, 3, …} and N=^{def}P ∪{0}. For m, n ∈ N, m ≥ n, we let [n, m] =^{def} {n, n + 1, …, m − 1, m} and [n] =^{def} [1, n] (where [0] =^{def} ∅). The cardinality of a set A will be denoted by |A|.

Given a set T we let S(T) be the set of all bijections π : T → T, and S_{n} =^{def} S([n]). If σ ∈ S_{n} then we write σ in disjoint cycle form (see, e.g., [17], p.17) and we usually omit writing the 1-cycles of σ. So, for example, if σ = (9, 7, 1, 3, 5)(2, 6) then σ(1) = 3, σ(2) = 6, σ(3) = 5, σ(4) = 4, etc... Given σ, τ ∈ S_{n} we let στ =^{def} σ ∘ τ (composition of functions) so that, for example, (1, 2)(2, 3) = (1, 2, 3).

We follow Chapter 3 of [17] for poset notation and terminology. In particular, given a poset (P, ≤) and u, v ∈ P we let [u, v] =^{def} {z ∈ P : u ≤ z ≤ v} and call this an interval of P. We say that v covers u, denoted u ⊲ v (or, equivalently, that u is covered by v) if |[u, v]| = 2. We say that u, v ∈ P are comparable if either u ≤ v or v ≤ u. If P has a minimum element, denoted ˆ0, then we call a subset of the form [ˆ0, u], for u ∈ P, a lower interval of P. Given any Q ⊆ P we will always consider Q as a poset with the partial ordering induced by P.

We follow §7.2 of [18] for any undefined notation and terminology concerning partitions. Let H =^{def} {(i, j) ∈ P^{2} : i ≤ j}, with the ordering induced by the product ordering on P^{2}. We call the finite order ideals of H shifted partitions. We denote by S the set of all finite order ideals of H. We will always assume that S is partially ordered by set inclusion. It is well known that this makes S into a distributive lattice. Note that there is an obvious bijection between partitions into distinct parts and shifted partitions, given by

(λ_{1}, …, λ_{k}) ↔ {(i, j) ∈ P^{2} : 1 ≤ i ≤ k, i ≤ j ≤ λ_{i} + i − 1}.

For this reason we will freely identify these objects, and use expressions such as “the shifted partition (5, 2, 1)”. We also identify each shifted partition with its diagram, which we draw rotated counterclockwise by \frac{3}{4}π radians with respect to the usual (Anglophone) convention. So, for example, the diagram of the shifted partition (5, 2, 1) is shown in Figure 1. We let δ_{n} =^{def} (n, n − 1, …, 2, 1).

[figure]

Figure 1

We follow [11] for general Coxeter groups notation and terminology. In particular, given a Coxeter system (W, S) and u ∈ W we denote by l(u) the length of u in W, with respect to S, and we let D(u) =^{def} {s ∈ S : l(us) < l(u)}. For u, v ∈ W we let l(u, v) =^{def} l(v) − l(u). We denote by e the identity of W, and we let T =^{def} {usu^{−1} : u ∈ W, s ∈ S} be the set of reflections of W. Given J ⊆ S we let W_{J} be the parabolic subgroup generated by J and

W^{J} =^{def} {u ∈ W : l(su) > l(u) for all s ∈ J}.  (1)

Note that W^{∅} = W. If W_{J} is finite then we denote by w_{0}(J) its longest element. We will always assume that W^{J} is partially ordered by Bruhat order. Recall (see, e.g., [11], §5.9) that this means that x ≤ y if and only if there exist r ∈ N and t_{1}, …, t_{r} ∈ T such that t_{r} ⋯ t_{1} x = y and l(t_{i} ⋯ t_{1} x) > l(t_{i−1} ⋯ t_{1}x) for i = 1, …, r. Given u, v ∈ W^{J}, u ≤ v, we let

[u, v]^{J} =^{def} {z ∈ W^{J} : u ≤ z ≤ v},

and [u, v] =^{def} [u, v]^{∅}.

The following result is due to Deodhar, and we refer the reader to [7, §§2-3] for its proof.

Theorem 2.1 Let (W, S) be a Coxeter system, and J ⊆ S. Then, for each x ∈ {−1, q}, there is a unique family of polynomials {R_{u,v}^{J,x}(q)}_{u,v∈W^{J}} ⊆ Z[q] such that, for all u, v ∈ W^{J}:

i) R_{u,v}^{J,x}(q) = 0 if u ≰ v;

ii) R_{u,u}^{J,x}(q) = 1;

iii) if u < v and s ∈ D(v) then

R_{u,v}^{J,x}(q) = \{ \begin{matrix}R_{us,vs}^{J,x}(q), & if us < u, \\ (q − 1)R_{u,vs}^{J,x}(q) + qR_{us,vs}^{J,x}(q), & if u < us ∈ W^{J}, \\ (q − 1 − x)R_{u,vs}^{J,x}(q), & if u < us ∉ W^{J}. \end{matrix}

The polynomials R_{u,v}^{J,x}(q), whose existence is guaranteed by the previous theorem, are called the parabolic R-polynomials of W^{J} of type x. It follows immediately from Theorem 2.1 and from well known facts (see, e.g., [11, §7.5]) that R_{u,v}^{∅,−1}(q) (= R_{u,v}^{∅,q}(q)) are the (ordinary) R-polynomials of W which we will denote simply by R_{u,v}(q), as customary. The parabolic R-polynomials can then be used to define and compute the parabolic Kazhdan-Lusztig polynomials of W^{J} of type x (see [7, Proposition 3.1]).

The parabolic R-polynomials are related to their ordinary counterparts also in the following way.

Proposition 2.2 Let (W, S) be a Coxeter system, J ⊆ S, and u, v ∈ W^{J}. Then we have that

R_{u,v}^{J,x}(q) = ∑_{w∈W_{J}}(−x)^{l(w)}R_{wu,v}(q),

for all x ∈ {−1, q}.

A proof of this result can be found in [7, Proposition 2.12].

There is one more property of the parabolic R-polynomials that we will use and that we recall for the reader’s convenience. A proof of it can be found in [8, Corollary 2.2].

Proposition 2.3 Let (W, S) be a Coxeter system, and J ⊆ S. Then

q^{l(u,v)}R_{u,v}^{J,x} (\frac{1}{q}) = (−1)^{l(u,v)}R_{u,v}^{J,q−1−x}(q)

for all u, v ∈ W^{J}, and x ∈ {−1, q}.

The purpose of this work is to study the parabolic R-polynomials for quotients W^{J} such that (W, W_{J}) is a Hermitian symmetric pair. These quotients have been classified (see, e.g., [3]) and there are five infinite families and two exceptional ones. Using (and abusing slightly) the standard notation for the classification of the finite Coxeter systems, the Hermitian symmetric pairs are: (A_{n}, A_{i−1} × A_{n−i}), (B_{n}, A_{n−1}), (B_{n}, B_{n−1}), (D_{n}, A_{n−1}), (D_{n}, D_{n−1}), (E_{6}, D_{5}), (E_{7}, E_{6}), (n ≥ 3, 1 ≤ i ≤ n). The parabolic R-polynomials for the pairs (A_{n}, A_{i−1} × A_{n−i}) (1 ≤ i ≤ n) have been computed in [4]. In this work we deal with the other ones.

We follow [2, Chap. 8] for combinatorial descriptions of the Coxeter systems of type B_{n} and D_{n} as permutation groups. In particular, we let S_{n}^{B} be the group of all bijections w of {−n, …, −1, 1, …, n} in itself such that w(−i) = −w(i) for all i ∈ [n], s_{j} =^{def} (j, j + 1)(−j, −j − 1) for j = 1, ..., n − 1, s_{0} =^{def} (1, −1), and

B_{n} =^{def} {s_{0}, ..., s_{n−1}}. If v ∈ S_{n}^{B} then we write v = [a_{1}, …, a_{n}] to mean that v(i) = a_{i},

for i = 1, …, n. It is well known that (S_{n}^{B}, B_{n}) is a Coxeter system of type B_{n} and that the following holds. Given v ∈ S_{n}^{B} we let

inv(v) =^{def} |{(i, j) ∈ [n]^{2} : i < j, v(i) > v(j)}|,

N_{1}(v) =^{def} |{i ∈ [n] : v(i) < 0}| and

N_{2}(v) =^{def} |{(i, j) ∈ [n]^{2} : i < j, v(i) + v(j) < 0}|.

Proposition 2.4 Let v ∈ S_{n}^{B}. Then l(v) = inv(v) + N_{1}(v) + N_{2}(v), and D(v) = {s_{i} ∈ B_{n} : v(i) > v(i + 1)}, where v(0) =^{def} 0.

Let A_{n−1} =^{def} {s_{1}, …, s_{n−1}}. Then (S_{n}^{B})_{A_{n−1}} ≅ S_{n} and it is clear from Proposition 2.4 that v ∈ (S_{n}^{B})^{A_{n−1}} if and only if v^{−1}(1) < v^{−1}(2) < ⋯ < v^{−1}(n). In this case

v^{−1}(1) < ⋯ < v^{−1}(k) < 0 < v^{−1}(k + 1) < ⋯ < v^{−1}(n)

where k = N_{1}(v), and we associate to v ∈ (S_{n}^{B})^{A_{n−1}} the shifted partition

Λ_{B}(v) =^{def} (v^{−1}(−1), v^{−1}(−2), …, v^{−1}(−k)).  (2)

The next result is known, and not hard to prove.

Proposition 2.5 The map Λ_{B} defined by (2) is a bijection between (S_{n}^{B})^{A_{n−1}} and {λ ∈ S : λ ⊆ δ_{n}}. Furthermore u ≤ v in (S_{n}^{B})^{A_{n−1}} if and only if Λ_{B}(u) ⊆ Λ_{B}(v), and l(v) = |Λ_{B}(v)|, for all u, v ∈ (S_{n}^{B})^{A_{n−1}}.

We find it sometimes convenient to identify a shifted partition in {λ ∈ S : λ ⊆ δ_{n}} with a lattice path with (1, 1) (up) and (1, −1) (down) steps starting at (0, 0) and having n steps. So, for example, the shifted partition (5, 2, 1) ⊆ δ_{7} corresponds to the lattice path illustrated in Figure 2.

[figure]

Figure 2

Proposition 2.6 Let v ∈ (S_{n}^{B})^{A_{n−1}} and i ∈ [n]. Then the i-th step (from the left) of Λ_{B}(v) (seen as a lattice path) is an up-step if and only if v(n + 1 − i) < 0.

Proof. From the definition of Λ_{B}(v) we have that the i-th step of Λ_{B}(v) is an up step if and only if n + 1 − i ∈ {v^{−1}(−1), …, v^{−1}(−k)}. But this, by the definition of k, happens if and only if v(n + 1 − i) < 0 , as desired. □

We let S_{n}^{D} be the subgroup of S_{n}^{B} defined by

S_{n}^{D} =^{def} {w ∈ S_{n}^{B} : N_{1}(w) ≡ 0 (mod 2)},

s̃_{0} =^{def} (1, −2)(2, −1), and D_{n} =^{def} {s̃_{0}, s_{1}, ..., s_{n−1}}. It is then well known that (S_{n}^{D}, D_{n}) is a Coxeter system of type D_{n}, and that the following holds (see, e.g., [2, §8.2]).

Proposition 2.7 Let v ∈ S_{n}^{D}. Then l(v) = inv(v) + N_{2}(v), and D(v) = {s_{i} ∈ D_{n} : v(i) > v(i + 1)} , where v(0) =^{def} −v(2).

Let v ∈ S_{n}^{D}. It is then clear from Proposition 2.7 that v ∈ (S_{n}^{D})^{A_{n−1}} if and only if v^{−1}(1) < ⋯ < v^{−1}(n). In this case

v^{−1}(1) < ⋯ < v^{−1}(k) < 0 < v^{−1}(k + 1) < ⋯ < v^{−1}(n)  (3)

where k = N_{1}(v), and we associate to v ∈ (S_{n}^{D})^{A_{n−1}} the shifted partition

Λ_{D}(v) =^{def} (v^{−1}(−1) − 1, v^{−1}(−2) − 1, …, v^{−1}(−k) − 1).  (4)

The next result is almost certainly known. However, for lack of an adequate reference, and for completeness, we include its proof here.

Proposition 2.8 The map Λ_{D} defined by (4) is a bijection between (S_{n}^{D})^{A_{n−1}} and {λ ∈ S : λ ⊆ δ_{n−1}}. Furthermore u ≤ v in (S_{n}^{D})^{A_{n−1}} if and only if Λ_{D}(u) ⊆ Λ_{D}(v), and l(v) = |Λ_{D}(v)|, for all u, v ∈ (S_{n}^{D})^{A_{n−1}}.

Proof. Since l(Λ_{B}(v)) = N_{1}(v) for all v ∈ (S_{n}^{B})^{A_{n−1}} and (S_{n}^{D})^{A_{n−1}} = {v ∈ (S_{n}^{B})^{A_{n−1}} : N_{1}(v) ≡ 0 (mod 2)} it follows immediately from Proposition 2.5 that Λ_{B} is a bijection between (S_{n}^{D})^{A_{n−1}} and {λ ∈ S : λ ⊆ δ_{n}, l(λ) ≡ 0 (mod 2)}. But the map (λ_{1}, …, λ_{k}) ↦ (λ_{1} − 1, …, λ_{k} − 1) is clearly a bijection between {λ ∈ S : λ ⊆ δ_{n}, l(λ) ≡ 0 (mod 2)} and {λ ∈ S : λ ⊆ δ_{n−1}} so the first statement follows. To prove the second statement note that, by Proposition 2.5 and the definition of Λ_{B}, it is equivalent to the statement that, for all u, v ∈ (S_{n}^{D})^{A_{n−1}} , u ≤ v in S_{n}^{D} if and only if u ≤ v in S_{n}^{B}. It is well known that if u ≤ v in S_{n}^{D} then u ≤ v in S_{n}^{B} (see, e.g., [2, Theorems 8.1.8 and 8.2.8]). So assume that u ≤ v in S_{n}^{B}. By [2, Theorems 8.1.8 and 8.2.8] it is enough to show that if a, b ∈ [n] are such that [−a, a] × [−b, b] is an empty rectangle for both u and v, and u[−a − 1, b + 1] = v[−a − 1, b + 1] (we refer the reader to [2, §8.2, p. 257] for the definition of this notation and terminology), then u[−1, b + 1] ≡ v[−1, b + 1] (mod 2). Note that, for any w ∈ (S_{n}^{D})^{A_{n−1}} and any a, b ∈ [n], w[−1, b+1] = max{0, N_{1}(w)−b}, and if [−a, a] × [−b, b] is empty for w then N_{1}(w) ≥ b (for if N_{1}(w) < b then w^{−1}(b) > 0 and hence, since [−a, a] × [−b, b] is empty for w, w^{−1}(b) > a and |w(1)| > b, hence a < w^{−1}(b) < w^{−1}(|w(1)|) ≤ 1, which is a contradiction). Hence, under our hypotheses, u[−1, b+1] = N_{1}(u)−b ≡ N_{1}(v)−b = v[−1, b+1] (mod 2), as desired. Finally, l_{D}(v) = l_{B}(v) − N_{1}(v) = |Λ_{B}(v)| − l(Λ_{B}(v)) = |Λ_{D}(v)| for any v ∈ (S_{n}^{D})^{A_{n−1}} by (4) and Propositions 2.4, 2.7, and 2.5. □

Proposition 2.9 Let v ∈ (S_{n}^{D})^{A_{n−1}} and i ∈ [n − 1]. Then the i-th step (from the left) of Λ_{D}(v) (seen as a lattice path) is an up-step if and only if v(n + 1 − i) < 0. In particular, v(1) < 0 if and only if Λ_{D}(v) has an odd number of up-steps.

Proof. This follows immediately from Proposition 2.6 and the fact that the i-th step (1 ≤ i ≤ n − 1) of (λ_{1}, …, λ_{k}) ⊆ δ_{n} is up if and only if the i-th step of (λ_{1} − 1, …, λ_{k} − 1) ⊆ δ_{n−1} is, for all (λ_{1}, …, λ_{k}) ∈ S. □

Let u ∈ S_{n}^{B}, and B_{n−1} =^{def} {s_{0}, …, s_{n−2}} ⊆ B_{n}. Then (S_{n}^{B})_{B_{n−1}} ≅ S_{n−1}^{B} and by

Proposition 2.4 we have that

(S_{n}^{B})^{B_{n−1}} = {w ∈ S_{n}^{B} : 0 < w^{−1}(1) < ⋯ < w^{−1}(n − 1)}.

Therefore u ∈ (S_{n}^{B})^{B_{n−1}} if and only if there exist i ∈ [n] and ε ∈ {1, −1} such that

u = [1, 2, …, i − 1, εn, i, …, n − 1].

Let, for brevity, u_{εi} =^{def} [1, 2, …, i − 1, εn, i, …, n − 1]. It then follows easily from the definition of Bruhat order that

u_{n} < u_{n−1} < ⋯ < u_{1} < u_{−1} < ⋯ < u_{−n}.

So (S_{n}^{B})^{B_{n−1}}, partially ordered by Bruhat order, is a chain with 2n elements.

Let v ∈ S_{n}^{D}, and D_{n−1} =^{def} {s̃_{0}, s_{1}, …, s_{n−2}} ⊆ D_{n}. Then (S_{n}^{D})_{D_{n−1}} ≅ S_{n−1}^{D} and

by Proposition 2.7 we have that

(S_{n}^{D})^{D_{n−1}} = {w ∈ S_{n}^{D} : w^{−1}(−2) < w^{−1}(1) < w^{−1}(2) < ⋯ < w^{−1}(n − 1)}.

Hence, if w ∈ (S_{n}^{D})^{D_{n−1}}, then 0 < w^{−1}(2) so

(S_{n}^{D})^{D_{n−1}} = {v_{εi} : i ∈ [n], ε ∈ {−1, 1}},

where

v_{εi} =^{def} \{ \begin{matrix}[ε1, 2, …, i − 1, ε n, i, …, n − 1], & if i ≥ 2, \\ [ε n, ε 1, 2, …, n − 1], & if i = 1, \end{matrix}

for i ∈ [n], ε ∈ {−1, 1}. Furthermore, it follows easily from the definition of Bruhat order (and from Proposition 2.7) that

v_{n} < v_{n−1} < ⋯ < v_{2} < v_{1} < v_{−2} < ⋯ < v_{−n},

v_{2} < v_{−1} < v_{−2}, and v_{−1} and v_{1} are incomparable.

3 Parabolic R-polynomials

In this section we obtain explicit combinatorial product formulas for the parabolic R-polynomials of Hermitian symmetric pairs. These show, in particular, that all the roots of these polynomials are (either zero or) roots of unity. As an application of our results, we derive explicit combinatorial product formulas for certain sums and alternating sums of ordinary Kazhdan-Lusztig R-polynomials.

Let u, v ∈ S_{n}^{B}. For j ∈ [n] let

b_{j}(u, v) =^{def} |{r ≥ j : v(r) < 0}| − |{r ≥ j : u(r) < 0}|.

For example, if u = [3, 4, 5, 6, −2, 7, −1] and v = [−4, 5, −3, 6, 7, −2, −1] then (b_{1}(u, v), …, b_{7}(u, v)) = (2, 1, 1, 0, 0, 1, 0). Note that it follows easily from Propositions 2.5 and 2.8 that if u, v ∈ (S_{n}^{B})^{A_{n−1}} (resp., (S_{n}^{D})^{A_{n−1}}) then b_{j}(u, v) ≥ 0 for j = 1, …, n if and only if u ≤ v in (S_{n}^{B})^{A_{n−1}} (resp., (S_{n}^{D})^{A_{n−1}}). We let

N(u, v) =^{def} {r ∈ [n] : u(r) v(r) < 0},

and

D(u, v) =^{def} {r ∈ N(u, v) : (−1)^{b_{r}(u,v)} < 0}.  (5)

Theorem 3.1 Let W^{J} ∈ {(S_{n}^{B})^{A_{n−1}}, (S_{n}^{D})^{A_{n−1}}} and u, v ∈ W^{J}, u ≤ v. Then

R_{u,v}^{J,−1}(q) = q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

where

b̃_{j}(u, v) =^{def} \{ \begin{matrix}b_{j}(u, v), & if u(j) > 0, \\ b_{j}(u, v) + 1, & if u(j) < 0. \end{matrix}  (6)

Proof. Let, for brevity, R_{u,v}(q) =^{def} R_{u,v}^{J,−1}(q) for u, v ∈ W^{J}. We proceed by induction on l(v) the result being trivially true if v = e. So suppose that l(v) ≥ 1. Let s ∈ D(v).

Suppose first that s = (i, i + 1)(−i − 1, −i) for some i ∈ [n − 1]. Then, by Propositions 2.4 and 2.7, v(i) > v(i + 1). Since v ∈ W^{J}, this implies that v(i) > 0 > v(i + 1). We have three cases to consider.

a) us < u.

Then, since u ∈ W^{J}, u(i) > 0 > u(i + 1). Therefore N(u, v) = N(us, vs) and b_{j}(u, v) = b_{j}(us, vs) for all j ∈ [n]. So D(u, v) = D(us, vs), b̃_{j}(u, v) = b̃_{j}(us, vs) for all j ∈ D(u, v), and by Theorem 2.1 and our induction hypothesis we have that

R_{u,v}(q) = R_{us,vs}(q)
= q^{l(us,vs)} ∏_{j∈D(us,vs)}(1 − q^{−b̃_{j}(us,vs)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

as desired.

b) u < us ∉ W^{J}.

Then, by Propositions 2.4 and 2.7, either u(i) < u(i + 1) < 0 or 0 < u(i) < u(i + 1). In the first case

N(u, vs) = (N(u, v) \ {i}) ∪ {i + 1}

and b_{j}(u, v) = b_{j}(u, vs) for j ∈ [n] \ {i + 1}, b_{i}(u, v) = b_{i+1}(u, vs). Therefore

D(u, vs) \ {i + 1} = D(u, v) \ {i}

and i ∈ D(u, v) if and only if i + 1 ∈ D(u, vs). Hence, by Theorem 2.1 and our induction hypothesis

R_{u,v}(q) = qR_{u,vs}(q)
= q q^{l(u,vs)} ∏_{j∈D(u,vs)}(1 − q^{−b̃_{j}(u,vs)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

as desired.

In the second case N(u, vs) = (N(u, v)\{i+1})∪{i}, and b_{j}(u, vs) = b_{j}(u, v) for j ∈ [n] \ {i + 1}, b_{i}(u, vs) = b_{i+1}(u, v). Therefore D(u, vs) \ {i} = D(u, v) \ {i + 1}, and i ∈ D(u, vs) if and only if i + 1 ∈ D(u, v), and the result follows as before.

c) u < us ∈ W^{J}.

Then, by Propositions 2.4 and 2.7, u(i) < 0 < u(i + 1). Therefore N(u, vs) = N(u, v) \ {i, i + 1}, N(us, vs) = N(u, v), b_{j}(us, vs) = b_{j}(u, vs) = b_{j}(u, v) for j ∈ [n] \ {i + 1}, and

b_{i+1}(us, vs) + 1 = b_{i}(u, v) = b_{i+1}(u, v) − 1.  (7)

So

D(u, vs) = D(u, v) \ {i, i + 1} = D(us, vs) \ {i, i + 1}

and i + 1 ∈ D(u, v) if and only if i ∉ D(u, v) if and only if i ∉ D(us, vs) if and only if i + 1 ∈ D(us, vs). Hence, we have from our induction hypothesis that

R_{u,vs}(q) = q^{l(u,vs)} ∏_{j∈D(u,vs)}(1 − q^{−b̃_{j}(u,vs)})
= q^{l(u,v)−1} ∏_{j∈D(u,v)\{i,i+1}}(1 − q^{−b̃_{j}(u,v)}),  (8)

and, if us ≤ vs,

R_{us,vs}(q) = q^{l(us,vs)} ∏_{j∈D(us,vs)}(1 − q^{−b̃_{j}(us,vs)})
= q^{l(u,v)−2} ∏_{j∈D(u,v)\{i,i+1}}(1 − q^{−b̃_{j}(u,v)}) A(u, v),

where

A(u, v) =^{def} \{ \begin{matrix}(1 − q^{−b̃_{i}(us,vs)}), & if i ∈ D(u, v), \\ (1 − q^{−b̃_{i+1}(us,vs)}), & if i ∉ D(u, v). \end{matrix}

Therefore, by Theorem 2.1,

R_{u,v}(q) = q^{l(u,v)−1}  ∏_{j∈D(u,v)\{i,i+1}}(1 − q^{−b̃_{j}(u,v)}) (q − 1 + A(u, v))

and the result follows since b̃_{i}(u, v) − 1 = b̃_{i}(us, vs) if i ∈ D(u, v) and b̃_{i+1}(us, vs) = b̃_{i+1}(u, v) − 1 if i ∉ D(u, v).

If us ≰ vs then from (7) and the comments at the beginning of this section we conclude that b_{i}(u, v) = 0, b_{i+1}(u, v) = 1. Hence i + 1 ∈ D(u, v) and by Theorem 2.1 and (8) we have that

R_{u,v}(q) = (q − 1)R_{u,vs}(q)
= (q − 1)q^{l(u,v)−1} ∏_{j∈D(u,v)\{i,i+1}}(1 − q^{−b̃_{j}(u,v)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

since b̃_{i+1}(u, v) = 1, and the result again follows.

Assume now that W^{J} = (S_{n}^{B})^{A_{n−1}} and that s = (−1, 1). Then, by Proposition 2.4, v(1) < 0. We have two cases to consider. a) us < u.

Then u(1) < 0. Therefore N(u, v) = N(us, vs), b_{j}(u, v) = b_{j}(us, vs) for all j ∈ [n], and hence D(u, v) = D(us, vs), and 1 ∉ D(u, v). So, by Theorem 2.1 and our induction hypothesis,

R_{u,v}(q) = R_{us,vs}(q)
= q^{l(us,vs)} ∏_{j∈D(us,vs)}(1 − q^{−b̃_{j}(us,vs)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

as desired.

b) u < us.

Then u(1) > 0 and necessarily us ∈ (S_{n}^{B})^{A_{n−1}}. Hence N(u, v) = N(us, vs), N(u, vs) = N(u, v) \ {1}, and b_{j}(u, v) = b_{j}(us, vs) = b_{j}(u, vs) for all j ∈ [n] \ {1},

b_{1}(u, vs) = b_{1}(u, v) − 1 = b_{1}(us, vs) + 1.  (9)

So

D(u, vs) = D(u, v) \ {1} = D(us, vs) \ {1},

and 1 ∈ D(u, v) if and only if 1 ∈ D(us, vs). Hence we have from our induction hypothesis that

R_{u,vs}(q) = q^{l(u,vs)} ∏_{j∈D(u,vs)}(1 − q^{−b̃_{j}(u,vs)})
= q^{l(u,v)−1} ∏_{j∈D(u,v)\{1}}(1 − q^{−b̃_{j}(u,v)}),  (10)

and, if us ≤ vs,

R_{us,vs}(q) = q^{l(us,vs)} ∏_{j∈D(us,vs)}(1 − q^{−b̃_{j}(us,vs)})
= q^{l(u,v)−2} ∏_{j∈D(u,v)\{1}}(1 − q^{−b̃_{j}(u,v)}) A(u, v),

where

A(u, v) =^{def} \{ \begin{matrix}1 − q^{−b̃_{1}(u,v)+1}, & if 1 ∈ D(u, v), \\ 1, & if 1 ∉ D(u, v). \end{matrix}  (11)

Therefore, by Theorem 2.1,

R_{u,v}(q) = q^{l(u,v)−1} ∏_{j∈D(u,v)\{1}}(1 − q^{−b̃_{j}(u,v)})(q − 1 + A(u, v)),

and the result follows from (11).

If us ≰ vs then from (9), and the comments at the beginning of this section, we conclude that b_{1}(u, v) = 1. Hence 1 ∈ D(u, v) and by Theorem 2.1 and (10) we have that

R_{u,v}(q) = (q − 1)R_{u,vs}(q)
= (q − 1) q^{l(u,v)−1} ∏_{j∈D(u,v)\{1}}(1 − q^{−b̃_{j}(u,v)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

since b̃_{1}(u, v) = 1, and the result again follows.

Assume now that W^{J} = (S_{n}^{D})^{A_{n−1}}, and that s = (−1, 2)(−2, 1). Then, by Proposition 2.7, v(1)+v(2) < 0. This, since v ∈ (S_{n}^{D})^{A_{n−1}}, implies that v(1), v(2) < 0 (for if v(1) < 0 < v(2) then v(−1) > 0 so, by (3), v(2) = v(−1) + 1 and hence v(1) + v(2) = 1 > 0, and similarly if v(1) > 0 > v(2)). We have two cases to consider.

a) us < u.

Then u(1), u(2) < 0. Therefore N(u, v) = N(us, vs), b_{j}(u, v) = b_{j}(us, vs) for all j ∈ [n], and hence D(u, v) = D(us, vs), and 1, 2 ∉ D(u, v), and we conclude as in the corresponding case for (S_{n}^{B})^{A_{n−1}}.

b) u < us.

Then, by Proposition 2.7, u(1)+u(2) > 0. There are then three subcases to consider.

i) u(1) < 0 < u(2).

Then, since u ∈ (S_{n}^{D})^{A_{n−1}}, u(2) = 1 + u(−1) and hence us ∉ (S_{n}^{D})^{A_{n−1}}. Therefore N(u, vs) = (N(u, v) \ {2}) ∪ {1}, b_{j}(u, v) = b_{j}(u, vs) for all j ∈ [n] \ {1, 2}, and b_{2}(u, v) = b_{1}(u, v) ≡ 0 (mod 2). So 1, 2 ∉ D(u, v) and hence D(u, vs) = D(u, v).

Therefore we have from our induction hypothesis and Theorem 2.1 that

R_{u,v}(q) = qR_{u,vs}(q)
= qq^{l(u,vs)} ∏_{j∈D(u,vs)}(1 − q^{−b̃_{j}(u,vs)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

as desired.

ii) u(1) > 0 > u(2).

Then u(2) = u(−1) + 1 and hence us ∉ (S_{n}^{D})^{A_{n−1}}. Therefore N(u, vs) = (N(u, v) \ {1})∪{2}, b_{j}(u, v) = b_{j}(u, vs) for all j ∈ [n]\{1, 2}, and b_{2}(u, vs)+1 = b_{1}(u, v)−1 ≡ 1 (mod 2). So 1, 2 ∉ D(u, v) and hence D(u, vs) = D(u, v), and we conclude as above.

iii) 0 < u(1), u(2).

Then u(2) = u(1) + 1 and hence us ∈ (S_{n}^{D})^{A_{n−1}}. Therefore N(us, vs) = N(u, v), N(u, vs) = N(u, v) \ {1, 2}, b_{j}(u, v) = b_{j}(u, vs) = b_{j}(us, vs) for all j ∈ [n] \ {1, 2}, and

b_{2}(u, vs) = b_{2}(u, v) − 1 = b_{2}(us, vs) + 1 = b_{1}(u, vs) ≡ 0 (mod 2), (12)
b_{2}(u, v) − 1 = b_{1}(u, v) − 2 = b_{1}(us, vs) + 2 ≡ 0 (mod 2).  (13)

So 2 ∈ D(u, v), D(us, vs) = D(u, v) and D(u, vs) = D(u, v) \ {1, 2}. Hence, by induction

R_{u,vs}(q) = q^{l(u,vs)} ∏_{j∈D(u,vs)}(1 − q^{−b̃_{j}(u,vs)})
= q^{l(u,v)−1} ∏_{j∈D(u,v)\{2}}(1 − q^{−b̃_{j}(u,v)}),

and, if us ≤ vs,

R_{us,vs}(q) = q^{l(us,vs)} ∏_{j∈D(us,vs)}(1 − q^{−b̃_{j}(us,vs)})
= q^{l(u,v)−2} (1 − q^{−b̃_{2}(u,v)+1}) ∏_{j∈D(u,v)\{2}}(1 − q^{−b̃_{j}(u,v)})

so

R_{u,v}(q) = (q − 1)R_{u,vs}(q) + qR_{us,vs}(q)
= q^{l(u,v)−1} (q − q^{−b̃_{2}(u,v)+1}) ∏_{j∈D(u,v)\{2}}(1 − q^{−b̃_{j}(u,v)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)})

as desired. If us ≰ vs then, by (12), (13) and the comments at the beginning of this section, b_{2}(u, v) = 1, hence

R_{u,v}(q) = (q − 1)R_{u,vs}(q)
= q^{l(u,v)−1} (q − 1) ∏_{j∈D(u,v)\{2}}(1 − q^{−b̃_{j}(u,v)})
= q^{l(u,v)} ∏_{j∈D(u,v)}(1 − q^{−b̃_{j}(u,v)}),

and the result again follows.

This concludes the induction step and hence the proof. □

We illustrate the preceding theorem with an example. Suppose W^{J} = (S_{7}^{B})^{A_{6}}, u = [3, 4, 5, 6, −2, 7, −1] and v = [−4, 5, −3, 6, 7, −2, −1]. Then (b̃_{1}(u, v), …, b̃_{7}(u, v)) = (2, 1, 1, 0, 1, 1, 1) so by Proposition 2.4 and Theorem 3.1 we have that R_{u,v}^{J,−1}(q) = q^{17−12}(1 − q^{−1})^{2}. On the other hand, if W^{J} = (S_{7}^{D})^{A_{6}} then R_{u,v}^{J,−1}(q) = q^{13−10}(1 − q^{−1})^{2}.

We now consider the Hermitian symmetric pairs (S_{n}^{B}, B_{n−1}) and (S_{n}^{D}, D_{n−1}).

Theorem 3.2 Let u, v ∈ (S_{n}^{B})^{B_{n−1}}, u < v. Then

R_{u,v}^{J,−1}(q) = q^{l(u,v)} (1 − q^{−1}).  (14)

Proof. We proceed by induction on l(v), the result being clear if l(v) = 1. So assume that l(v) ≥ 2 and let u < v. If l(u, v) = 1 then the result is easy to check, so assume l(u, v) ≥ 2. Let s ∈ D(v). Then from Theorem 2.1 and our induction hypothesis we conclude that

R_{u,v}^{J,−1}(q) = \{ \begin{matrix}q^{l(us,vs)}(1 − q^{−1}), & if us < u, \\ (q − 1) q^{l(u,vs)}(1 − q^{−1}) + q q^{l(us,vs)}(1 − q^{−1}), & if u < us ∈ (S_{n}^{B})^{B_{n−1}}, \\ q q^{l(u,vs)}(1 − q^{−1}), & if u < us ∉ (S_{n}^{B})^{B_{n−1}}, \end{matrix}

(note that, if us ∈ (S_{n}^{B})^{B_{n−1}}, then us < vs since (S_{n}^{B})^{B_{n−1}} is a chain), and (14) follows. □

Recall the notation v_{εi} (ε ∈ {−1, 1}, i ∈ [n]) introduced in §2 for the elements of (S_{n}^{D})^{D_{n−1}}.

Theorem 3.3 Let u, v ∈ (S_{n}^{D})^{D_{n−1}}, u < v. Then

R_{u,v}^{J,−1}(q) = \{ \begin{matrix}q^{l(u,v)}(1 − q^{−1})(1 − q^{− \frac{l(u,v)}{2}} ), & if l(u) + l(v) = 2n − 2, \\ q^{l(u,v)}(1 − q^{−1}), & otherwise. \end{matrix}

Proof. We proceed by induction on l(v), the result being clear if l(v) = 1. So assume that l(v) ≥ 2. If [u, v]^{D_{n−1}} is a chain then the result follows similarly as in the proof of Theorem 3.2. If [u, v]^{D_{n−1}} is not a chain then u = v_{i} and v = v_{−j} for some i, j ∈ [2, n]. Since s_{j−1} ∈ D(v_{−j}) and D(v_{i}) = {s_{i}} we conclude from Theorem 2.1 that

R_{v_{i},v_{−j}}(q) = \{ \begin{matrix}R_{v_{i+1},v_{−j+1}}(q), & if j = i + 1, \\ (q − 1) R_{v_{i},v_{−j+1}}(q) + q R_{v_{i−1},v_{−j+1}}(q), & if j = i, \\ q R_{v_{i},v_{−j+1}}(q), & otherwise, \end{matrix}

and the result follows from our induction hypothesis (note that v_{i−1} ≤ v_{−i+1} if and only if i ≠ 2). □

The results in this section, together with those in [4, §3] and Proposition 2.3, complete the computation of the parabolic R-polynomials of Hermitian symmetric pairs.

A consequence of these results is the following one, which implies that all the roots of these polynomials are (either zero or) roots of unity.

Corollary 3.4 Let (W, W_{J}) be a Hermitian symmetric pair, and u, v ∈ W^{J}, u < v. Then there exist b̃_{1}(u, v), …, b̃_{r}(u, v) ∈ P ( r = r(u, v) ∈ P) such that

R_{u,v}^{J,q}(q) = (−1)^{l(u,v)} ∏_{i=1}^{r}(1 − q^{b̃_{i}(u,v)}),  (15)

and

R_{u,v}^{J,−1}(q) = q^{l(u,v)} ∏_{i=1}^{r}(1 − q^{−b̃_{i}(u,v)}).  (16)

Proof. This follows immediately from Theorems 3.1, 3.2, and 3.3, Proposition 2.3, the main result of [4, §3], and computer calculations. □

The parabolic Kazhdan-Lusztig R-polynomials for the exceptional Hermitian symmetric pairs have been computed by implementing in Maple 6 the recursion given by Theorem 2.1. These implementations used two Maple packages for handling finite Coxeter groups (Version 2.3) and posets (Version 2.2) developed by John Stembridge.

It would be interesting to have a unified proof of this result, and a unified interpretation of the integers b̃_{1}(u, v), …, b̃_{r}(u, v) appearing in (15) and (16).

In the case of a lower interval, more can be said. For a finite Coxeter group W let E(W) ⊂ P be its set of exponents.

Corollary 3.5 Let (W, W_{J}) be a Hermitian symmetric pair and v ∈ W^{J}. Then there exists T (v) ⊆ E(W) such that

R_{e,v}^{J,q}(q) = (−1)^{l(v)} ∏_{j∈T (v)}(1 − q^{j}),

and

R_{e,v}^{J,−1}(q) = q^{l(v)} ∏_{j∈T (v)}(1 − q^{−j}).

Proof. The result follows immediately from Corollary 3.5 of [4] for the Hermitian symmetric pairs of type A, from Theorems 3.2 and 3.3 if W^{J} ∈ {(S_{n}^{B})^{B_{n−1}}, (S_{n}^{D})^{D_{n−1}}}, from Theorem 3.1 if W^{J} ∈ {(S_{n}^{B})^{A_{n−1}}, (S_{n}^{D})^{A_{n−1}}}, and from computer calculations for the exceptional Hermitian symmetric pairs. □

It is an open problem, in the theory of the (ordinary) R-polynomials, to know if given u, v ∈ W there exists w ∈ W such that R_{u,v}(q) = R_{e,w}(q) ([1]). The last two results (and simple examples) show that, in general, this is false for the parabolic R-polynomials of Hermitian symmetric pairs.

As a further consequence of our main results we obtain combinatorial closed product formulas for certain sums and alternating sums of ordinary R-polynomials.

Corollary 3.6 Let (W, W_{J}) be a Hermitian symmetric pair, and u, v ∈ W^{J}, u < v. Then there exist b̃_{1}(u, v), …, b̃_{r}(u, v) ∈ P ( r = r(u, v) ∈ P) such that

∑_{w∈W_{J}}(−x)^{l(w)}R_{wu,v}(q) = (q − x − 1)^{l(u,v)} ∏_{i=1}^{r}(1 − (\frac{x^{2}}{q})^{b̃_{i}(u,v)}),

for all x ∈ {−1, q}.

Proof. This follows immediately from Corollary 3.4 and Proposition 2.2. □

Note that the integers b̃_{1}(u, v), …, b̃_{r}(u, v) in Corollary 3.6 are explicitly determined in Theorems 3.1, 3.2 and 3.3, and Theorem 3.1 of [4] (see also Corollary 3.8 below).

We conclude by showing that, for the Hermitian symmetric pairs (S_{n}^{B}, A_{n−1}), (S_{n}^{D}, A_{n−1}) and (S_{n}, A_{n−1} \{s_{i}}) (n ≥ 3, 1 ≤ i ≤ n−1) (the “interesting” Hermitian symmetric pairs, according also to [3, §1, p.279]) there is a unified interpretation of the exponents appearing in Corollary 3.4.

Let W^{J} ∈ {(S_{n}^{B})^{A_{n−1}}, (S_{n+1}^{D})^{A_{n}}, (S_{n})^{A_{n−1}\{s_{i}}}} (1 ≤ i ≤ n − 1) and u ∈ W^{J}. By (2), (4), and (2) of [4], and the comments following Propositions 2.5 and 2.8 of [4] we may associate to u a lattice path, which we will denote, for simplicity, by Λ(u), with (1, 1) and (1, −1) steps, starting at (0, 0) and having n steps. Furthermore, by Propositions 2.5, 2.8 and 2.8 of [4], we have that, for all u, v ∈ W^{J}, u ≤ v if and only if Λ(u) lies (weakly) below Λ(v) (write Λ(u) ≤ Λ(v), if this is the case).

Let λ, μ be two such lattice paths, with μ ≤ λ. Let j ∈ [n] and consider the j-th step of λ (from the left). We say that such a step is shifted-allowable (or, s-allowable, for short) with respect to μ if the j-th step of μ is not parallel to it, and ã_{j}(μ, λ) is odd, where ã_{j}(μ, λ) is the vertical distance (divided by two, since it is always even) between the (right end of the) j-th step of λ and the (right end of the) j-th step of μ. Let |λ \ μ| =^{def} ∑_{j=1}^{n} ã_{j}(μ, λ).

Proposition 3.7 Let u, v ∈ (S_{n}^{B})^{A_{n−1}}, u ≤ v. Then

b_{i}(u, v) = ã_{n+1−i}(Λ_{B}(u), Λ_{B}(v)),  (17)

for i = 1, …, n. Furthermore n+1−i ∈ D(u, v) if and only if the i-th step of Λ_{B}(v) is s-allowable with respect to Λ_{B}(u).

Proof. Let i ∈ [n]. Clearly, the vertical height of the path Λ_{B}(v) after i steps, h_{i}(Λ_{B}(v)), equals the difference between the number of up-steps and that of downsteps among the first i steps of Λ_{B}(v). But, by Proposition 2.6, the j-th step of Λ_{B}(v) is an up-step if and only if v(n + 1 − j) < 0. Therefore

h_{i}(Λ_{B}(v)) = |{j ∈ [i] : v(n + 1 − j) < 0}| − |{j ∈ [i] : v(n + 1 − j) > 0}|
= 2|{j ∈ [i] : v(n + 1 − j) < 0}| − i.

Hence

ã_{i}(Λ_{B}(u), Λ_{B}(v)) = \frac{1}{2}(h_{i}(Λ_{B}(v)) − h_{i}(Λ_{B}(u)))

= |{j ∈ [i] : v(n + 1 − j) < 0}| − |{j ∈ [i] : u(n + 1 − j) < 0}|
= b_{n+1−i}(u, v).

Furthermore, again by Proposition 2.6, n + 1 − i ∈ N(u, v) if and only if the i-th steps of Λ_{B}(v) and Λ_{B}(u) are not parallel, and the result follows from the definition of D(u, v) and (17). □

We can now give the following unified interpretation of the exponents appearing in Corollary 3.4.

Corollary 3.8 Let W^{J} ∈ {(S_{n}^{B})^{A_{n−1}}, (S_{n+1}^{D})^{A_{n}}, (S_{n})^{A_{n−1}\{s_{i}}}} for some 1 ≤ i ≤ n−1 and u, v ∈ W^{J}, u ≤ v. Then

R_{u,v}^{J,q}(q) = (−1)^{|λ\μ|} ∏_{j}(1 − q^{\overline{a}_{j}(μ,λ)})

where μ =^{def} Λ(u), λ =^{def} Λ(v), j runs over all the s-allowable steps of λ with respect to μ, and

\overline{a}_{j}(μ, λ) =^{def} \{ \begin{matrix}ã_{j}(μ, λ), & if the j-th step of μ is down, \\ ã_{j}(μ, λ) + 1, & if the j-th step of μ is up. \end{matrix}  (18)

Proof. The result follows immediately from Theorem 3.1, and Propositions 2.3 and 3.7 if W^{J} = (S_{n}^{B})^{A_{n−1}}. If W^{J} = (S_{n+1}^{D})^{A_{n}} then it follows similarly using the facts that if v ∈ (S_{n+1}^{D})^{A_{n}} then Λ_{D}(v) (seen as a lattice path) consists of the first n steps of Λ_{B}(v), and 1 ∉ D(u, v) if u, v ∈ (S_{n+1}^{D})^{A_{n}}.

Finally, suppose W^{J} = (S_{n})^{A_{n−1}\{s_{i}}} for some i ∈ [n − 1]. Let [n]_{−}^{+}(μ, λ) be the set of all j ∈ [n] such that the j-th step of λ is up and the j-th step of μ is down, and define [n]_{+}^{−}(μ, λ) similarly. Then, by Corollary 3.4 of [4],

R_{u,v}^{J,q}(q) = (−1)^{|λ\μ|} ∏_{j∈[n]_{−}^{+}(μ,λ)}(1 − q^{ã_{j}(μ,λ)}).  (19)

On the other hand, by (18)

∏_{j}(1 − q^{\overline{a}_{j}(μ,λ)}) = ∏_{j}(1 − q^{ã_{j}(μ,λ)}) ∏_{j}(1 − q^{ã_{j}(μ,λ)+1})  (20)

where the first (resp., second, third) product is over all j ∈ [n]_{−}^{+}(μ, λ) ∪ [n]_{+}^{−}(μ, λ) (resp., [n]_{−}^{+}(μ, λ), [n]_{+}^{−}(μ, λ)) such that ã_{j}(μ, λ) ≡ 1 (mod 2). Because μ and λ end at the same point (see the comments following Proposition 2.8 in [4]) ã_{n}(μ, λ) = 0 so there is a bijection j ↦ j′ from [n]_{−}^{+}(μ, λ) to [n]_{+}^{−}(μ, λ) such that ã_{j}(μ, λ) = ã_{j′−1}(μ, λ) for all j ∈ [n]_{−}^{+}(μ, λ) (note that 1 ∉ [n]_{+}^{−}(μ, λ) since μ ≤ λ). Hence

∏_{j}(1 − q^{ã_{j}(μ,λ)+1}) = ∏_{j}(1 − q^{ã_{j−1}(μ,λ)}) = ∏_{j}(1 − q^{ã_{j}(μ,λ)})  (21)

where the first two products are over all j ∈ [n]_{+}^{−}(μ, λ) such that ã_{j}(μ, λ) ≡ 1 (mod 2) while the third product is over all j ∈ [n]_{−}^{+}(μ, λ) such that ã_{j}(μ, λ) ≡ 0 (mod 2). Therefore, by (20) and (21)

∏_{j}(1 − q^{\overline{a}_{j}(μ,λ)}) = ∏_{j∈[n]_{−}^{+}(μ,λ)}(1 − q^{ã_{j}(μ,λ)})

where the first product is over all j ∈ [n]_{−}^{+}(μ, λ) ∪ [n]_{+}^{−}(μ, λ) such that ã_{j}(μ, λ) ≡ 1 (mod 2), and the result follows from (19). □

Acknowledgments. Most of the computer calculations in this paper have been carried out using a Maple package for finite Coxeter groups and a Maple package for handling posets developed by John Stembridge.

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