Eulerian polynomials of spherical type

The Eulerian polynomial of a finite Coxeter system (W, S) of rank n records, for each k ∈ {1, . . . , n}, the number of elements w ∈ W with an ascent set {s ∈ S | l(ws) > l(w)} of size k, where l(w) denotes the length of w with respect to S. The classical Eulerian polynomial occurs when the Coxeter group has type An, so W is the symmetric group on n + 1 letters. Victor Reiner gave a formula for arbitrary Eulerian polynomials and showed how to compute them in the classical cases. In this note, we compute the Eulerian polynomial for any spherical type. Let M be a Coxeter matrix of rank n. This means M is a symmetric n× n matrix with entries in N such that Mii = 1 and Mij > 1 if i 6= j. We also refer to M as a diagram, that is, an edge-labeled graph with nodes {1, . . . , n} and edge {i, j} labeled Mij whenever Mij > 2. Our setting will involve Coxeter groups as introduced in [3]. Accordingly, we let (W,S) be a Coxeter system of type M . Then {1, . . . , n} and the set S = {s1, . . . , sn} of simple reflections are in bijective correspondence and we will often identify the two, so S can be viewed as the set of nodes of M . We also write W (M) instead of W to record the dependence on M . If W (M) is finite, then M is called spherical. The connected spherical diagrams M are An (n ≥ 1), Bn (n ≥ 2), Dn (n ≥ 4), En (n = 6, 7, 8), F4, G2, Hn (n = 3, 4), and I (m) 2 (m ≥ 3). The double occurrences in this list are A2 = I (3) 2 , B2 = I (4) 2 , and G2 = I (6) 2 . The nodes of these diagrams are labeled as in [3]. In this note, we assume that M is spherical. Two great assets of the study of Coxeter groups are the reflection representation ρ and the root system Φ. Both are related to the vector space V = ⊕iRαi with formal basis αi (1 ≤ i ≤ n) supplied with the symmetric bilinear form (·, ·) determined by (αi, αj) = −2 cos(2π/Mij) for 1 ≤ i, j ≤ n. The reflection representation of W is the group homomorphism ρ from W to the orthogonal group on V with respect to (·, ·) for which ρ(s)αj = αj − (αj , αs)αs, where j and s are nodes of M . This representation is faithful. 2 Arjeh M. Cohen As M is spherical, (·, ·) is positive definite, so W may be viewed as a finite real orthogonal group in n dimensions. Now Φ = ⋃ s∈S Wαs is a root system in the sense of [6]; its members are called roots. The elements αs for s ∈ S are called the simple roots. In the case of a Weyl group, a root system in the sense of [3] can be obtained from Φ by adjusting the length of certain roots. The set of positive roots of Φ is defined to be Φ = Φ ∩ (⊕sR≥0αs). It is well known that Φ is the disjoint union of Φ and −Φ. For j ∈ {1, . . . , n}, define pj to be the number of elements w ∈ W such that {s ∈ S | ρ(w)αs ∈ Φ } has size j. This number is related to the descent statistics discussed in [2, 10]. The Eulerian polynomial of type M is P (M, t) = n