Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 1, 115-140.

Green Functions and a Conjecture of Geck and Malle^{1}

Toshiaki Shoji

Department of Mathematics, Science University of Tokyo Noda, Chiba 278-8510, Japan

Abstract. For any special unipotent class C of a split reductive group G over a finite field F_{q}, a special piece Ĉ is defined. It is known that the cardinality of Ĉ(F_{q}) is a polynomial in q. Geck and Malle proposed a conjectural algorithm for computing these polynomials, and verified it in the case of exceptional groups. In this paper we show, in the case of classical groups, that their conjecture is reduced to another conjecture concerning Springer representations of W. We verify this conjecture in the case where G is of type B_{n}, C_{n} or D_{n} with n ≤ 6.

1. Introduction and the statement of results

1.1. Let W be a finite Coxeter group, and W^{∧} the set of irreducible characters of W. To each χ ∈ W^{∧} one can attach a polynomial D_{χ} ∈ R[t] (where t is an indeterminate), i.e., the generic degree of χ. The a-function a(χ) is defined as the biggest integer s ≥ 0 such that t^{s} divides the polynomial D_{χ}. Let V be an RW -module affording the reflection character of W. The b-function b(χ) is defined as the smallest integer r ≥ 0 such that χ occurs with non-zero multiplicity in V^{⊗r}. We always have a(χ) ≤ b(χ), and χ is called special if the equality holds. The set W^{∧} is partitioned into various two-sided cells of W. Each two-sided cell contains a unique special character of W.

Let S_{W} be the coinvariant algebra of the symmetric algebra S(V). Then S_{W} = ⊕_{i≥0}S_{W}^{i} is a graded regular RW -module. We define, for any character φ of W, a polynomial R(φ) ∈ Z[t] by

1This paper is a contribution to the Joint Research Project “Representation Theory of Finite and Algebraic Groups” 1997–99 under the Japanese-German Cooperative Science Promotion Program supported by JSPS and DFG.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

R(φ) = ∑_{i≥0}⟨φ, S_{W}^{i} ⟩_{W} t^{i},

where ⟨ , ⟩_{W} denotes the inner product of class functions of W, (here the RW -module S_{W}^{i} is also regarded as a character of W). For χ ∈ W^{∧}, R(χ) is called the fake degree of χ.

1.2. Let G be a connected reductive algebraic group over k, where k is an algebraic closure of a finite field. Let W be the Weyl group of G. By the Springer correspondence there exists a natural injective map from W^{∧} to the set I of all the pairs (C, L), where C is a unipotent class in G and L is a G-equivariant irreducible (Q̄_{l}-) local system on C. We denote by I_{0} the subset of I obtained as the image of this map. If L is a constant sheaf Q̄_{l}, any pair (C, L) is contained in I_{0}. A unipotent class C is called special if (C, Q̄_{l}) corresponds to a special character. Then special unipotent classes are in bijective correspondence with special characters of W. This was known by Lusztig [8] for good characteristic case, and was recently verified also for bad characteristic case by Geck and Malle [4]. We define, for a special unipotent class C, a special piece Ĉ as a subset of the unipotent variety G_{uni} consisting of all the elements in the closure C̄ of C which are not in the closure of any special unipotent class C′ ≠ C, C′ ⊂ C̄. Hence Ĉ is an irreducible, locally closed subvariety in G_{uni}. It is known by [14] that special pieces form a partition of G_{uni}. (Note that Th.1.4 and Th.1.5 in Chap. III of [14] substantially contains the proof of the above result. However the notion of special pieces is implicit there).

Let F_{q} be a finite field of q elements with ch F_{q} = p. We assume that G has a split F_{q}-structure with Frobenius map F : G → G. Then all the unipotent classes are F -stable. It is known, once p is fixed, that the cardinality of F -fixed points in a unipotent class C is expressed by a polynomial in q. The classification of unipotent classes however depends on p. Nevertheless, Lusztig has shown in [13] the following fact: Let C be the special unipotent class corresponding to a special character χ of W, and Ĉ the special piece associated to C. Then there exists a polynomial f_{χ} ∈ Z[t] such that

|Ĉ^{F} | = f_{χ}(q),

i.e., the polynomial expressing the cardinality |Ĉ^{F} | is independent of the characteristic of the base field.

1.3. The above result of Lusztig is based on a case by case argument. In [5], Geck and Malle proposed a general algorithm for producing polynomials f_{χ}(q) without referring to unipotent classes. Their algorithm is an analogue of the algorithm for computing Green functions (see Section 2), and is given as follows: For i ∈ I_{0}, we denote by χ_{i} the corresponding irreducible character of W. We put a(i) = a(χ_{i}). Let us define a matrix Ω = (ω_{ij})_{i,j∈I_{0}} by

ω_{ij} = t^{N}R(χ_{i} ⊗ χ_{j} ⊗ ε) ∈ Z[t],

where N is the number of reflections in W, and ε is the sign character of W. We consider the following system of equations for unknown λ_{ij}, p_{ij} (i, j ∈ I_{0}),

(1.3.1)  \begin{cases}λ_{i′j′} = 0 & unless a(i′) = a(j′) \\ p_{j′j} = 0 & unless a(j′) > a(j) or j = j′ \\ p_{jj} = t^{a(j)} & for all j ∈ I_{0} \\ ω_{ij} = ∑_{i′,j′∈I_{0}}p_{i′i}λ_{i′j′}p_{j′j} & for all i, j ∈ I_{0} \end{cases}

We choose a total order on I_{0} which is compatible with the order reverse to the preorder induced from a(i), and define matrices Λ = (λ_{i′j′})_{i′,j′∈I_{0}} and P = (p_{j′j})_{j′,j∈I_{0}} along this order. Then the last equations in (1.3.1) can be written as

tP ΛP = Ω.

We define an equivalence relation ∼ on I_{0} by i ∼ j if a(i) = a(j), and consider P, Λ as block matrices with respect to this relation. Then P is a block upper triangular matrix, with diagonal block consisting of identity matrix multiplied by t^{a(i)}, and Λ is a block diagonal matrix. Moreover, they proved that Λ is non-singular. Hence the system of equations (1.3.1) has a unique solution with λ_{ij}, p_{ij} ∈ Q(t), rational functions on t. Under this setting, Geck and Malle stated the following conjecture.

Conjecture 1.4. (Geck and Malle [5]) Let G be as before and W the Weyl group of G. Then (i) λ_{ij}, p_{ij} ∈ Q[t] for i, j ∈ I_{0}, (ii) ∑_{i} λ_{ii} = t^{2N}, where the sum is taken over all i ∈ I_{0} such that χ_{i} is special. (iii) For i ∈ I_{0} such that χ_{i} is special, λ_{ii} coincides with f_{χ_{i}}(t) given in 1.2, i.e., if i = (C, Q̄_{l})

with C special, we have

|Ĉ^{F} | = λ_{ii}(q).

Remarks 1.5. (i) In the case of type A_{n}, the equations (1.3.1) are exactly the same as the equations for computing Green functions for G, and the conjecture follows from the properties of Green functions. Geck and Malle verified the conjecture in [5] in the case of all exceptional groups by using the computer.

(ii) The statements (i) and (ii) in the conjecture make sense also for finite Coxeter groups, or even for certain finite complex reflection groups under an appropriate setting. (Note that in the original conjecture, λ_{ij} and p_{ij} are required only in R[t]. But in the case of Weyl groups, Q[t] would be more appropriate.) Geck and Malle verified that even in those cases, the equations (1.3.1) produce polynomials λ_{ii} = f_{χ} ∈ Z[t] for each special character χ ∈ W^{∧}, satisfying the equation (ii). Hence those polynomials f_{χ}(t) are regarded as those expressing the cardinality of “special pieces” in G^{F} even if G does not exist.

1.6. Let G be as in 1.2. For a unipotent class C in G, choose u ∈ C and let A_{G}(u) be the component group of Z_{G}(u). Then G-equivariant irreducible local systems on C are in bijective correspondence with irreducible characters of A_{G}(u). Hence the pair (C, L) ∈ I is also represented by a pair (u, ρ) with ρ ∈ A_{G}(u)^{∧}. Let B_{u} be the variety of Borel subgroups of G containing u. We have an action of W × A_{G}(u) on the l-adic cohomology group H^{m}(B_{u}) = H^{m}(B_{u}, Q̄_{l}) (the Springer action of W and the natural action of A_{G}(u)). Assume that (C, L) ∈ I is represented by a pair (u, ρ). Then the ρ-isotypic subspace H^{m}(B_{u})_{ρ} of H^{m}(B_{u}) is a W -module. Set d_{u} = dim B_{u}. It is known that the pair (C, L) is in I_{0} if and only if H^{2d_{u}}(B_{u})_{ρ} ≠ 0, and in that case W × A_{G}(u)-module H^{2d_{u}}(B_{u})_{ρ} is written as χ ⊗ ρ for an irreducible character χ of W. The correspondence (C, L) ↦ χ gives rise to a bijection between I_{0} and W^{∧}, which is nothing but the Springer correspondence. Note that the condition (C, L) ∈ I_{0} is also equivalent to the condition H^{i}(B_{u})_{ρ} ≠ 0 for some i ≥ 0 by [10, 24.4]. Concerning the relationship between Springer representations and the a-function on W^{∧}, the followings are known.

1.6.1. ([1]) Assume that (u, 1) corresponds to χ ∈ W^{∧}. Then d_{u} = b(χ) ≥ a(χ). In particular, if χ is special, then we have d_{u} = a(χ).

1.6.2. ([7], [12]) More generally, if (u, ρ) corresponds to χ ∈ W^{∧}, then we have d_{u} ≥ a(χ).

We can state the following conjecture, which gives a connection between the a-function and Springer representations occurring in lower cohomology.

Conjecture 1.7. Let G be as in 1.2. Assume that χ ∈ W^{∧} corresponds to (C, L), and that χ_{1} ∈ W^{∧} corresponds to (C_{1}, L_{1}) = (u_{1}, ρ_{1}) with C_{1} ⊂ C̄.

(i) If χ occurs in H^{i}(B_{u_{1}})_{ρ_{1}} for some i, then a(χ_{1}) ≥ a(χ).

(ii) In the setting in (i), assume further that C is special with L = Q̄_{l} and that C_{1} ⊄ Ĉ. Then a(χ_{1}) > a(χ).

In the remainder of this section, we assume that p is good for G. Then in the case of exceptional groups the first statement of Conjecture 1.7 is obtained from the following stronger fact which is checked by making use of the tables of Springer correspondence (see, e.g. [3]).

Proposition 1.8. Let G be of exceptional type, and assume that p is good for G. Assume further that χ ∈ W^{∧} corresponds to (C, L), and that χ_{1} corresponds to (C_{1}, L_{1}) with C_{1} ⊊ C̄. Then we have a(χ_{1}) ≥ a(χ).

In the case of exceptional groups, the second statement of Conjecture 1.7 is also verified by using the table of Green functions. The author is indebted to Frank Lübeck for checking this by using CHEVIE. Note that in contrast to Proposition 1.8, the second statement does not hold if one drops the assumption in (i).

We note also that Proposition 1.8 does not hold in the case of classical groups in general (see 4.13 for counterexamples). In Section 4, we show some properties of Springer representations, and using them we verify Conjecture 1.7 for G of type B_{n}, C_{n} or D_{n} with n ≤ 6, without computing Green functions. The main purpose of this note is to show the following weaker version of Conjecture 1.4 in the case of classical groups, assuming Conjecture 1.7.

Theorem 1.9. Let G be of classical type, and assume that p is odd. Assume further that Conjecture 1.7 holds for G. Then λ_{ij}, p_{ij} ∈ Q[t, t^{−1}]. Moreover, the statements (ii), (iii) in Conjecture 1.4 hold for λ_{ii}.

The idea of the proof is to compare the equations in (1.3.1) with the equations used for computing Green functions, which are obtained by expressing Green functions in terms of certain functions, associated to pairs (C, L) ∈ I_{0}, on the set of unipotent elements. These functions form a basis of the space V_{0} of functions generated by Green functions. We construct a new basis of V_{0} which has closer relations with the a-function. The theorem is obtained by expressing Green functions in terms of this new basis. It would be interesting to find a geometric interpretation of this basis.

The author is grateful to Meinolf Geck and Gunter Malle for valuable discussions concerning their result.

2. Green functions

2.1. In this subsection, we shall recall the definition of Green functions, and give an algorithm for computing them. We follow the notation in 1.2. Besides it, we assume that p is good for G, (and q ≡ 1 (mod 3) if G is of type E_{8}), and that G^{F} is of split type. Then for each unipotent class C, there exists a good representative u ∈ C^{F} as in [16], [2], a “split element” in C, which is unique up to G^{F} -conjugacy modulo the center of G. Then F acts trivially on the component group A_{G}(u). The set of G^{F} -conjugacy classes in C^{F} is in bijective correspondence with the set A_{G}(u)/∼ of conjugacy classes in A_{G}(u). We denote by u_{a} a representative of the G^{F} -conjugacy class in C^{F} corresponding to a ∈ A_{G}(u)/∼. For each w ∈ W, the Green function Q_{w} : G_{uni}^{F} → Q̄_{l} is defined by

(2.1.1)  Q_{w}(g) = ∑_{m=0}^{d_{u}} Tr ((w, a), H^{2m}(B_{u}))q^{m},

if g ∈ G_{uni}^{F} is G^{F} -conjugate to u_{a}. Hence Q_{w} is a G^{F} -invariant function on G_{uni}^{F}. We define, for each χ ∈ W^{∧}, a function Q_{χ} on G_{uni}^{F} by

Q_{χ} = |W |^{−1} ∑_{w∈W} χ(w)Q_{w}.

Now assume that χ corresponds to i = (C, L) ∈ I_{0} under the Springer correspondence. We consider the intersection cohomology IC(C̄, L) on G. Then it is known, by Borho-MacPherson [1] that the χ-isotypic part H^{2m}(B_{u_{1}})_{χ} is expressed as

(2.1.2)  H^{2m}(B_{u_{1}})_{χ} = V_{χ} ⊗ H_{u_{1}}^{2m−2d_{u}} IC(C̄, L),

where V_{χ} denotes the irreducible W -module affording χ, and u ∈ C. (Here K = IC(C̄, L) is regarded as a complex on G by extending by 0 outside of C̄, and H_{u_{1}}^{j}K denotes the stalk at u_{1} of the j-th cohomology sheaf of the complex K.) Hence Q_{χ} is also expressed, for a split element g ∈ G_{uni}^{F}, as

(2.1.3)  Q_{χ}(g) = ∑_{m≥0}(dim H_{g}^{2m−2d_{u}} IC(C̄, L))q^{m}.

In what follows, we denote Q_{χ} by Q_{i} if χ corresponds to i = (C, L) ∈ I_{0}.

2.2. Let us define, for each i = (C, L) ∈ I, a G^{F} -invariant function ψ_{i} on G_{uni}^{F} by

ψ_{i}(g) = \begin{cases}ρ(a) & if g is G^{F} -conjugate to u_{a}, \\ 0 & if g ∉ C^{F} , \end{cases}

where (C, L) is represented by (u, ρ).

Let V be the Q̄_{l}-space of G^{F} -invariant functions on G_{uni}^{F}. We define an inner product on V by

⟨f, h⟩= ∑_{g∈G_{uni}^{F}} f(g)h(g).

Set μ_{ij} =⟨ψ_{i}, ψ_{j}⟩. We note that

2.2.1. For each i, j ∈ I_{o}, there exists a polynomial μ_{ij} ∈ Q[t] such that μ_{ij} = μ_{ij}(q). If i = (C, Q̄_{l}), then we have μ_{ii}(q) = |C^{F} |.

In fact, if (u, ρ) corresponds to i ∈ I_{0}, ρ is obtained as the pull back of a character of the component group associated to the adjoint group. But it is known that if G is of adjoint type, A_{G}(u) is isomorphic to the product of various symmetric groups S_{i} (1 ≤ i ≤ 5). Hence ρ(a) ∈ Z for any a ∈ A_{G}(u). On the other hand, the classification of unipotent classes and the structure of A_{G}(u) are independent of p whenever p is good. Furthermore the cardinality of the G^{F} -conjugacy class containing u_{a} is expressed as f_{a}(q) for some polynomial f_{a} ∈ Q[t]. The first statement of 2.2.1 follows from these facts. The second statement is clear from the definition.

It is easily verified that the set {ψ_{i} | i ∈ I} forms a basis of V. In turn it is proved by Lusztig [10] that the set {ψ_{i} | i ∈ I_{0}} forms a basis of the subspace V_{0} of V spanned by Green functions Q_{i} (i ∈ I_{0}). (Here Q_{i} denotes the function Q_{χ} if χ corresponds to i). Now one can express Q_{i} as

(2.2.2)  Q_{i} = ∑_{j∈I_{0}} π_{ji}ψ_{j}

with π_{ji} ∈ Q̄_{l}. We note that

2.2.3. There exists a polynomial π_{ij} ∈ Z[t] such that π_{ij} = π_{ij}(q).

In fact, assume that j = (u_{1}, ρ_{1}) ∈ I_{0}. Then in view of (2.1.1), π_{ji} can be expressed as

(2.2.4)  π_{ji} = ∑_{m≥0}⟨H^{2m}(B_{u_{1}}), χ ⊗ ρ_{1}⟩q^{m},

where ⟨H^{2m}(B_{u_{1}}), χ ⊗ ρ_{1}⟩ denotes the multiplicity of χ ⊗ ρ_{1} in H^{2m}(B_{u_{1}}). It is known that the structure of H^{2m}(B_{u_{1}}) as W × A_{G}(u_{1})-module is independent of p. Hence 2.2.3 holds.

We write C = C_{i} if i = (C, L). We define a partial order on the set of C_{i} (i ∈ I) by C_{i} ≤ C_{j} if C_{i} is contained in the closure of C_{j}. Then it is easy to see that μ_{ij} = 0 unless C_{i} = C_{j}. Moreover, it follows from (2.1.2) and (2.2.4) that we have π_{ij} = 0 unless C_{j} < C_{i} or i = j. If i = j, we have π_{ii} = q^{d_{u}} for i = (u, ρ).

Now the orthogonality relations for Green functions can be transformed to the following formula for Q_{i}, (see, e.g. [17]),

⟨Q_{i}, Q_{j}⟩= ω_{ij}(q).

The following result gives an algorithm for computing Green functions of G^{F}.

Proposition 2.3. (Lusztig [10]) The polynomials π_{ij}, μ_{ij} ∈ Q[t] are the unique solution of the following system of equations with unknown π̃_{ij}, μ̃_{ij},

(2.3.1)  \begin{cases}μ̃_{i′j′} = 0 & unless C_{i′} = C_{j′}, \\ π̃_{j′j} = 0 & unless C_{j′} < C_{j} or j = j′, \\ π̃_{jj} = t^{d_{u}} & for j = (u, ρ) ∈ I_{0}, \\ ω_{ij} = ∑_{i′,j′∈I_{0}}π̃_{i′i}μ̃_{i′j′} π̃_{j′j} & for all i, j ∈ I_{0}. \end{cases}

In fact, as discussed in 1.3, if we choose a total order on I_{0} which is compatible with the preorder for C_{i}, and define matrices M = (μ̃_{i′j′}) and Π = (π̃_{j′j}) along this order, the last equation in (2.3.1) can be written as

tΠMΠ = Ω.

As in 1.3, we can consider M and Π as block matrices with respect to the equivalence relation ≈ (i ≈ j if C_{i} = C_{j}). Then M and Π have the same shape as Λ and P. Moreover M is non-singular since this holds for t = q. Hence (2.3.1) has a unique solution μ̃_{ij}, π̃_{ij} ∈ Q(t). But the argument in 2.2 shows that μ̃_{ij}(t) (resp. π̃_{ij}(t)) coincides with μ_{ij}(t) (resp. π_{ij}(t)) if t is a power of a good prime. Hence μ̃_{ij} = μ_{ij} and π̃_{ij} = π_{ij} as asserted.

3. Proof of Theorem 1.9

3.1. Before going into details of the proof of Theorem 1.9, we prepare some combinatorial lemmas. In this section we regard q also as an indeterminate. Let X = {1, 2, …, k} for some integer k > 0. We fix positive integers d_{1}, d_{2}, …, d_{k}. We denote by P_{X} the set of all the subsets of X. For each I ∈ P_{X}, let us define O_{I} ∈ Z[q] by

O_{I} = ∏_{i∈I}(q^{d_{i}} − 1) ∏_{i∈X−I}(q^{d_{i}} + 1),

We define, for each I ∈ P_{X}, a function Ψ_{I} : P_{X} → {±1} by

Ψ_{I}(J) = (−1)^{|I∩J|}.

For abbreviation, we write Ψ_{i} = Ψ_{{i}}. Then we can write Ψ_{I} = ∏_{i∈I} Ψ_{i}. For each I ∈ P_{X}, set Y_{I} = ∑_{J∈P_{X}} O_{J}Ψ_{I}(J). Then we have

Lemma 3.2. Y_{I} = 2^{k} ∏_{i∈X−I}q^{d_{i}}.

Proof. We show the lemma by induction on k = |X|. The statement of the lemma is easily verified if k = 1. So, we assume that the lemma holds for any proper subset X′ of X. First consider the case where I is a proper subset of X. Then there exists x ∈ X such that x ∉ I. We decompose P_{X} as P_{X} = P′ ∪ P′′, where P′ = P_{X′} with X′ = X − {x} and P′′ is the set of all the subsets of X containing x. We have

Y_{I} = ∑_{J∈P′} O′_{J}(q^{d_{x}} + 1)(−1)^{|I∩J|} + ∑_{J′∈P′′} O′′_{J′}(q^{d_{x}} − 1)(−1)^{|I∩J′|},

where O′_{J} coincides with O_{J}^{(X′)}, a similar object as O_{J} defined by replacing X by X′, and O′′_{J′} coincides with O_{J′−{x}}^{(X′)} under a bijection P′′ ≃ P_{X′}, (J′ ↔ J′ − {x}). Since |I ∩ J′| = |I ∩ (J′ − {x})|, we have

Y_{I} = 2q^{d_{x}}Y_{I}^{(X′)},

where Y_{I}^{(X′)} is a similar object as Y_{I} defined by using P_{X′}. Hence by induction, we obtain the required formula.

Next consider the case where I = X. Again we decompose P_{X} as P_{X} = P′ ∪ P′′, where P′ = P_{X′} with X′ = {1, 2, …, k − 1}, and P′′ is the set of all the subsets of X containing k. Then we have

Y_{I} = ∑_{J∈P′} O′_{J}(q^{d_{k}} + 1)(−1)^{|J|} + ∑_{J′∈P′′} O′′_{J′}(q^{d_{k}} − 1)(−1)^{|J′|}.

As before, O′_{J} coincides with O_{J}^{(X′)} and O′′_{J′} coincides with O_{J′−{k}} under a bijection P′′ ≃ P_{X′}. Then the last formula is equal to

2 ∑_{J∈P_{X′}} O′_{J}(−1)^{|J|} = 2Y_{X′}^{(X′)}.

Hence by induction, we have Y_{I} = 2^{k} as asserted. This proves the lemma. □

3.3. Let V_{X} be the space of R(q)-valued functions on P_{X}. We define an inner product on V_{X} by

⟨f, h⟩_{X} = ∑_{J∈P_{X}} O_{J}f(J)h(J)

for f, h ∈ V_{X}. The inner product ⟨Ψ_{I}, Ψ_{J}⟩ is computed as follows. Since Ψ_{I} = ∏_{i∈I} Ψ_{i}, we see that Ψ_{I}Ψ_{J} = Ψ_{I⊖J}, where I ⊖ J denotes the symmetric difference I ∪ J − I ∩ J. Hence we can write

(3.3.1)  ⟨Ψ_{I}, Ψ_{J}⟩_{X} = Y_{I⊖J}.
Let us define, for each J ∈ P_{X}, a function Θ_{J} ∈ V_{X} by
(3.3.2)  Θ_{J} = ∑_{J′⊂J}(−1)^{|J|+|J′|}q_{J′}Ψ_{J′},

where q_{J′} = ∏_{j∈J′} q^{d_{j}}. We show Lemma 3.4. Let I, J ∈ P_{X} be such that |I| < |J| or |I| = |J| and I ≠ J. Then we have ⟨Ψ_{I}, Θ_{J}⟩_{X} = 0. In particular, ⟨Θ_{I}, Θ_{J}⟩_{X} = 0 for any I, J ∈ P_{X} such that I ≠ J.

Proof. It is enough to show the first assertion. We have

⟨Ψ_{I}, Θ_{J}⟩_{X} = ∑_{J′⊂J}(−1)^{|J|+|J′|}q_{J′}⟨Ψ_{I}, Ψ_{J′}⟩

= ∑_{J′⊂J}(−1)^{|J|+|J′|}q_{J′}Y_{I⊖J′},

by (3.3.1). By our assumption, there exists x ∈ J such that x ∉ I. Let P_{J} be the set of all the subsets in J. Then P_{J} is written as P_{J} = P′_{J} ∪ P′′_{J}, where P′_{J} (resp. P′′_{J}) is the subset of P_{J} consisting of J′ ⊂ J such that x ∉ J′ (resp. x ∈ J′). We have P′_{J} ≃ P′′_{J}, J′ ↔ J′′ = J′ − {x}, and

(3.4.1)  (−1)^{|J|+|J′′|}q_{J′′}Y_{I⊖J′′} = −(−1)^{|J|+|J′|}q^{d_{x}}q_{J′}Y_{(I⊖J′)∪x}.

But, by Lemma 3.2, we have Y_{(I⊖J′)∪x} = q^{−d_{x}}Y_{I⊖J′}. Hence the right hand side of (3.4.1) is equal to −(−1)^{|J|+|J′|}q_{J′}Y_{I⊖J′}. This implies that ⟨Ψ_{I}, Θ_{J}⟩_{X} = 0, and we obtain the formula. This proves the lemma. □

3.5. Let W be the Weyl group of type B_{n} or D_{n}. A notion of “symbols” was introduced by Lusztig, and irreducible characters of W are parametrized by certain symbols as follows, (see [8, chap.4]). First assume that W is of type B_{n}. Each irreducible character χ of W is parametrized by an ordered pair (α; β) of partitions of n. Here by allowing 0 in the entries of α or β, we may write those partitions as

α : 0 ≤ α_{1} ≤ α_{2} ≤ ⋯ ≤ α_{m+1},
β : 0 ≤ β_{1} ≤ β_{2} ≤ ⋯ ≤ β_{m}

for some m > 0 such that ∑ α_{i} + ∑ β_{j} = n. We further assume that α_{1} ≠ 0 or β_{1} ≠ 0. Let us define sequences S = {λ_{1}, λ_{2}, …, λ_{m+1}}, T = {μ_{1}, μ_{2}, …, μ_{m}} by λ_{i} = α_{i} + (i − 1), μ_{j} = β_{j} + (j − 1). Then Λ = Λ(χ) = \binom{S}{T} is called a (reduced) symbol of defect 1 and rank n corresponding to χ of W. We denote by Φ_{n,1} the set of symbols of rank n and defect 1. Thus we have Φ_{n,1} ≃ W^{∧}.

Next assume that W is of type D_{n}. Each irreducible character χ of W is parametrized by an unordered pair (α; β) of partitions of n, (here if α = β, the pair (α, β) corresponds to two distinct irreducible characters). By allowing 0 in the entries of α or β, we may write them as

α : 0 ≤ α_{1} ≤ α_{2} ≤ ⋯ ≤ α_{m},
β : 0 ≤ β_{1} ≤ β_{2} ≤ ⋯ ≤ β_{m}

for some m > 0 such that ∑ α_{i} +∑ β_{j} = n. As before, we may assume that α_{1} ≠ 0 or β_{1} ≠ 0. Let us define sequences S = {λ_{1}, λ_{2}, …, λ_{m}}, T = {μ_{1}, μ_{2}, …, μ_{m}} by λ_{i} = α_{i} + (i − 1), μ_{j} = β_{j} + (j − 1). Then an unordered pair Λ = Λ(χ) = \binom{S}{T} is called a (reduced) symbol of defect 0 and rank n corresponding to χ ∈ W^{∧}. We denote by Φ̃_{n,0} the set of symbols of rank n and defect 0, where the symbols \binom{S}{T} with S = T are counted twice. Thus we have Φ̃_{n,0} ≃ W^{∧}.

Using the notion of symbols, the partition of W^{∧} into families is easily described as follows: Take χ, χ′ ∈ W^{∧} and let Λ(χ) = \binom{S}{T} and Λ(χ′) = \binom{S′}{T′} be associated to χ, χ′. Then χ and χ′ are in the same family if and only if S ∪ T = S′ ∪ T′ and S ∩ T = S′ ∩ T′. Moreover, the condition for χ being special is given as follows: In the case where W is of type B_{n}, χ ∈ W^{∧} is special if and only if λ_{1} ≤ μ_{1} ≤ λ_{2} ≤ μ_{2} ≤ · · ·. In the case where W is of type D_{n}, χ is special if and only if λ_{1} ≤ μ_{1} ≤ λ_{2} ≤ μ_{2} ≤ ⋯ or μ_{1} ≤ λ_{1} ≤ μ_{2} ≤ λ_{2} ≤ · · ·.

The a-function on W^{∧} is described as follows: For a given Λ = \binom{S}{T} as above, we arrange the elements in S ∪ T as ν_{1} ≤ ν_{2} ≤ ⋯ ≤ ν_{2m+1} (resp. ν_{1} ≤ ν_{2} ≤ ⋯ ≤ ν_{2m}) if W is of type B_{n} (resp. if W is of type D_{n}), respectively. We prepare a sequence

ν_{1}^{0} ≤ ν_{2}^{0} ≤ ⋯ ≤ ν_{2m+1}^{0} = 0 ≤ 0 ≤ 1 ≤ 1 ⋯ ≤ m − 1 ≤ m − 1 ≤ m

in the case where W is of type B_{n}, and

ν_{1}^{0} ≤ ν_{2}^{0} ≤ ⋯ ≤ ν_{2m}^{0} = 0 ≤ 0 ≤ 1 ≤ 1 ≤ ⋯ ≤ m − 1 ≤ m − 1

in the case where W is of type D_{n}. Let a(Λ) = a(χ) for Λ = Λ(χ). Then we have

(3.5.1)  a(Λ) = ∑_{i<j} inf{ν_{i}, ν_{j}} − ∑_{i<j} inf{ν_{i}^{0}, ν_{j}^{0}}.

In fact, the formula for a(Λ) is given in [8, chap.4] in a slightly different form. The formula (3.5.1) for type B_{n} is found in [11, 4.11]. The similar formula for type D_{n} is also deduced easily from the one in [8].

3.6. Let G = Sp_{N}(k) or SO_{N}(k) with ch k ≠ 2. The unipotent classes and the structure of the centralizer of unipotent elements are described as follows. Assume that G = Sp_{N}(k). Then the set of unipotent classes of G is in bijection with the set X_{N} of partitions λ = (1^{r_{1}}, 2^{r_{2}}, …) of N such that r_{i} is even for odd i. We denote by u_{λ} a unipotent element in G corresponding to λ ∈ X_{N}. Then we have

Z_{G}(u_{λ}) ≃ ∏_{i: odd} Sp_{r_{i}} × ∏_{i: even} O_{r_{i}}.

Hence the component group A_{G}(u_{λ}) = Z_{G}(u_{λ})/Z_{G}^{0}(u_{λ}) is isomorphic to a product Z/2Z × ⋯ × Z/2Z, where factors are parametrized by the set Δ_{λ} = {i : even | r_{i} > 0}. We denote by α_{i} the generator of Z/2Z corresponding to i ∈ Δ_{λ}. It follows that the sets A_{G}(u_{λ}) and A_{G}(u_{λ})^{∧} are parametrized by the set P(Δ_{λ}) of all the subsets of Δ_{λ}. For each I ⊂ Δ_{λ}, ρ_{I} ∈ A_{G}(u_{λ})^{∧} is given by

(3.6.1)  ρ_{I}(α_{i}) = \begin{cases}1 & if i ∉ I, \\ −1 & if i ∈ I. \end{cases}

Next assume that G = SO_{N}(k). Then the set of unipotent classes in G is in bijection with the set X′_{N} of partitions λ = (1^{r_{1}}, 2^{r_{2}}, …) of N such that r_{i} is even for even i, except that to λ = (2^{r_{2}}, 4^{r_{4}}, …) ∈ X′_{N} there correspond two unipotent classes. Again let u_{λ} stand for a unipotent element corresponding to λ ∈ X′_{N}. Then we have

Z_{G}(u_{λ}) ≃ ∏_{i: even} Sp_{r_{i}} × {(x_{i}) ∈ ∏_{i: odd} O_{r_{i}} | ∏ det x_{i} = 1}.

Hence A_{G}(u_{λ}) is isomorphic to an index 2 subgroup of ∏_{i:odd} O_{r_{i}}/SO_{r_{i}} ≃ Z/2Z×⋯×Z/2Z, where factors are parametrized by the set Δ_{λ} = {i : odd | r_{i} > 0} and A_{G}(u_{λ}) is defined by the condition that ∑ u_{i}α_{i} ∈ A_{G}(u) if and only if ∑ u_{i} ≡ 0 (mod 2). (Here α_{i} is the generator of Z/2Z corresponding to i ∈ Δ_{λ}). As in the previous case, the set A_{G}(u_{λ}) is parametrized by the set P_{ev}(Δ_{λ}) = {I ⊂ Δ_{λ} | |I| : even}, and A_{G}(u_{λ})^{∧} is parametrized by the set P(Δ_{λ})/∼, where ∼ is the equivalence relation defined by I ∼ Δ_{λ} − I. For each I ∈ P(Δ_{λ})/∼, ρ_{I} is defined similarly.

We now consider the F_{q}-structure on G. We choose a split element u_{λ} in G^{F}. Let C_{λ} be the conjugacy class containing u_{λ}. Then the set of G^{F} -conjugacy classes in C_{λ}^{F} is parametrized by A_{G}(u_{λ}), hence by subsets of Δ_{λ}. We denote by u_{I} ∈ C_{λ}^{F} a representative corresponding to I ⊂ Δ_{λ}. We define a subset Δ_{λ}^{0} of Δ_{λ} by

Δ_{λ}^{0} = {i ∈ Δ_{λ} | r_{i} ≡ 2 (mod 4)}.

For each i ∈ Δ_{λ}^{0}, we put d_{i} = r_{i}/2. For I ⊂ Δ_{λ} let O(u_{I}) be the G^{F} -conjugacy class containing u_{I}. Then the cardinality |O(u_{I})| is a polynomial in q. Furthermore the following formula is easily verified.

3.6.2. There exists a rational function F_{λ}(t) such that

|O(u_{I})| = F_{λ}(q) ∏_{i∈I∩Δ_{λ}^{0}}(q^{d_{i}} − 1) ∏_{i∈Δ_{λ}^{0}−I}(q^{d_{i}} + 1).

3.7. We now recall combinatorial objects introduced by Lusztig [9] to describe the generalized Springer correspondence of classical groups. (Here we only need the Springer correspondence). For an even integer N ≥ 2, let Ψ_{N,1} be the set of pairs \binom{A}{B}, called u-symbols, subject to the following conditions: (i) A is a finite subset of {0, 1, 2, … }, B is a finite subset of {1, 2, … } such that |A| = |B| + 1, (ii) A, B contain no consecutive integers, (iii) ∑_{a∈A} a + ∑_{b∈B} b = \frac{1}{2}N + \frac{1}{2}(|A| + |B|)(|A| + |B| − 1). Moreover a u-symbol \binom{A}{B} is called reduced if 0 ∉ A or 0 ∉ B. Unless otherwise stated, we assume that u-symbols are reduced. We call Ψ_{N,1} the set of (reduced) u-symbols of defect 1 and rank N. Two u-symbols \binom{A}{B}, \binom{A′}{B′} are said to be similar if A ∪ B = A′ ∪ B′ and A ∩ B = A′ ∩ B′. It is known that there exists a natural bijection between the set of similarity classes of Ψ_{N,1} and the set X_{N}, (see 4.3). Hence we can identify similarity classes and unipotent classes in Sp_{N}.

Next, for any odd integer N ≥ 3, let Ψ′_{N,1} be the set of pairs \binom{A}{B}, also called u-symbols, subject to the following conditions: (i)’ A, B are finite subsets of {0, 1, 2 … } such that |A| = |B| + 1, (ii) the same as above, (iii)’ ∑_{a∈A} a + ∑_{b∈B} b = \frac{1}{2}N + \frac{1}{2}((|A| + |B| − 1)^{2} − 1). Also for an even integer N ≥ 4, let Ψ̃′_{N,0} be the set of unordered pairs \binom{A}{B}, where the conditions are the same as (i)’, (ii), (iii)’ except that |A| = |B|, and that \binom{A}{B} is counted twice in the case where A = B. A u-symbol \binom{A}{B} is called reduced if 0 ∉ A ∩ B. As before any elements in Ψ′_{N,1} or Ψ̃′_{N,0} is assumed to be reduced. We define similarity classes for Ψ′_{N,1} or Ψ̃′_{N,0} as in the case of Ψ_{N,1}. Then it is known that there exists a natural bijection from the set of similarity classes in Ψ′_{N,1} (resp. Ψ̃′_{N,0}) to the set X′_{N} for N : odd (resp. N : even). Hence we can identify similarity classes and unipotent classes in these cases also.

3.8. Each similarity class contains a unique u-symbol \binom{A}{B} of the form A = \{a_{1} < a_{2} < ⋯ \}, B = {b_{1} < b_{2} < … } such that a_{1} ≤ b_{1} ≤ a_{2} ≤ b_{2} ≤ ⋯, (or b_{1} ≤ a_{1} ≤ b_{2} ≤ a_{2} ≤ ⋯ in the case of Ψ̃′_{N,0}). Such an element is called a distinguished u-symbol. Let ξ = ξ_{λ} = \binom{A}{B} be a distinguished u-symbol corresponding to λ ∈ X_{N}. We consider the symmetric difference A ⊖ B. A non-empty subset Γ in A ⊖ B is said to be an interval if it is of the form {i, i + 1, …, j − 1, j} with i − 1 ∉ A ⊖ B, j + 1 ∉ A ⊖ B, (and furthermore i ≠ 0 in the case of Ψ_{N,1}). Let A_{λ} be the set of intervals of A ⊖ B. Then the cardinality of A_{λ} is the same as |Δ_{λ}|, and we can identify A_{λ} with Δ_{λ} in the following way. We arrange the elements in A_{λ} in an increasing order Γ_{1} < Γ_{2} < ⋯ < Γ_{l}, and also write Δ_{λ} = {i_{1} < i_{2} < ⋯ < i_{l}}, where l = |A_{λ}| = |Δ_{λ}|. We associate i_{j} to Γ_{j} (j = 1, …, l). Then we have |Γ_{j}| = r_{i_{j}}.

In the case of Ψ_{N,1}, for each I ⊂ A_{λ}, a u-symbol ξ_{I} = \binom{A_{I}}{B_{I}} is obtained from ξ by exchanging the entries A ∩ Γ and B ∩ Γ for Γ ∈ I. (The entries {0, 1, 2, …, k} which do not belong to any interval remain unchanged). If \binom{A_{I}}{B_{I}} ∈ Ψ_{N,1}, this gives an element in the similarity class of ξ. In the case of Ψ′_{N,1} or Ψ̃′_{N,0}, for each subset I ⊂ A_{λ}, a u-symbol ξ_{I} = \binom{A_{I}}{B_{I}} or \binom{B_{I}}{A_{I}} is obtained from ξ, and it gives an element in the similarity class of ξ if it is in Ψ′_{N,1} or Ψ̃′_{N,0}. (We regard \binom{A_{I}}{B_{I}} as an unordered pair). In both cases, all the elements in the similarity class of ξ are obtained in this way. Hence one can identify elements in the similarity class of ξ with a subset of A_{G}(u_{λ})^{∧}.

The following result gives a complete description of the Springer correspondence of classical groups.

3.8.1. Let π : W^{∧} → I_{0} be the Springer correspondence. Then under the above identification, the set I_{0} coincides with Ψ_{2n,1}, Ψ′_{2n+1,1}, Ψ̃_{2n,0} according as G = Sp_{2n}, SO_{2n+1} or SO_{2n}. The correspondence π is given as follows.

(i) G = Sp_{2n}. π : Φ_{n,1} \underrightarrow{∼} Ψ_{2n,1} is given by

\binom{λ_{1} < λ_{2} < ⋯ < λ_{m+1}}{μ_{1} < μ_{2} < ⋯ < μ_{m}} ↦ \binom{λ_{1} < λ_{2} + 1 < ⋯ < λ_{m+1} + m}{μ_{1} + 1 < μ_{2} + 2 < ⋯ < μ_{m} + m}.

(ii) G = SO_{2n+1}. π : Φ_{n,1} \underrightarrow{∼} Ψ′_{2n+1,1} is given by

\binom{λ_{1} < λ_{2} < ⋯ < λ_{m+1}}{μ_{1} < μ_{2} < ⋯ < μ_{m}} ↦ \binom{λ_{1} < λ_{2} + 1 < ⋯ < λ_{m+1} + m}{μ_{1} < μ_{2} + 1 < ⋯ < μ_{m} + (m − 1)}.

(iii) G = SO_{2n}. π : Φ̃_{n,0} \underrightarrow{∼} Ψ̃′_{2n,0} is given by

\binom{λ_{1} < λ_{2} < ⋯ < λ_{m}}{μ_{1} < μ_{2} < ⋯ < μ_{m}} ↦ \binom{λ_{1} < λ_{2} + 1 < ⋯ < λ_{m} + (m − 1)}{μ_{1} < μ_{2} + 1 < ⋯ < μ_{m} + (m − 1)}.

3.9. Let ξ = \binom{A}{B} be a (not necessarily distinguished) u-symbol belonging to the class of u_{λ}. For any i ∈ Δ_{λ} such that |Γ_{i}| is even, we can construct a u-symbol ξ_{i} by replacing A ∩ Γ_{i} and B ∩ Γ_{i} in the entries of ξ. (If ξ ∈ Ψ_{N,1}, Ψ′_{N,1} or Ψ̃_{N,0}, ξ_{i} is again contained in the same set.) Let us write A = {a_{1} < a_{2} < ⋯ } and B = {b_{1} < b_{2} < ⋯ }. Assume that |Γ_{i}| = 2d. Then we can write A ∩ Γ_{i} = {a_{j}, a_{j+1}, ⋯, a_{j+d−1}}, B ∩ Γ_{i} = {b_{k}, b_{k+1}, ⋯, b_{k+d−1}} for some j, k. We have the following lemma.

Lemma 3.10. Let Λ = π^{−1}(ξ) and Λ_{i} = π^{−1}(ξ_{i}) be symbols obtained from ξ and ξ_{i} via the Springer correspondence. Let δ = 1 (resp. δ = 0) in the case of Sp_{N} (resp. SO_{N}), respectively.

(i) Assume that j = k + δ. Then a(Λ_{i}) = a(Λ). (ii) Assume that j > k + δ. Then

\begin{cases}a(Λ_{i}) ≥ a(Λ) + d & if a_{j} < b_{k}, \\ a(Λ_{i}) ≤ a(Λ) − d & if a_{j} > b_{k}. \end{cases}

(iii) Assume that j < k + δ. Then

\begin{cases}a(Λ_{i}) ≤ a(Λ) − d & if a_{j} < b_{k}, \\ a(Λ_{i}) ≥ a(Λ) + d & if a_{j} > b_{k}. \end{cases}

Proof. First consider the case G = Sp_{N}. We assume that a_{j} = x < x + 1 = b_{k} since the other case is discussed by exchanging Λ and Λ_{i}. Then we can write

Λ = (\begin{matrix}⋯ & x + 1 − j, & x + 2 − j, & ⋯, & x + d − j, & ⋯ \\ ⋯ & x + 1 − k, & x + 2 − k, & ⋯, & x + d − k, & ⋯ \end{matrix}) ,
Λ_{i} = (\begin{matrix}⋯ & x + 2 − j, & x + 3 − j, & ⋯, & x + d + 1 − j, & ⋯ \\ ⋯ & x − k, & x + 1 − k, & ⋯, & x + d − 1 − k, & ⋯ \end{matrix}) .

If j = k + 1, we see easily that a(Λ_{i}) = a(Λ). Assume that j ≥ k + 2. Then the entries in Λ can be written in an increasing order as

⋯ ≤ x + 1 − j < x + 2 − j < ⋯ < x + 1 − k ≤ ⋯,

which we denote by {ν_{i}}. The a-function a(Λ) can be computed by the formula (3.5.1). Let {ν′_{i}} be the sequence obtained from {ν_{i}} by replacing x + 1 − j by x + 2 − j and x + 1 − k by x−k. Let α be the value defined by a similar formula as (3.5.1) by using {ν′_{i}} instead of {ν_{i}}. Then we have α ≥ a(Λ)+1. Applying a similar procedure as above for x+2−j, ⋯, x+d−j, successively, we see that a(Λ_{i}) ≥ a(Λ) + d.

In the case where j ≤ k, a similar argument as above shows that a(Λ_{i}) ≤ a(Λ) − d. Next consider the case where G = SO_{N}. We assume that a_{j} = x < x + 1 = b_{k}. Then we

have

Λ = (\begin{matrix}⋯ & x + 1 − j, & x + 2 − j, & ⋯ & x + d − j, & ⋯ \\ ⋯ & x + 2 − k, & x + 3 − k, & ⋯ & x + d + 1 − k, & ⋯ \end{matrix}) ,
Λ_{i} = (\begin{matrix}⋯ & x + 2 − j, & x + 3 − j, & ⋯ & x + d + 1 − j, & ⋯ \\ ⋯ & x + 1 − k, & x + 2 − k, & ⋯ & x + d − k, & ⋯ \end{matrix}) .

Then if j = k, we have a(Λ_{i}) = a(Λ). The other cases are dealt with in a similar way. This proves the lemma. □

In the special case where ξ is a distinguished u-symbol, we have the following result which is straightforward from Lemma 3.10.

Corollary 3.11. Assume that ξ is distinguished. Then we have a(Λ_{i}) ≤ a(Λ) − d unless j = k + δ, in which case we have a(Λ_{i}) = a(Λ).

3.12. Let Δ_{λ}^{1} be the complement of Δ_{λ}^{0} in Δ_{λ}. Hence Δ_{λ} = Δ_{λ}^{0} ∐ Δ_{λ}^{1}. For each I ⊂ Δ_{λ}^{1}, let Δ′_{I} be the set of i ∈ Δ_{λ}^{0} such that the interval Γ_{i} satisfies the relation (i) in Lemma 3.10 with respect to ξ = ξ_{I}. Let Δ_{I} = Δ_{λ}^{0} − Δ′_{I}. Then it follows from Lemma 3.10 that there exists I^{∗} ⊂ Δ_{I} satisfying the following property: For each J ⊂ Δ′_{I}, K ⊂ Δ_{I}, let ξ_{IJK} be the u-symbol corresponding to I ∪ J ∪ (K ⊖ I^{∗}) ∈ Δ_{λ}. Set Λ_{IJK} = π^{−1}(ξ_{IJK}). Then a(Λ_{IJ∅}) is maximal among all a(Λ_{IJK}) for K ⊂ Δ_{I}. Note that, if ξ is distinguished, then in view of Corollary 3.11 we have I^{∗} = ∅, and so Λ_{∅∅∅} corresponds to the u-symbol ξ. More generally as a corollary to Lemma 3.10, we have the following.

Corollary 3.13. Let the notations be as above. Then (i) a(Λ_{IJK}) = a(Λ_{IJ′K}) for J, J′ ⊂ Δ′_{I}.

(ii) a(Λ_{IJK}) < a(Λ_{IJK′}) if K′ ⊊ K ⊂ Δ_{I}.

(iii) d_{u_{λ}} − a(Λ_{IJK}) ≥ ∑_{j∈K} d_{j}.

In fact, (i) and (ii) follows from Lemma 3.10 directly. Moreover, we have

a(Λ_{IJ∅}) − a(Λ_{IJK}) ≥ ∑_{j∈K} d_{j}

by a repeated use of Lemma 3.10. Since d_{u_{λ}} − a(Λ_{IJ∅}) ≥ 0 by 1.6.2, we obtain (iii). (The property (iii) in the corollary is not used later).

3.14. As described in 3.6, the set A_{G}(u_{λ})^{∧} is parametrized by P(Δ_{λ}) or P(Δ_{λ})/∼. Let ρ_{I} ∈ A_{G}(u_{λ})^{∧} be as in (3.6.1). Then for i = (u_{λ}, ρ_{I}) ∈ I_{0}, we obtain a G^{F} -invariant function ψ_{i} as defined in 2.2, which we denote by ψ_{I}. Now I ⊂ Δ_{λ} can be written uniquely in the form I = I_{1} ∪ I_{2} ∪ (I_{3} ⊖ I_{1}^{∗}), where I_{1} ⊂ Δ_{λ}^{1}, I_{2} ⊂ Δ′_{I_{1}}, I_{3} ⊂ Δ_{I_{1}} as in 3.12. We define a G^{F} -invariant function θ_{I} on G_{uni}^{F} by θ_{I} = ψ_{I_{1}}ψ_{I_{2}}θ′_{I_{3}}, where

θ′_{I_{3}} = ∑_{J′⊂I_{3}}(−1)^{|I_{3}|+|J′|}q_{J′}ψ_{J′⊖I_{1}^{∗}} = ψ_{I_{1}^{∗}} ∑_{J′⊂I_{3}}(−1)^{|I_{3}|+|J′|}q_{J′}ψ_{J′}

with q_{J′} = ∏_{j∈J′} q^{d_{j}}. The following property is easily verified by using Corollary 3.13 (ii).

3.14.1. θ_{I} is written in the form θ_{I} = q_{I_{3}}ψ_{I} + α, where α is a linear combination of ψ_{J}, with coefficients in Z[q], such that J ⊊ I and that a(J) > a(I).

When i = (u_{λ}, ρ_{I}) varies over all the elements in I_{0}, the function θ_{I} gives rise to a basis of V_{0} (cf. 2.2). For each i = (u_{λ}, ρ_{I}) ∈ I_{0}, we define a(I) as a(I) = a(i). Then we have the following lemma.

Lemma 3.15. Let I, J ⊂ Δ_{λ}. Then ⟨θ_{I}, θ_{J}⟩= 0 unless a(I) = a(J).

Proof. To simplify the notation, we write ρ_{I}(u_{K}), O(u_{K}) for a unipotent element u_{K} as ρ_{I}(K), O(K). For I, J ⊂ Δ_{λ}, we can write

⟨ψ_{I}, ψ_{J}⟩= ∑_{K} |O(K)|ρ_{I}(K)ρ_{J}(K),

where K runs over all the elements in P(Δ_{λ}) (resp. P_{ev}(Δ_{λ})) in the case where G = Sp_{N} (resp. G = SO_{N}), respectively. We decompose I and J as I = I_{1} ∪ I_{2} ∪ (I_{3} ⊖ I_{1}^{∗}), J = J_{1} ∪ J_{2} ∪ (J_{3} ⊖ J_{1}^{∗}), where I_{1}, J_{1} ∈ Δ_{λ}^{1}, I_{2} ∈ Δ′_{I_{1}}, I_{3} ∈ Δ_{I_{1}}, J_{2} ∈ Δ′_{J_{1}}, J_{3} ∈ Δ_{J_{1}}. We also decompose K as K = K_{1} ∪ K′_{2} with K_{1} ⊂ Δ_{λ}^{1} and K′_{2} ∈ Δ_{λ}^{0}. By 3.6.2, |O(K)| only depends on the K′_{2} parts. Hence, the last formula is equal to (up to non-zero scalar)

(3.15.1)  ∑_{K_{1}∈Δ_{λ}^{1}} ρ_{I_{1}}(K_{1})ρ_{J_{1}}(K_{1}) · ∑_{K′_{2}∈Δ_{λ}^{0}} |O(K′_{2})|ρ_{I′}(K′_{2})ρ_{J′}(K′_{2}),

where I′ = I_{2} ∪ (I_{3} ⊖ I_{1}^{∗}), J′ = J_{2} ∪ (J_{3} ⊖ J_{1}^{∗}). Moreover, the first part ∑ ρ_{I_{1}}(K_{1})ρ_{J_{1}}(K_{1}) coincides with the inner product for characters ρ_{I_{1}} and ρ_{J_{1}} of the elementary abelian 2-group generated by Δ_{λ}^{1}. Hence it is 0 unless I_{1} = J_{1}.

We assume that I_{1} = J_{1}. We decompose K′_{2} ∈ Δ_{λ}^{0} as K′_{2} = K_{2}∪K_{3} with K_{2} ∈ Δ′_{I_{1}} = Δ′_{J_{1}}, K_{3} ∈ Δ_{I_{1}} = Δ_{J_{1}}. Then

∑_{K′_{2}∈Δ_{λ}^{0}} |O(K′_{2})|ρ_{I′}(K′_{2})ρ_{J′}(K′_{2})

= c ∑_{K_{2}∈Δ′_{I_{1}}} |O(K_{2})|ρ_{I_{2}}(K_{2})ρ_{J_{2}}(K_{2}) ∑_{K_{3}∈Δ_{I_{1}}} |O(K_{3})|ρ_{I_{3}⊖I_{1}^{∗}}(K_{3})ρ_{J_{3}⊖I_{1}^{∗}}(K_{3}),

since ρ_{I_{3}⊖I_{1}^{∗}}(K_{3}) = ρ_{I_{3}}(K_{3})ρ_{I_{1}^{∗}}(K_{3}). Here c is a non-zero constant depending only on I_{1}.

We now compute the inner product⟨ψ_{I}, θ_{J}⟩under the assumption |I| ≤ |J|. The previous computation shows that

⟨ψ_{I}, θ_{J}⟩= c ∑_{K_{2}∈Δ_{I_{1}}} |O(K_{2})|ρ_{I_{2}}(K_{2})ρ_{J_{2}}(K_{2}) ∑_{K_{3}∈Δ_{I_{1}}} |O(K_{3})|ρ_{I_{3}}(K_{3})θ′_{J_{3}}(K_{3}).

The last sum of the right hand side coincides with ⟨Ψ_{I_{3}}, Θ_{J_{3}}⟩_{X} in the notation of 3.3, applied for X = Δ_{I_{1}} under the identification Ψ_{I_{3}} ↔ ψ_{I_{3}}, Θ_{J_{3}} ↔ θ′_{J_{3}}. Hence by Lemma 3.4, we know that ⟨Ψ_{I_{3}}, Θ_{J_{3}}⟩_{X} = 0, and so ⟨ψ_{I}, θ_{J}⟩ = 0 unless I_{3} = J_{3}. It follows that ⟨θ_{I}, θ_{J}⟩ = 0 unless I_{1} = J_{1} and I_{3} = J_{3}. Now assume that ⟨θ_{I}, θ_{J}⟩ ≠ 0. Then I and J differs only by I_{2} and J_{2} parts. But then by Corollary 3.13 (i), we see that a(I) = a(J). This proves the lemma. □

3.16. For i = (u, ρ_{I}) ∈ I_{0}, we write θ_{I} as θ_{i}. Also in this case we write q_{i} for q_{I_{3}} = ∏_{j∈I_{3}} q^{d_{j}}. We consider the expression Q_{i} = ∑_{j} π_{ji}ψ_{j} as in (2.2.2). For each i ∈ I_{0}, let us define a G^{F} -invariant function θ̃_{i} on G_{uni}^{F} by

(3.16.1)  θ̃_{i} = ∑_{j} π_{ji}q^{−a(i)}q_{j}^{−1}θ_{j},

where the sum is taken over all j such that a(j) = a(i). If i ≠ j, then π_{ji} = 0 unless C_{j} < C_{i}. This implies that the θ̃_{i} (i ∈ I_{0}) form a basis of V_{0}. In view of 3.14.1, θ̃_{i} is a linear combination of ψ_{j} with coefficients in Z[q, q^{−1}]. Hence, if we set λ_{ij} = ⟨θ̃_{i}, θ̃_{j}⟩, 2.2.1 implies that λ_{ij} ∈ Q[q, q^{−1}]. Moreover, it follows from Lemma 3.15 that λ_{ij} = 0 unless a(i) = a(j). We can express Q_{i} in terms of θ̃_{j} as Q_{i} = ∑_{j∈I_{0}} p_{ji}θ̃_{j}. Now Conjecture 1.7 (i) implies that a(j) ≥ a(i) unless π_{ji} = 0. Hence in view of 3.14.1, we see that p_{ji} = 0 unless a(j) > a(i) or i = j, and that p_{ii} = q^{a(i)}. Moreover, we have p_{ji} ∈ Q[q, q^{−1}]. Finally, the fourth formula in (1.3.1) is obtained from the relation ⟨Q_{i}, Q_{j}⟩= ω_{ij}(q). Summing up the above argument, we have the following.

Proposition 3.17. Let us write λ_{ij} =⟨θ̃_{i}, θ̃_{j}⟩, and Q_{i} = ∑_{j∈I_{0}} p_{ji}θ̃_{j}. Assume that Conjecture 1.7 holds for G. Then λ_{ij}, p_{ij} satisfy the relations in (1.3.1). Moreover, we have λ_{ij}, p_{ij} ∈ Q[q, q^{−1}].

Now in order to prove Theorem 1.9, it is enough to show the following lemma.

Lemma 3.18. Let λ_{ii} be as above. Then (i) ∑_{i} λ_{ii} = q^{2N}, where the sum is taken over all i ∈ I_{0} such that χ_{i} is special.

(ii) For i = (C, Q̄_{l}) ∈ I_{0} with C special, λ_{ii} coincides with |Ĉ^{F} |.

Proof. First we argue on the group G_{C}, the algebraic group defined over C. Let C = C_{i} be the corresponding special unipotent class in G_{C} with u ∈ C. Then by [6], Ĉ is a rational homology manifold, and so the restriction IC(C̄, C)|Ĉ of IC(C̄, C) to Ĉ is a constant sheaf C. It follows from (2.1.2) that H^{2m}(B_{u_{1}})_{χ⊗ρ_{1}} is non-zero only when m = d_{u} and ρ_{1} = 1. Moreover in this case⟨H^{2m}(B_{u_{1}}), χ ⊗ ρ_{1}⟩= 1. The corresponding fact also holds for G. Hence we see that π_{ji} = q^{d_{u}} for j = (C_{j}, Q̄_{l}) such that C_{j} ⊂ Ĉ_{i}. On the other hand, by Conjecture 1.7 (ii), if C_{j} ⊂ C̄_{i} and C_{j} ⊄ Ĉ_{i}, and if a(j) = a(i), we must have π_{ji} = 0. Moreover by [11], if j = (C_{j}, Q̄_{l}) with C_{j} ⊂ Ĉ_{i}, then a(i) = a(j). Hence in (3.16.1) the sum is taken exactly over j = (C_{j}, Q̄_{l}) ∈ I_{0} such that C_{j} ⊂ Ĉ_{i}. Furthermore by a remark in 3.12, we see that q_{j} = 1 and θ_{j} = 1_{C_{j}} in this case. Hence θ̃_{i} = ∑_{j} θ_{j} coincides with the characteristic function 1Ĉ_{i} on Ĉ_{i}^{F}. This implies that λ_{ii} = ⟨θ̃_{i}, θ̃_{i}⟩ = |Ĉ_{i}^{F} |. Thus we have ∑_{i} λ_{ii} = ∑_{i} |Ĉ_{i}^{F} | = q^{2N}. This proves the lemma, and so the theorem follows. □

3.19. It is likely that a similar construction of a basis θ̃_{i} will work also for the case of exceptional groups. As an example, we consider the case of F_{4}. So, assume that G is of type F_{4}, and let C be a unipotent class in G. If the a-function is constant for all χ_{i} ∈ W^{∧} belonging to C, we may choose θ_{i} = ψ_{i}. In the case of F_{4}, there exist two unipotent classes where the a-function is non-constant. They are given as follows; (here we follow the notation in [3]).

C = F_{4}(a_{2}) : a(ϕ_{9,2}) = 2, a(ϕ′′_{2,4}) = 1,
C = A_{2} :  a(ϕ′′_{8,9}) = 9, a(ϕ′′_{1,2}) = 4.

In the case where C = F_{4}(a_{2}), we have A_{G}(u) ≃ Z/2Z, and |Z_{G}(u)^{F} | = |Z_{G}(u′)^{F} | = 2q^{8} for two representatives u, u′ of G^{F} -classes in C^{F}. Therefore⟨ψ_{i}, ψ_{i′}⟩= 0 for i, i′ ∈ I_{0} belonging to C, and so we may choose θ_{i}, θ_{i′} so that they coincide with ψ_{i}, ψ_{i′}. In the case where C = A_{2}, we have A_{G}(u) ≃ Z/2Z, and

|Z_{G}(u)^{F} | = 2q^{17}(q^{2} − 1)(q^{3} − 1), |Z_{G}(u′)^{F} | = 2q^{17}(q^{2} − 1)(q^{3} + 1),

for two representatives u, u′ in G^{F} -classes in C^{F}. Let i = (u, 1), i′ = (u, −1) ∈ I_{0}, where −1 denotes the non-trivial character of A_{G}(u). We put θ_{i} = q^{3}ψ_{i} − ψ_{i′}, θ_{i′} = ψ_{i′}. Then ⟨θ_{i}, θ_{i′}⟩ = 0, and one can define θ̃_{i} (i ∈ I_{0}) in a similar way as (3.16.1). In this way, we can verify the same statement as in Theorem 1.9 for G = F_{4}.

4. Springer representations

4.1. In this section we shall prove some properties of Springer representations of W related to the a-function in the case of classical groups. This makes it possible to check Conjecture 1.7 in the case of low rank classical groups without computing Green functions. In what follows we assume that G = Sp_{N} or SO_{N} with rank G = n. First we recall some results from [16]. Let P be a maximal parabolic subgroup of G with a Levi subgroup L of the same type as G. We consider the variety B_{u} as before for a fixed unipotent element u = u_{λ} with λ ∈ X_{N} or X′_{N}. Let P_{u} be the variety of parabolic subgroups of G containing u and conjugate to P. We have a natural surjective map π : B_{u} → P_{u}. According to [16], the map π has the following locally trivial filtration: There exists a filtration of P_{u},

P_{u} = Y_{0} ⊃ Y_{1} ⊃ Y_{2} ⊃ ⋯

such that for each x ∈ Y_{i} − Y_{i+1}, the fibre π^{−1}(x) is isomorphic to B_{u′}^{L}, the variety of Borel subgroups of L containing a unipotent element u′ = u_{λ′} in L. Here λ′ ∈ X_{N−2} (resp. λ′ ∈ X′_{N−2}) is obtained from the Young diagram of λ by (under the notation in [16])

Case I-(a_{1}), (a_{2}) replacing two rows of length k by two rows of length k − 1, for k ∈ Δ_{λ},

Case I-(b_{1}), (b_{2}) replacing one row of length k by a row of length k − 2, for k ∈ Δ_{λ},

Case II replacing two rows of length k by two rows of length k − 1, for k ∉ Δ_{λ}.

Moreover, Y_{i} − Y_{i+1} has the following form: It is isomorphic to an affine space A^{r} for some r in Case II, Case I-(a_{1}) or Case I-(b_{2}), to A^{r} ∐ A^{r} in Case I-(a_{2}), and to A^{r} − A^{r−1} in Case I-(b_{1}). The Case I-(a_{2}) occurs only when r_{k} is even for k ∈ Δ_{λ}, and the Case I-(b_{2}) occurs only when r_{k} is odd for k ∈ Δ_{λ}. Set X_{i} = π^{−1}(Y_{i} − Y_{i+1}). Then in each case, we have X_{i} ≃ (Y_{i} − Y_{i+1}) × B_{u′}^{L}, or its étale covering version holds. Let W′ be the Weyl group of L, regarded as a Weyl subgroup of W of type C_{n−1} or D_{n−1}. The cohomology group H_{c}^{m}(X_{i}) has a natural structure of W′×A_{G}(u)-module, which is compatible with the action of W′×A_{G}(u) on H^{m}(B_{u}) obtained by restriction from W × A_{G}(u). Then we have

Proposition 4.2. ([16, Prop. 2.4]) Let X = X_{i} be as above for some i. Then the following formulas hold for any m ≥ 0, where d is a positive integer depending only on X.

(i) Case II H_{c}^{m}(X) ≃ H^{m−d}(B_{u′}^{L}),

(ii) Case I-(a_{1}) H_{c}^{m}(X) ≃ H^{m−d}(B_{u′}^{L}),

(iii) Case I-(a_{2}) H_{c}^{m}(X) ≃ H^{m−d}(B_{u′}^{L}) ⊕ H^{m−d}(B_{u′}^{L}),

(iv) Case I-(b_{1}) H_{c}^{m}(X) ≃ H^{m−d}(B_{u′}^{L})^{τ} ⊕ H^{m−d+1}(B_{u′}^{L})^{τ},

(v) Case I-(b_{2}) H_{c}^{m}(X) ≃ H^{m−d}(B_{u′}^{L}). The isomorphisms are those of W′-modules. τ is the element α′_{k} · α′_{k−2} ∈ A_{L}(u′), where α′_{k}, α′_{k−2} are generators of A_{L}(u′) corresponding to k, k − 2 ∈ Δ_{λ′} as in 3.6, and the right hand side stands for the τ-invariant subspace. The action of A_{G}(u) on H_{c}^{m}(X) is described by the action of A_{L}(u′) as follows: In the cases (i) and (ii), the actions on both sides are compatible with the natural map A_{G}(u) → A_{L}(u′) given by α_{i} ↦ α′_{i}. In (iii), the same is true for α_{i} (i ≠ k), and α_{k} permutes two components of the right hand side. In (iv), the actions are compatible with the map A_{G}(u) → A_{L}(u′)/⟨τ⟩. In (v), the same as (i), (ii) is true for α_{i} (i ≠ k), while α_{k} acts as α′_{k−2} on the right hand side.

4.3. For a locally closed subvariety X′ of B_{u}, let H_{c}^{∗}(X′) = ∑(−1)^{m}H_{c}^{m}(X′). Note that we have H^{∗}(B_{u}) = ⨁_{m≥0} H^{2m}(B_{u}). Then for X as in 4.1, H_{c}^{∗}(X) is a virtual W′ × A_{G}(u)- module. We denote by H_{c}^{∗}(X)_{ρ} its ρ-isotypic subspace for each ρ ∈ A_{G}(u)^{∧}. It follows from Proposition 4.2 that we have

(4.3.1)  H_{c}^{∗}(X)_{ρ} ≃ \begin{cases}H^{∗}(B_{u′}^{L})_{ρ′} & case (i) or (ii), \\ H^{∗}(B_{u′}^{L})_{ρ′} ⊕ H^{∗}(B_{u′}^{L})_{ρ′′} & case (iii), \\ H^{0}(B_{u′}^{L})_{ρ′} & case (iv), \\ H^{∗}(B_{u′}^{L})_{ρ′} & case (v), \end{cases}

where in the case (i), ρ′ ∈ A_{L}(u′)^{∧} is obtained from ρ by the isomorphism A_{G}(u) ≃ A_{L}(u′) if r_{k−1} ≠ 0, and by setting ρ′(α′_{k−1}) = 1 if r_{k−1} = 0, and ρ′(α′_{i}) = ρ(α_{i}) for i ≠ k. While in the case (ii), ρ′ is obtained by ρ′(α′_{k}) = ρ(α_{k}) if r_{k} ≥ 3, and similarly for i ≠ k. In the case (iii), ρ′, ρ′′ ∈ A_{L}(u′) are obtained by ρ′(α′_{k}) = 1, ρ′(α′_{k}) = −1 if ρ_{k} ≥ 4, and similarly for i ≠ k. In the case (iv), H^{0}(B_{u′}^{L}) ≃ C is a trivial W′-module, and so H_{c}^{∗}(X)_{ρ} = 0 except when ρ(α_{k}) = ρ(α_{k−2}) and ρ(α_{i}) = 1 for i ≠ k, k − 2. In the case (v), H_{c}^{∗}(X) ≠ 0 only when ρ(α_{k}) = ρ(α_{k−2}), and then we have ρ′(α′_{k}) = ρ(α_{k}).

Summing up the above arguments, we have an isomorphism of W′-modules,

(4.3.2)  H^{∗}(B_{u})_{ρ} ≃ ⨁_{(u′,ρ′)} m_{u′,ρ′}H^{∗}(B_{u′}^{L})_{ρ′} ⊕ U,

where U is a trivial W′-module and m_{u′,ρ′} denotes the multiplicity of the W′-modules. The sum is taken over all the pairs (u′, ρ′) as in (i), (ii), (iii) and (v) in (4.3.1).

4.4. We consider a pair (u, ρ) such that H^{∗}(B_{u})_{ρ} ≠ 0. Then by 1.6, (u, ρ) ∈ I_{0}. Let χ ∈ W^{∧} be the character corresponding to (u, ρ) via the Springer correspondence, and for (u′, ρ′) appearing in the right hand side of (4.3.2) let χ′ ∈ (W′)^{∧} be the corresponding character. We shall investigate the relationship between a(χ) and a(χ′). In order to avoid too much complication, we pose the following assumption on (u, ρ). (In fact, a weaker assumption is enough for the discussion below. See Remark 4.12.)

4.4.1. ρ ∈ A_{G}(u)^{∧} is the character satisfying the condition that ρ(α_{i}) = 1 for any i ∈ Δ_{λ} such that r_{i} is odd.

In the following, we denote by D(u) = D(λ) the Young diagram associated to u = u_{λ}. Let u = u_{λ} with λ = (1^{r_{1}}, 2^{r_{2}}, …, t^{r_{t}}). So, t is the length of the longest rows in D(λ). We define an integer t(u), for G = Sp_{2n} or SO_{2n}, by

t(u) = \begin{cases}r_{t} − 2 & if t ∈ Δ_{λ} and r_{t} ≥ 2, \\ r_{t} − 1 & otherwise, \end{cases}

and for G = SO_{2n+1} by

t(u) = \begin{cases}r_{t} − 2 & if r_{t} ≥ 2 \\ 0 & otherwise. \end{cases}

We have the following lemma.

Lemma 4.5. Assume that (u, ρ) satisfies the condition 4.4.1.

(i) For any (u′, ρ′) occurring in the expression in (4.3.2), we have a(χ) ≥ a(χ′) + t(u). (ii) Assume further that u′ is obtained from u by replacing the rows, which are not largest

among the rows in D(u). Then we have a(χ) > a(χ′) + t(u).

Proof. We give the proof only in the case where G = Sp_{2n} or SO_{2n}. The case where G = SO_{2n+1} is dealt with by a trivial modification. (The difference of these two cases arises from the maps in 3.8.1.) Let ξ be the u-symbol associated to (u, ρ). Then by our assumption 4.4.1, any interval of odd length is in the same position as in the distinguished u-symbol associated to u, (see 3.9). We consider each (u′, ρ′) separately for the cases (i), (ii), (iii) and (v) in Proposition 4.2. Let ξ′ be the u-symbol associated to (u′, ρ′).

First assume that (u′, ρ′) is as in (i). Then ξ is written as

ξ = (\begin{matrix}… & a, & a + 2, & … & a + 2r − 2, & … \\ … & a, & a + 2, & … & a + 2r − 2, & … \end{matrix}) ,

where 2r is the number of rows of length k in D(λ). Then ξ′ is obtained from ξ by replacing a by a − 1. (The choice of a is taken subject to the condition that it produces a u-symbol in Ψ_{N−2,1} or Ψ̃′_{N−2,0}). Hence the corresponding symbol Λ′ = Λ(χ′) is also obtained from Λ = Λ(χ) by replacing the element b corresponding to a by b−1. In the case where a+2r −2 is the largest number in the entries of ξ, we have t(u) = 2r − 1. In this case the order of elements a ≤ a ≤ a + 2 ≤ ⋯ ≤ a + 2r − 2 remains unchanged by the transformation from ξ to ξ′. Hence we have a(χ′) = a(χ) − 2r + 1 = a(χ) − t(u). If a + 2r − 2 is not the largest number, it may happen that a ≤ a goes to b_{1} < b_{2} by the transformation ξ → ξ′ and Λ′ is obtained from Λ by replacing b_{2} by b_{2} − 1. Since the order of the other parts remains unchanged by 4.4.1, we have

a(χ′) = a(χ) − ♯{x ∈ ξ | x ≥ a + 2}
< a(χ) − t(u).

Next assume that (u′, ρ′) is as in (ii). Then ξ contains an interval Γ = {a, a + 1, …, a + r_{k} − 1} corresponding to k ∈ Δ_{λ}, and ξ′ is obtained from ξ by replacing a + 1 by a. Hence if Γ lies in the top part of ξ, we see that a(χ′) = a(χ) − r_{k} + 2. The general case is also dealt with easily, and we have a(χ′) < a(χ) − t(u).

Next assume that (u′, ρ′) is as in (iii). As in the case (ii), ξ contains an interval Γ = {a, a+1, …, a+r_{k} −1}. Let ξ′, ξ′′ be the u-symbols associated to (u′, ρ′), (u′, ρ′′), respectively. Then ξ′ is obtained from ξ by replacing a + 1 by a, and ξ′′ is obtained from ξ′ by exchanging lower entries and upper entries of Γ′ = {a + 2, a + 3, …, a + r_{k} − 2} contained in ξ′. (Note that |Γ′| is even.) Hence a similar argument as in the case (ii) can be applied.

Finally assume that (u′, ρ′) is as in (v). Let Γ be an interval {a, a + 1, …, a + r_{k} − 1} contained in ξ and corresponding to k ∈ Δ_{λ}. In the case where r_{k−1} = 0, by using the fact that ρ(α_{k}) = ρ(α_{k−2}) if k − 2 ∈ Δ_{λ}, we see that ξ′ is obtained from ξ by replacing a by a − 1. Hence a similar argument as in the case (i) shows the required inequality. However, a somewhat different phenomenon occurs in the case where r_{k−1} ≠ 0. In this case r_{k−1} is even since k − 1 ∉ Δ_{λ}, and ξ can be expressed as

ξ = (\begin{matrix}… & a − 2r′, & … & a − 4, & a − 2, & a + 1, & … \\ … & a − 2r′, & … & a − 4, & a − 2, & a, & … \end{matrix}) ,

or in the form obtained by exchanging a and a + 1 in the above form. Here 2r′ = r_{k−1}. (We assumed here that r_{k} ≥ 3. But the argument below works as well for the case r_{k} = 1.) We may assume that ξ is as above. Then ξ′ is given as

ξ′ = (\begin{matrix}⋯ & a − 2r′ + 1, & ⋯ & a − 3, & a − 1, & a + 1, & ⋯ \\ ⋯ & a − 2r′ − 1, & ⋯ & a − 5, & a − 3, & a − 1, & ⋯ \end{matrix}) ,

i.e., the last element a in Γ is replaced by a − 1, and each pair (a − 2i, a − 2i) is changed to a pair (a − 2i − 1, a − 2i + 1). Note that again we used the property that ρ(α_{k}) = ρ(α_{k−2}) if k − 2 ∈ Δ_{λ}. We express ξ as ξ = \binom{A}{B} with A = \{⋯ ≤ a_{2} ≤ a_{1}\} and B = \{⋯ ≤ b_{2} ≤ b_{1}\}. Then by our assumption 4.4.1, a + 1 and a can be written as a + 1 = a_{j}, a = b_{j+e} for some j with e ∈ {1, 0, −1}. Moreover in this case Λ and Λ′ can be expressed as

Λ = (\begin{matrix}⋯ & b − r′, & ⋯ & b − 2, & b − 1, & b + 1, & ⋯ \\ ⋯ & b − r′ + e, & ⋯ & b − 2 + e, & b − 1 + e, & b + e, & ⋯ \end{matrix}) ,
Λ′ = (\begin{matrix}⋯ & b − r′ + 1, & ⋯ & b − 1, & b, & b + 1, & ⋯ \\ ⋯ & b − r′ − 1 + e, & ⋯ & b − 3 + e, & b − 2 + e, & b − 1 + e, & ⋯ \end{matrix})

for some b (b is an element corresponding to a under the map π in 3.8.1). Let Z = {z_{1} ≤ z_{2} ≤ ⋯ } be the sequence consisting of all the entries in Λ arranged in an increasing order, and Z′ the similar sequence for Λ′. Then Z′ is obtained from Z by replacing b + 1 by b if e = 1, by replacing b − r′ by b − r′ − 1 if e = 0, and by replacing b − r′ − 1, b − r′, b − 1 by b − r′ − 2, b − r′ − 1, b, respectively if e = −1. It follows that

a(χ′) = \begin{cases}a(χ) − 2j & if e = 1, \\ a(χ) − 2r′ − 2j + 1 & if e = 0, \\ a(χ) − 4r′ − 2j + 3 & if e = −1. \end{cases}

In any case we have a(χ′) < a(χ) − t(u). The lemma is now proved. □

4.6. We consider the setting in Conjecture 1.7. So u_{1} ∈ C_{1}, u ∈ C with C_{1} < C, and let χ, χ_{1} ∈ W^{∧} be characters corresponding to (u, ρ), (u_{1}, ρ_{1}) respectively. Let Λ(χ) = \binom{A}{B} be a symbol associated to χ, and let c be the largest element in the entries in Λ(χ) such that the replacement c → c − 1 produces a symbol Λ′ of rank n − 1. Set

s(χ) = ♯{x ∈ A ∪ B | x ≥ c} − 1.

Then if we denote by χ′ the character of W′ corresponding to Λ′, we have

(4.6.1)  a(χ) = a(χ′) + s(χ).

Moreover, χ′ appears in the restriction of χ to W′. We have the following proposition.

Proposition 4.7. Let the notations be as above. Suppose that Conjecture 1.7 holds for W′. Let ρ_{1} ∈ A_{G}(u_{1})^{∧} be the character satisfying the condition 4.4.1. Assume that χ occurs in H^{∗}(B_{u_{1}})_{ρ_{1}}. Then we have a(χ_{1}) ≥ a(χ) + t(u_{1}) − s(χ).

Proof. Assume that χ occurs in H^{i}(B_{u_{1}})_{ρ_{1}} for some i ≥ 0. Let χ′ ∈ (W′)^{∧} be as in 4.6. We may assume that χ′ is not the trivial character. In fact, if χ′ = 1_{W′}, then χ = 1_{W} and the proposition is easily verified for χ. Hence by (4.3.2), there exists a pair (u′_{1}, ρ′_{1}) such that χ′ occurs in H^{∗}(B_{u′_{1}})_{ρ′_{1}}. Let χ′_{1} be the character corresponding to (u′_{1}, ρ′_{1}). Then by our assumption, we have a(χ′_{1}) ≥ a(χ′). On the other hand, by applying Lemma 4.5 to χ_{1} and χ′_{1}, we have

a(χ_{1}) ≥ a(χ′_{1}) + t(u_{1}).

It follows that

a(χ_{1}) ≥ a(χ′) + t(u_{1}) = a(χ) + t(u_{1}) − s(χ).

by (4.6.1). Hence the proposition holds. □

The following special case seems worth mentioning.

Proposition 4.8. Under the same assumption as in Proposition 4.7, let u_{1} = u_{λ_{1}} with λ_{1} = (1^{r_{1}}, 2^{r_{2}}, …, t^{r_{t}}) be such that t, t − 2 ∈ Δ_{λ_{1}}, r_{t} = 1 and that ρ_{1}(α_{t}) ≠ ρ_{1}(α_{t−2}), (if r_{t−2} = 0, we assume that ρ_{1}(α_{t}) ≠ 1). Further assume that s(χ) = 0. Then we have a(χ_{1}) > a(χ).

Proof. Let (u′_{1}, ρ′_{1}) be as in the proof of Proposition 4.7. We note that u′_{1} satisfies the assumption in Lemma 4.5 (ii), since r_{t} = 1 and ρ_{1}(α_{t}) ≠ ρ_{1}(α_{t−2}), (see 4.3). In this case t(u_{1}) = 0 and we have a(χ_{1}) > a(χ′_{1}) by Lemma 4.5 (ii). The proposition then follows from a similar argument as in the previous proposition. □

4.9. By making use of the following formula of Spaltenstein on Springer representations, we can exploit another type of criterion for the above inequality. Let G be of type B_{n} or C_{n}. For each χ ∈ W^{∧}, we define the parity p(χ) by the condition that p(χ) = 0 (resp. p(χ) = 1) if χ(−1) = χ(1) (resp. χ(−1) = −χ(1)). For a pair (u, ρ) ∈ I_{0}, let χ_{u,ρ} be the corresponding character of W. Then by Spaltenstein [15], the following formula holds.

4.9.1. Assume that χ ∈ W^{∧} occurs in H^{2i}(B_{u})_{ρ}. Then we have

i ≡ d_{u} + p(χ) + p(χ_{u,ρ}) (mod 2).

We now consider the split F_{q}-structure on G with Frobenius map F. We choose u ∈ G_{uni}^{F} as in 2.1. For an F -stable locally closed subvariety X′ of B_{u}, let F^{∗} be the map on H_{c}^{i}(X′) induced from F. We denote by H_{c}^{i}(X′)^{0} (resp. H_{c}^{i}(X′)^{1}) the sum of generalized eigenspaces of F^{∗} corresponding to the eigenvalues q^{2j} (resp. q^{2j+1}) for j ≥ 0. Note that the eigenvalue of F^{∗} on H^{2i}(B_{u}) is q^{i}. Hence 4.9.1 implies that χ occurs in either H^{∗}(B_{u})_{ρ}^{0} or H^{∗}(B_{u})_{ρ}^{1}, and does not occur simultaneously.

Let X = X_{i} ⊂ B_{u} be as in 4.1. It is possible to choose a filtration so that all the X are F -stable. Since taking 0- or 1-part is an exact functor, we have, for ε ∈ {0, 1},

H^{∗}(B_{u})^{ε} = ⨁ H_{c}^{∗}(X)^{ε},

where the sum is taken over all the pieces X appearing in the filtration in 4.1. The isomorphisms in Proposition 4.2 are all F -stable, where u′ can be taken to be a split element except in the case (iv). In each case (i) ∼ (v) (in the last case (v) we need to assume that r_{k−1} = 0), there exists a maximal choice of X so that d = 2d_{u} − 2d_{u′}. We have the following lemma.

Lemma 4.10. Let G be of type B_{n} or C_{n}. Assume that χ ∈ W^{∧} occurs in H^{∗}(B_{u_{1}})_{ρ_{1}}^{ε} for ε ∈ {0, 1}. Let X be a piece satisfying the condition d = 2d_{u_{1}} − 2d_{u′_{1}}. Let (u′_{1}, ρ′_{1}) ∈ (W′)^{∧} be the pair corresponding to X in (4.3.1). Set χ_{1} = χ_{u_{1},ρ_{1}} as before, and χ′_{1} = χ_{u′_{1},ρ′_{1}}. Let χ′∈ (W′)^{∧} be a character occurring in the restriction of χ to W′, satisfying the condition that

(4.10.1)  p(χ) + p(χ_{1}) ≢ p(χ′) + p(χ′_{1}) (mod 2).

Then χ′ does not occur in H_{c}^{∗}(X)_{ρ_{1}}^{ε}.

Proof. Suppose that χ′ occurs in H_{c}^{∗}(X)_{ρ_{1}}^{ε}. Then χ′ occurs in H^{∗}(B_{u′}^{L})_{ρ′_{1}}^{ε′} with ε′ ≡ ε+d_{u_{1}}−d_{u′_{1}}. By 4.9.1 we have

ε′ ≡ d_{u′_{1}} + p(χ′) + p(χ′_{1}).

On the other hand, since χ occurs in H^{∗}(B_{u_{1}})_{ρ_{1}}^{ε}, we have

ε ≡ d_{u_{1}} + p(χ) + p(χ_{1}).

This contradicts (4.10.1). □

Proposition 4.11. Let G be of type B_{n} or C_{n}. Under the same assumption as in Proposition 4.7, let u_{1} =u_{λ_{1}} with λ_{1} =(1^{r_{1}}, 2^{r_{2}}, …, t^{r_{t}}) be such that (a) t ∈ Δ_{λ_{1}} with r_{t} =2, or (b) t ∈ Δ_{λ_{1}} with r_{t} = 1, r_{t−1} = 0. Let χ′, χ′_{1} ∈ (W′)^{∧} be as in 4.6 for χ, χ_{1}, respectively. Assume further that χ, χ_{1}, χ′, χ′_{1} satisfy the condition (4.10.1). Then we have a(χ_{1}) > a(χ) + t(u_{1}) − s(χ).

Proof. Let (u′_{1}, ρ′_{1}) be the pair corresponding to χ′_{1}. Then u′_{1} is obtained from u_{1} by removing two boxes from the top and second top rows of D(u_{1}) in the case (a), and removing two boxes from the top row of D(u_{1}) in the case (b), respectively. In view of (4.3.1), there exists a piece X corresponding to (u′_{1}, ρ′_{1}). In our case, such X is unique (it is the last one in the filtration of B_{u_{1}}), and it satisfies the assumption of Lemma 4.10. Hence we see that χ′ does not occur in H_{c}^{∗}(X)_{ρ_{1}}^{ε}. It follows that χ′ occurs in another H_{c}^{∗}(X′)_{ρ_{1}}^{ε}. If we denote by (u′′_{1}, ρ′′_{1}) the pair corresponding to X′, (u′′_{1}, ρ′′_{1}) satisfies the assumption in Lemma 4.5 (ii). Hence we have a(χ_{1}) > a(χ′′_{1}) + t(u_{1}), where χ′′_{1} ∈ (W′)^{∧} corresponds to (u′′_{1}, ρ′′_{1}). Now the assertion follows by a similar argument as in Proposition 4.8. □

Remark 4.12. Until now, we have assumed the condition 4.4.1 for (u_{1}, ρ_{1}). But this condition is not essential. What is necessary in the proof of Lemma 4.5 is actually that (in the notation there) in the u-symbol ξ, any interval of odd length has the same shape as in the case of distinguished u-symbol. So, Proposition 4.8 and Proposition 4.11 can be applied, for example, to the case of (u_{1}, ρ_{1}) with u_{1} = (1^{3}35), and ρ_{1}(α_{3}) = ρ_{1}(α_{5}) = −1.

4.13. By applying the previous results, one can verify Conjecture 1.7 for G of type B_{n}, C_{n} or D_{n} with n ≤ 6. First consider the statement (i) in the conjecture. The following gives a list of the cases where Proposition 1.8 can not be applied, i.e, C_{u_{1}} < C_{u}, but a(χ_{1}) < a(χ). In the following, we give Λ_{1} = Λ(χ_{1}) and Λ = Λ(χ). Note that for B_{n}, (n ≤ 4), C_{n}, (n ≤ 4) and D_{n}, (n ≤ 6) there exist no such χ and χ_{1}.

C_{5} :  u_{1} = (1^{2}2^{2}4),  u = (2^{3}4).
Λ_{1} = \binom{1 3 4}{0 1},  Λ = \binom{0 2 4}{1 2}, \binom{0 1 2}{2 4},
a(Λ_{1}) = 5,  a(Λ) = 6.
C_{6} :  u_{1} = (1^{4}2^{2}4),  u = (1^{2}2^{3}4).
Λ_{1} = \binom{1 2 4 5}{0 1 2}, Λ = \binom{0 1 3 5}{1 2 3}, \binom{0 1 2 3}{1 3 5},
a(Λ_{1}) = 10,  a(Λ) = 11.
B_{5} :  u_{1} = (12^{2}3^{2}),  u = (1^{2}3^{3}).
Λ_{1} = \binom{0 1 2}{2 4},  Λ = \binom{0 2 3}{1 3}, \binom{1 2 3}{0 3},
a(Λ_{1}) = 6,  a(Λ) = 7.
B_{6} :  u_{1} = (12^{2}35),  u = (1^{2}3^{2}5).
Λ_{1} = \binom{0 1 2}{2 5},  Λ = \binom{0 2 4}{1 3}, \binom{1 2 4}{0 3}, \binom{0 3 4}{1 2}, \binom{1 3 4}{0 2},
a(Λ_{1}) = 6,  a(Λ) = 7.

In each of the above cases, it is enough to show that χ does not occur in H^{∗}(B_{u_{1}})_{ρ_{1}}. In the case of type C_{5} or C_{6}, we have s(χ) = t(u_{1}) = 0. In these cases, Proposition 4.8 (or Proposition 4.7) can be applied to show that χ does not occur in H^{∗}(B_{u_{1}})_{ρ_{1}}. Note that if χ ∈ W^{∧} is given by a pair (α, β) of partitions of n, it is easy to see that p(χ) = (−1)^{|β|}. In the case of type B_{6}, we have t(u_{1}) = 1, and s(χ) = 0 for the first two χ, and s(χ) = 1 for the other two χ. For each of these 4 cases, we can check that (4.10.1) holds. Hence by applying Proposition 4.11, one knows that a(χ_{1}) ≥ a(χ) if χ occurs in H^{∗}(B_{u_{1}})_{ρ_{1}}. This contradicts a(χ_{1}) < a(χ). Finally consider the case of type B_{4}. In this case, neither Proposition 4.8 nor Proposition 4.11 can be applied. However, in this case, we consider Λ′ = Λ(χ′) of the following type,

Λ′ = \binom{0 1 3}{1 3} for Λ = \binom{0 2 3}{1 3},

and

Λ′ = \binom{0 2 3}{0 3} = \binom{1 2}{2} for Λ = \binom{1 2 3}{0 3},

respectively. According to each step of the filtration of B_{u_{1}}, we have two kinds of (u′_{1}, ρ′_{1}). The corresponding Λ′_{1} = Λ(χ′_{1}) are

Λ′_{1} = \binom{0 1 2}{2 3} or Λ′_{1} = \binom{0 1 2}{1 4}.

The pieces X corresponding to the pairs (u′_{1}, ρ′_{1}) always satisfy the condition in Lemma 4.10. Moreover, for each χ′ and χ′_{1} as above, we see that (4.10.1) holds. Hence by Lemma 4.10, χ′ does not occur in any of H_{c}^{∗}(X)_{ρ_{1}}^{ε}, where X runs over all the pieces in the filtration of B_{u_{1}}. This means that χ does not occur in H^{∗}(B_{u_{1}})_{ρ_{1}} as asserted. Now the statement (i) of the conjecture is verified for these cases.

4.14. Next consider the statement (ii) of the conjecture. So assume that u is special and χ is the special character corresponding to (u, 1). Let χ_{1} correspond to (u_{1}, ρ_{1}) with C_{1} < C such that C_{1} ⊄ Ĉ. The following is the list of pairs χ and χ_{1} such that a(χ_{1}) ≤ a(χ). There are no such pairs for C_{n}, (n ≤ 3), B_{n}, (n ≤ 3) and D_{n}, (n ≤ 5). We note that in the list below, Λ_{1} and Λ are always in the same family and so we have a(Λ_{1}) = a(Λ).

C_{4} : u_{1} = (2^{2}4), Λ_{1} = \binom{2 3}{0},  u = (4^{2}),  Λ = \binom{0 3}{2},

C_{5} : u_{1} = (2^{2}6), Λ_{1} = \binom{2 4}{0},  u = (46),  Λ = \binom{0 4}{2},
u_{1} = (1^{2}2^{2}4), Λ_{1} = \binom{1 3 4}{0 1},  u = (1^{2}4^{2}), Λ = \binom{0 1 4}{1 3},

C_{6} : u_{1} = (2^{2}8), Λ = \binom{2 5}{0},  u = (48),  Λ = \binom{0 5}{2},
u_{1} = (246), Λ_{1} = \binom{3 4}{0},  u = (6^{2}),  Λ = \binom{0 4}{3},
u_{1} = (1^{2}2^{2}6), Λ_{1} = \binom{1 3 5}{0 1},  u = (1^{2}46), Λ = \binom{0 1 5}{1 3},

u_{1} = (1^{4}2^{2}4), Λ_{1} = \binom{1 2 4 5}{0 1 2},  u = (1^{4}4^{2}), Λ_{1} = \binom{0 1 2 5}{1 2 4},

B_{4} : u_{1} = (135), Λ_{1} = \binom{0 1}{4},  u = (1^{2}7), Λ = \binom{0 4}{1},
u_{1} = (1^{3}3^{2}), Λ_{1} = \binom{0 1 2}{1 4},  u = (1^{4}5), Λ = \binom{0 1 4}{1 2},

B_{5} : u_{1} = (137), Λ_{1} = \binom{0 1}{5},  u = (1^{2}9), Λ = \binom{0 5}{1},
u_{1} = (15^{2}), Λ_{1} = \binom{0 2}{4},  u = (137), Λ = \binom{0 4}{2},
u_{1} = (1^{3}35), Λ_{1} = \binom{0 1 2}{1 5},  u = (1^{4}7), Λ = \binom{0 1 5}{1 2},
u_{1} = (12^{2}3^{2}), Λ_{1} = \binom{0 1 2}{2 4},  u = (1^{2}2^{2}5), Λ = \binom{0 2 4}{1 2},
u_{1} = (1^{5}3^{2}), Λ_{1} = \binom{0 1 2 3}{1 2 5},  u = (1^{6}5), Λ = \binom{0 1 2 5}{1 2 3},

B_{6} : u_{1} = (139), Λ_{1} = \binom{0 1}{6},  u = (1^{2}, 11), Λ = \binom{0 6}{1},
u_{1} = (157), Λ_{1} = \binom{0 2}{5},  u = (139), Λ = \binom{0 5}{2},
u_{1} = (1^{3}37), Λ_{1} = \binom{0 1 2}{1 6},  u = (1^{4}9), Λ = \binom{0 1 6}{1 2},
u_{1} = (12^{2}35), Λ_{1} = \binom{0 1 2}{2 5},  u = (1^{2}2^{2}7), Λ = \binom{0 2 5}{1 2},
u_{1} = (13^{4}), Λ_{1} = \binom{0 1 2}{3 4},  u = (1^{2}3^{2}5), Λ = \binom{0 2 4}{1 3},
u_{1} = (1^{7}3^{2}), Λ_{1} = \binom{0 1 2 3 4}{1 2 3 6}, u = (1^{8}5), Λ = \binom{0 1 2 3 6}{1 2 3 4},

D_{6} : u_{1} = (13^{2}5), Λ_{1} = \binom{0 1}{3 4},  u = (1^{2}5^{2}), Λ = \binom{0 3}{1 4}.

In order to verify the statement (ii) in the conjecture, it is enough to show that χ does not occur in H^{∗}(B_{u_{1}})_{ρ_{1}} for each case listed above. In all the cases in C_{4} ∼ C_{6} and D_{6}, Proposition 4.8 can be applied to show that χ does not occur in H^{∗}(B_{u_{1}})_{ρ_{1}}. On the other hand, in all the cases in B_{4} ∼ B_{6}, Proposition 4.11 can be applied to show the required property. Note in these cases, the situation described in Remark 4.12 appears, and we have to apply the proposition in this modified form. In this way, the statement (ii) in the conjecture is verified.

References

[1] Borho, W.; MacPherson, R.: Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes. C.R. Acad. Sci. Paris, sér. A, 292 (1981), 707–710.

[2] Beynon, W. M.; Spaltenstein, N.: Green functions of finite Chevalley groups of type E_{n}(n = 6, 7, 8). J. Algebra 88 (1984), 584–614.

[3] Carter, R.: Finite groups of Lie type, Conjugacy classes and complex characters. Wiley-Interscience, New York 1985.

[4] Geck, M.; Malle, G.: On the existence of a unipotent support for the irreducible characters of finite groups of Lie type. To appear in Trans. Amer. Math. Soc.

[5] Geck, M.; Malle, G.: On special pieces in the unipotent variety. Preprint.

[6] Kraft, H.; Procesi, C.: A special decomposition of the nilpotent cone of a classical Lie algebra. Astérisque 173–174 (1989), 271–279.

[7] Kashiwara, M.; Tanisaki, T.: The characteristic cycles of holonomic systems on a flag manifold. Invent. Math. 75 (1984), 327–358.

[8] Lusztig, G.: Characters of Reductive groups over a Finite field. Ann. of Math. Stud. 107, Princeton Univ. Press, Princeton 1984.

[9] Lusztig, G.: Intersection cohomology complexes on a reductive group. Invent. Math. 75 (1984), 205–272.

[10] Lusztig, G.: Character sheaves V. Adv. in Math. 61 (1986), 103–155.

[11] Lusztig, G.: On the character values of finite Chevalley groups at unipotent elements. J. Algebra 104 (1986), 146–194.

[12] Lusztig, G.: Unipotent support for irreducible representations. Adv. in Math. 94 (1992), 139–179.

[13] Lusztig, G.: Notes on unipotent classes. Asian J. Math. 1 (1997), 194–207.

[14] Spaltenstein, N.: Classes unipotentes et Sous-groupes de Borel. Lecture Notes in Math. 946, Springer-Verlag, Berlin-Heidelberg-New York 1982.

[15] Spaltenstein, N.: On Springer representations of Weyl groups containing −1. Arch. Math. (Basel) 44 (1985), 26–28.

[16] Shoji, T.: On the Green polynomials of classical groups. Invent. Math. 74 (1983), 239– 267.

[17] Shoji, T.: Green functions of reductive groups over a finite field. Proc. Sympos. Pure Math. 47 (1987), 289–301.

Received May 6, 1998