THE GEOMETRY OF FIXED POINT VARIETIES ON AFFINE FLAG MANIFOLDS

DANIEL S. SAGE

To appear in Transactions of the AMS, 1998.

Abstract. Let G be a semisimple, simply connected, algebraic group over an algebraically closed field k with Lie algebra 𝔤. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of 𝔤⊗k((π)), i.e. fixed point varieties on affine flag manifolds. We define a natural class of k^{∗}-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair (N, f) consisting of N ∈ 𝔤 ⊗ k((π)) and a k^{∗}-action f of the specified type which guarantees that f induces an action on the variety of parahoric subalgebras containing N.

For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the k^{∗}-fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of 𝔤.

1. Introduction

Let G be a semisimple, simply connected, algebraic group over an algebraically closed field k, and let 𝔤 be its Lie algebra. The set of Borel subalgebras of 𝔤 forms a smooth, projective variety which we denote by ℬ. For any nilpotent element N in 𝔤, the set of Borel subalgebras containing N is a closed subvariety ℬ_{N} of ℬ. These varieties have been studied extensively and have interesting applications to representation theory. For example, Springer and several others, including Ginzburg, Kazhdan, Lusztig, Slodowy, Joseph, Borho, and MacPherson, have defined representations of the Weyl group W in the homology (or cohomology) of the ℬ_{N}’s. Moreover, all irreducible representations of W may be obtained from the Springer representations in the top homology of the ℬ_{N}’s as N runs over the nilpotent conjugacy classes of 𝔤. In fact, when G = SL_{n}(k), these representations are precisely the irreducible representations of W.

Kazhdan and Lusztig have introduced the study of the affine analogue of these fixed point varieties on the flag manifold, which they anticipate will have applications to the character theory of semisimple p-adic groups. They have also used affine techniques to extend to all nilpotent orbits of 𝔤 a map defined by Carter and Elkington from a certain class of nilpotent orbits into the conjugacy classes of W. Let F = k((π)) be the field of formal power series in one variable with A = k[[π]] its ring of integers. We then extend scalars to form G(F), G(A), 𝔤_{F} = 𝔤 ⊗ _{k}F, and 𝔤_{A} = 𝔤 ⊗_{k}A. Fix a Borel subalgebra 𝔟 ∈ ℬ. An Iwahori subalgebra of 𝔤_{F} is any G(F)-conjugate of the pullback 𝔟̂ of 𝔟 under the natural projection 𝔤_{A} → 𝔤; we denote the space of Iwahori subalgebras by ℬ̂. If B ⊂ G corresponds to 𝔟 with preimage B̂ ⊂ G(A) under the map G(A) → G, then ℬ̂ may be identified with the quotient space G(F)/B̂. The set ℬ̂ has the structure of an infinite-dimensional projective variety over k, i.e. it is an increasing union of finite-dimensional projective varieties. (Kazhdan and Lusztig work in the case k = C, and their proofs of the results quoted also apply generally in characteristic zero. However, for this paper, it will suffice to know that for classical groups in positive characteristic, these spaces have ind-variety structures and moreover that certain subspaces are algebraic varieties. We will show these results directly.)

1991 Mathematics Subject Classification. 14L30, 20G25.

Key words and phrases. fixed point varieties on affine flag manifolds, Iwahori subalgebras, parahoric subalgebras, lattices.

We can also generalize the classical partial flag varieties, consisting of the subalgebras of 𝔤 conjugate to a fixed parabolic subalgebra, to the affine context. A parahoric subalgebra of 𝔤_{F} is a subalgebra containing an Iwahori subalgebra. Let S be a set of simple reflections for the affine Weyl group Ŵ, and for each I ⊆ S, let Ŵ_{I} be the subgroup of Ŵ generated by I. The parahoric subalgebras containing 𝔟̂ are precisely the algebras 𝔭̂_{I} = ∑_{w∈Ŵ_{I}} w · 𝔟̂; their stabilizers under the adjoint action are the parahoric subgroups P̂_{I} = B̂W_{I} B̂. The map I ↦ 𝔭̂_{I} gives a one-to-one correspondence between subsets of S and conjugacy classes of parahoric subalgebras. The set 𝒫̂_{I} of G(F)-conjugates of 𝔭̂_{I} is an infinite-dimensional projective variety which may be identified with the quotient space G(F)/P̂_{I}. These are the affine flag manifolds of G(F). The variety 𝒫̂_{S} is just a point, so we will only consider the varieties of proper parahoric subalgebras.

For each (nontrivial) affine flag manifold 𝒫̂ and any element N in the Lie algebra 𝔤_{F}, we define 𝒫̂_{N} to be the space of parahoric subalgebras of the specified type containing N. This is a closed subvariety of 𝒫̂. We shall restrict our attention to the topologically nilpotent (or nil) elements N, those elements such that (ad N)^{m} → 0 in the power series topology on End 𝔤_{F} as m → ∞. More concretely, these are the elements of 𝔤_{F} which may be conjugated into some N′ ∈ 𝔤_{A} whose image under projection to 𝔤 is nilpotent. Kazhdan and Lusztig have shown in [KL] that if 𝔤 is simple, then for N nil, the closed subvariety 𝒫̂_{N} is finite-dimensional (although possibly having infinitely many irreducible components) precisely for N regular semisimple. If we further assume that N is elliptic, i.e. the connected centralizer of N in G(F) is an anisotropic maximal torus, then 𝒫̂_{N} is a projective variety in the usual sense.

The affine fixed point varieties 𝒫̂_{N} are much more complicated than their classical counterparts, and not much is known about their topology. Unlike the P_{N}’s, they need not be rational nor do they necessarily have cell decompositions. For example, even in the low dimensional case 𝔰𝔭_{6}(F), Bernstein and Kazhdan have found N such that an irreducible component of ℬ̂_{N} admits a dominant morphism onto an elliptic curve [KL]. In fact, even the dimensions of these varieties were not known in general until recently, when Bezrukavnikov proved a conjectural formula of Kazhdan and Lusztig.

Lusztig and Smelt have defined k^{∗}-actions on certain of these fixed point varieties for the special linear groups [LS]. They used these actions to obtain interesting formulas for the Euler characteristics of these subvarieties for the varieties of maximal parahoric and Iwahori subalgebras. We have been studying those 𝒫̂_{N} which admit a class of algebraic k^{∗}-actions which generalize the Lusztig-Smelt actions. In section 2, we define a natural action of the group Q′_{G} = G_{ad}(k) × k^{∗} on the affine flag manifolds of G(F). An algebraic homomorphism k^{∗} → Q′_{G} defines a k^{∗}-action on 𝒫̂ which we call a Q′_{G}-action. The group Q′_{G} also acts on 𝔤_{F} by (g, λ) · N = Ad(g)N^{λ} for N ∈ 𝔤_{F}, where N^{λ} is the image of N under the k^{∗}-action on 𝔤_{F} induced by mapping the uniformizing parameter π of F to λπ. We say that a k^{∗}-action f almost commutes with an element N ∈ 𝔤_{F} if for some nonzero integer s, N is an eigenvector of f(λ) with eigenvalue λ^{s} for all λ ∈ k^{∗}. We show that if (N, f) is an almost commuting pair, then f induces an action on 𝒫̂_{N}.

In order to study these actions, we need a more concrete description of our infinite-dimensional varieties, analogous to the identification of the varieties of parabolic subalgebras of 𝔤 with certain varieties of partial flags. Let 𝔤 be a classical Lie algebra, and take the standard representation 𝔤 ↪ 𝔤𝔩(V). Set n equal to the dimension of V. The varieties of maximal parahoric subalgebras simply consist of certain classes of A-lattices in V_{F}. The full affine flag manifold ℬ̂ is a space of complete chains of lattices L_{0} ⊃ L_{1} ⊃ ⋯ ⊃ L_{n}, where L_{i} is of k-codimension 1 in L_{i−1} for each i, L_{n} = πL_{0}, and the lattices L_{i} have certain properties depending on 𝔤. The 𝒫̂ for other parahorics are sets of partial lattice chains. Since the Lie algebra action of 𝔤_{F} on 𝒫̂ corresponds to the natural action of 𝔤_{F} on lattice chains, the variety 𝒫̂_{N} may be identified with those partial lattice chains fixed by N. We can now define a k^{∗}-action on 𝒫̂_{N} by defining a k^{∗}-action on V_{F} which preserves lattices (and hence lattice chains) of the specified types and then descends to the fixed point subvarieties.

In sections 3 and 4, we identify the affine flag manifolds for 𝔰𝔩(V_{F}) and 𝔰𝔭(V_{F}) with varieties of lattice chains. We then apply the results of section 2 to obtain complete and concrete descriptions, in terms of the characteristic polynomial, of the nil-elliptic conjugacy classes containing a representative N whose fixed point varieties admit k^{∗}-actions induced by a Q′_{G}-action. (For the symplectic algebras, we assume that the characteristic of k is not 2.) Moreover, for each such conjugacy class, we find class representatives in a uniform way for which the lattices fixed by both N and the k^{∗}-action can be determined explicitly.

These actions are useful and interesting because they enable one to determine the Euler characteristics of the 𝒫̂_{N}’s. By a theorem of Bialynicki-Birula [B-B1], if the multiplicative group k^{∗} acts on an algebraic variety Y, then the Euler characteristics of Y and Y^{k^{∗}} are the same, where Y^{k^{∗}} denotes the fixed point set of the action. Therefore, we need only understand the much simpler variety of k^{∗}- fixed partial lattice chains to calculate the Euler characteristics of the 𝒫̂_{N}’s. In the most favorable cases, including every nil-elliptic conjugacy class of 𝔰𝔩(V_{F}) obtained in the above classification, these k^{∗}-actions only fix diagonal lattices relative to a fixed k-basis e_{1}, …, e_{n} of V, i.e. lattices of the form ℒ_{r} = Aπ^{r_{1}}e_{1} + ⋯ + Aπ^{r_{n}}e_{n} for r ∈ Z^{n}. This implies that the fixed point set (𝒫̂_{N})^{k^{∗}} is discrete and in fact finite, since N is nil-elliptic. We have thus reduced the computation of the Euler characteristic of 𝒫̂_{N} to a purely combinatorial problem of counting lattice chains. For those pairs (N, f) such that f fixes only diagonal lattices, we have discovered simple formulas for the Euler characteristics of 𝒫̂_{N} for each class of parahoric subalgebras as products of binomial coefficients. These formulas can be interpreted as the number of simplices of type corresponding to the parahoric class within a certain simplex in an apartment of the affine building of G. It is possible to apply similar techniques to the orthogonal algebras; this will be the subject of a future paper.

If an algebraic k^{∗}-action on a smooth projective variety Y has isolated fixed points, the cells of the Bialynicki-Birula plus decomposition of Y are isomorphic to affine space [B-B2]. The cells are indexed by the fixed points of the action, and the cell corresponding to a fixed point x is {y ∈ Y | lim_{λ→0} λ · y = x}, where the limit is understood in the sense of [B-B3]. It is natural to conjecture that when the fixed point set (𝒫̂_{N})^{k^{∗}} is finite, the variety 𝒫̂_{N} has such a cell decomposition. For 𝔰𝔩(V), an argument used by Lusztig and Smelt to obtain this result for Iwahori and maximal parahoric subalgebras applies to show that this is indeed true. In particular, the étale cohomology of 𝒫̂_{N} vanishes in odd degrees in these cases.

These k^{∗}-actions also have applications to the topology of the classical fixed point varieties. This is in very much the same spirit as Kazhdan and Lusztig’s use of nil-elliptic classes in 𝔤_{F} to study nilpotent classes in 𝔤. Let X denote the variety of conjugates of the maximal parahoric subalgebra 𝔤_{A}. The natural ℬ(k)-fibration ℬ̂ →^{ρ} X is given in terms of lattice chains by (L_{0}, …, L_{n}) ↦^{ρ} L_{0}. The fibers of this map are the complete flags (of the type corresponding to 𝔤) in the k-vector space L_{0}/πL_{0}, and the N-fixed lattice chains in the fiber may be interpreted as ℬN̄, where N̄ is the induced nilpotent transformation on L_{0}/πL_{0}. If (ℬ̂_{N})^{k^{∗}} is finite and L_{0} ∈ (X_{N})^{k^{∗}}, then the Euler characteristic of ℬN̄ is the number of lattice chains in (ℬ̂_{N})^{k^{∗}} lying above L_{0}. Similar considerations apply to the partial flag varieties. In section 3, we show that every nilpotent conjugacy class of 𝔰𝔩(V) may be realized as the class of such an N̄. We thus obtain a combinatorial description of the Euler characteristics of the classical fixed point varieties for 𝔰𝔩(V) without using their explicit cell decomposition or arguments involving induction on unipotent classes. We use a similar technique in section 4 to find combinatorial interpretations of the Euler characteristics of the classical fixed point varieties for 𝔰𝔭(V) for certain nilpotent classes.

This paper is based on part of my doctoral thesis at the University of Chicago [S]. It is a great pleasure to thank my advisor, Robert Kottwitz, for many extremely helpful comments and suggestions. I am also happy to acknowledge the support of a National Science Foundation Graduate Fellowship and a University of Chicago McCormick Fellowship.

2. k^{∗}-actions on fixed point varieties of affine flag manifolds

Let k be an algebraically closed field, and let G = G(k) be a semisimple, simply connected, algebraic group over k with Lie algebra 𝔤. Let F = k((π)) be the field of formal power series in one variable with ring of integers A = k[[π]]. The local field F is furnished with its usual valuation so that v(π) = 1. We now extend scalars of the group and Lie algebra to the power series field and ring.

Let p : 𝔤_{A} → 𝔤 be the natural projection. An Iwahori (or minimal parahoric) subalgebra of 𝔤_{F} is a G(F)-conjugate of the pullback via p of any Borel subalgebra of 𝔤. The parahoric subalgebras of 𝔤_{F} are those subalgebras which contain an Iwahori subalgebra. Iwahori and parahoric subgroups of G(F) may be defined in the same manner. Alternatively, a parahoric subgroup is the stabilizer of a parahoric subalgebra under the adjoint action. If 𝔭̂ is a parahoric subalgebra with stabilizer P̂, then the set of parahoric subalgebras conjugate to 𝔭̂ may be identified with the quotient space G(F)/P̂. This space has the natural structure of an infinitedimensional projective algebraic variety over k, i.e. it is an increasing union of finite-dimensional projective varieties. (The results on the variety structure of these spaces are proved by Kazhdan and Lusztig for k = C (and their proofs work generally in characteristic zero). For classical groups, we may give these spaces the structure of ind-varieties, using their concrete identifications with lattices and lattice chains as explained in section three. For our computations of the Euler characteristic of certain fixed point subvarieties, it will suffice to know that the spaces in question are indeed algebraic varieties in positive characteristic. Since we will show this directly, in the present paper we will not seek to extend the general results of Kazhdan and Lusztig to characteristic p.)

Fix a Borel subalgebra 𝔟 of 𝔤 with B the corresponding Borel subgroup, and let T be a maximal torus of G contained in B. The affine Weyl group is a Coxeter group defined by Ŵ = N(T (F))/T (A), where N(T (F)) is the normalizer of T (F) in G(F). Let S be a set of simple reflections for Ŵ. Then the conjugacy classes of parahoric subalgebras correspond bijectively to the subsets of the simple reflections S. Given I ⊆ S, let Ŵ_{I} be the subgroup of Ŵ generated by I. We obtain a standard Iwahori subalgebra 𝔟̂ (with respect to the choice of 𝔟) by pulling back 𝔟 to 𝔤_{A}. There is precisely one parahoric subalgebra from each conjugacy class containing 𝔟̂; these are just the subalgebras 𝔭̂_{I} = ∑_{w∈Ŵ_{I}} w · 𝔟̂, and with B̂ the stabilizer of 𝔟̂, their stabilizers are P̂_{I} = B̂Ŵ_{I} B̂. The corresponding varieties of conjugate subalgebras will be denoted 𝒫̂_{I}. If I ⊆ J, then P̂_{I} ⊆ P̂_{J}, inducing a natural map 𝒫̂_{I} → 𝒫̂_{J}; this is a fiber bundle with a classical (partial) flag variety as fiber. Note that 𝒫̂_{S} is just a point. From now on, we shall restrict attention to the nontrivial affine flag manifolds or equivalently, to the proper parahoric subalgebras.

We are interested in defining actions of k^{∗} on the subvarieties of the affine flag manifolds containing a fixed element N of 𝔤_{F}. As a first step, we will define a natural class of k^{∗}-actions on the affine flag manifolds themselves. The multiplicative group k^{∗} acts by k-algebra automorphisms on F via the formula λ · π^{m} = λ^{m}π^{m} for all λ ∈ k^{∗} and m ∈ Z. This induces an action of k^{∗} on G(F). Since k^{∗} preserves the standard Iwahori subgroup B̂ and acts trivially on the affine Weyl group, the action descends to the varieties of parahoric subalgebras. Also, G(F) acts on itself by left multiplication. It is now easy to see that the semidirect product G(F) ⋊ k^{∗} acts on G(F) by (g, λ) · h = gh^{λ} where g, h ∈ G(F) and λ ∈ k^{∗} and induces an action on the affine flag manifolds. Since the center Z(F) of G(F) is contained in every parahoric subgroup, we finally obtain an action of Q_{G} = G_{ad}(F)⋊k^{∗} on each 𝒫̂_{I}, where G_{ad} is the adjoint form of G. We have G_{ad}(F)^{k^{∗}} = G_{ad}(k), so the group Q_{G} contains the direct product Q′_{G} = G_{ad}(k) × k^{∗}. We can now define the class of k^{∗}-actions which we will study.

Definition. A k^{∗}-action f on an affine flag manifold for the group G is called a Q′_{G}-action if it is obtained from an algebraic homomorphism k^{∗} →^{f} Q′_{G}.

Remark. We can similarly define Q_{G}-actions, but in this paper, we will restrict attention to Q′_{G}-actions.

The group Q_{G} also acts on 𝔤_{F} by (g, λ) · N = Ad(g)N^{λ} for N ∈ 𝔤_{F}.

Definition. Given N ∈ 𝔤_{F}, we say that a k^{∗}-action f almost commutes with N if for some nonzero integer s, N is an eigenvector of f(λ) with eigenvalue λ^{s}, i.e. f(λ) · N = λ^{s}N for all λ ∈ k^{∗}. The integer s is called the exponent of the pair (N, f).

This property guarantees that a Q_{G}-action f on a conjugacy class of parahoric subalgebras 𝒫̂ descends to 𝒫̂_{N}, the subvariety of parahorics containing N.

Lemma 2.1. Let 𝒫̂ be an affine flag manifold, and suppose that N almost commutes with the Q_{G}-action f. Then f induces an action on 𝒫̂_{N}.

Proof. Take 𝔭̂ ∈ 𝒫̂_{N}, and set f(λ) = (g(λ), λ^{p}) for some p ∈ Z. (The map of k^{∗} into the second factor of Q_{G} is a homomorphism from k^{∗} to itself.) Then N = λ^{−s}f(λ) · N = Ad g(λ)N^{λ^{p}} ∈ Ad g(λ)𝔭̂^{λ^{p}} = f(λ) · 𝔭̂ as desired.

Elements of 𝔤_{F} which almost commute with a Q_{G}-action must satisfy the following condition:

Theorem 2.2. Let ρ : G → GL(V) be a representation of G, inducing the Lie algebra representation dρ : 𝔤_{F} → 𝔤𝔩(V_{F}). Suppose that N almost commutes with the Q_{G}-action f(λ) = (g(λ), λ^{p}) with exponent s. Then the characteristic polynomial char_{dρ(N)}(μ) is a homogeneous polynomial of degree n = dim V in μ and π^{q} for some rational number q ∈ ⋃_{i=1}^{n} \frac{1}{i}Z; if dρ(N) is not nilpotent, then p ≠ 0 and q = s/p.

Proof. Set char_{dρ(N)}(μ) = ∑_{i=0}^{n} c_{i}μ^{n−i} for some c_{i} ∈ F. Since dρ(Ad g(λ)N^{λ^{p}}) and dρ(N^{λ^{p}}) are GL(V_{F})-conjugate, we see that ∑_{i=0}^{n} c_{i}^{λ^{p}}μ^{n−i} = (char_{dρ(N)}(μ))^{λ^{p}} = char_{dρ(N^{λ^{p}})}(μ) = char_{dρ(λ^{s}N)}(μ) = ∑_{i=0}^{n} λ^{si}c_{i}μ^{n−i}. The claim is trivial if dρ(N) is nilpotent, so we may assume that c_{j} ≠ 0 for some j ≥ 1. The equations c_{i}^{λ^{p}} = λ^{si}c_{i} imply that pv(c_{i}) = si for each i with c_{i} ≠ 0; furthermore, c_{i} is homogeneous, i.e. a Laurent series in π with at most one nonvanishing term. This shows that char_{dρ(N)}(μ) is homogeneous of degree n in μ and π^{q}, where q = s/p, and since sj/p is an integer, q ∈ \frac{1}{j}Z.

Definition. If N is not nilpotent, we call the ratio s/p the loop index of the pair (N, f).

Corollary 2.3. Let \overline{F} be an algebraic closure of F. Then all nonzero eigenvalues of dρ(N) are homogeneous elements of \overline{F} with valuation equal to the loop index.

Proof. By the theorem, char_{dρ(N)}(μ) = ∑_{i=0}^{n} c_{i}(π^{q})^{i}μ^{n−i} for some c_{i} ∈ k. The roots of char_{dρ(N)}(μ) in the algebraic closure \overline{F} are just xπ^{q} where x ∈ k is a root of the polynomial ∑_{i=0}^{n} c_{i}μ^{n−i} ∈ k[μ].

Recall that N ∈ 𝔤_{F} is called nil (or topologically nilpotent) if lim_{m→∞}(ad N)^{m} = 0. We are primarily interested in defining k^{∗}-actions on the fixed point variety 𝒫̂_{N} when N is regular semisimple and nil, since in this case, 𝒫̂_{N} is finite-dimensional [KL]. For N of this type in the case of the special linear and symplectic Lie algebras, we will prove a converse statement to theorem 2.2, namely, that if the charactersitic polynomial of a nil, regular semisimple conjugacy class has the form given in theorem 2.2, then we can find an element of the class together with a Q_{G}-action almost commuting with it.

Now assume that G is a classical group, and let ρ : G → GL(V) be the standard representation of G, inducing the Lie algebra monomorphism 𝔤 ↪ 𝔤𝔩(V). In order to define explicit Q′_{G}-actions when G is a classical group, it will be convenient to consider homomorphisms k^{∗} → Q′ =^{def} GL(V) × k^{∗}. Such a homomorphism gives rise to a Q′_{SL(V)}-action via the projection GL(V) → P GL(V) = P SL(V), and if G is not the special linear group, we can obtain a Q′_{G}-action in a similar way after placing suitable restrictions on the homomorphism.

More generally, if V is an n-dimensional vector space over k, Q′ acts on V_{F} by k-linear automorphisms. We call a k^{∗}-action f on V_{F} a Q′-action if it comes from an algebraic homomorphism k^{∗} →^{f} Q′. For n ≥ 2, this is equivalent to f mapping A-submodules of V_{F} to A-submodules and the existence of a k-basis \underline{e} = (e_{1}, …, e_{n}) of V such that every element of the topological spanning set \underline{ê} = \{π^{m}e_{i}|m ∈ Z, i ∈ [1, n]} of V_{F} is an eigenvector of f(λ) for each λ ∈ k^{∗}. In fact, the eigenvalue of π^{m}e_{i} must have the form λ^{ν(·,i)}, where ν(·, i) : Z → Z is an affine function with slope independent of i, i.e. ν(m, i) = pm + c_{i} for some integers p and c_{i} [S, theorem 2.5]. (A specific example of a k^{∗}-action of this form was first introduced by Lusztig and Smelt [LS].)

We say that an endomorphism N ∈ 𝔤𝔩(V_{F}) almost commutes with f if for some nonzero integer s, f(λ)N = λ^{s}Nf(λ) for all λ ∈ k^{∗}. The group Q′ acts on 𝔤𝔩(V_{F}), and noting that f(λ) · N = f(λ)Nf(λ)^{−1}, we see that this is compatible with our previous definition. Fix a basis \underline{e} such that the elements of \underline{ê} are eigenvectors of the f(λ)’s. Let (a_{ij}) ∈ 𝔤𝔩_{n}(F) be the matrix for N determined by \underline{e}, and write a_{ij} = ∑_{l≫−∞} a_{ij}^{l}π^{l} where a_{ij}^{l} ∈ k. Equating the π^{m+l}e_{j} terms in f(λ)Nπ^{m}e_{i} = λ^{s}Nf(λ)π^{m}e_{i} for all m and i gives the transformation laws ν(m, j)+s = ν(m+l, i) for all m ∈ Z and all i, j, and l such that a_{ij}^{l} ≠ 0.

The characteristic polynomial determines the GL(V_{F})-conjugacy classes of semisimple endomorphisms. We can now characterize the regular semisimple conjugacy classes obtained from almost commuting pairs (N, f) with f a Q′-action.

Theorem 2.4. Let 𝒞 be a regular semisimple conjugacy class in 𝔤𝔩(V_{F}). Then there is a representative N ∈ 𝒞 almost commuting with a Q′-action if and only if the characteristic polynomial char_{𝒞} (μ) is a homogeneous polynomial of degree n in μ and π^{q} for some nonzero rational number q ∈ \frac{1}{n−1}Z ∪ \frac{1}{n}Z.

Proof. The forward implication follows just as in theorem 2.2. We need only note that since at most one of the eigenvalues is 0, either the π^{qn}μ^{0} or the π^{q(n−1)}μ term of the characteristic polynomial has nonzero coefficient, implying that q ∈ \frac{1}{n−1}Z ∪ \frac{1}{n}Z.

Conversely, assume that for some nonzero q ∈ \frac{1}{n−1}Z ∪ \frac{1}{n}Z, we have char_{𝒞} (μ) = ∑_{i=0}^{n} c_{i}(π^{q})^{i}μ^{n−i} with each c_{i} ∈ k and c_{0} = 1. Define a matrix N = (a_{ij}) by

a_{ij} = \begin{cases}1 & if j = i + 1 \\ (−1)c_{i}π^{iq} & if j = 1 \\ 0 & otherwise. \end{cases}

Note that if c_{i} ≠ 0, then qi ∈ Z. Thus, a_{ij} ∈ F for each i and j.

If the determinant of N is nonzero, set ν(m, i) = nm − inq; otherwise, set ν(m, i) = (n − 1)m − i(n − 1)q. It is easy to check that if we define f(λ)π^{m}e_{i} = λ^{ν(m,i)}π^{m}e_{i}, where (e_{1}, …, e_{n}) is the standard basis of k^{n}, then N and f almost commute with exponent nq (resp. (n − 1)q). (The existence of a Q′-action almost commuting with N is also guaranteed by theorem 2.20 of [S].)

Finally, it follows easily from the definition of the determinant as a sum over the symmetric group S_{n} that char_{N} = char_{𝒞}. The conjugacy class 𝒞 is regular semisimple, so char_{𝒞} has n distinct roots, and since any endomorphism in 𝔤𝔩(V_{F}) with n distinct eigenvalues must be diagonalizable, we have N ∈ 𝒞.

It is known that an endomorphism is nil if and only if there exists N′ ∈ 𝔤𝔩(V_{A}) conjugate to N such that the image of N′ under projection to 𝔤𝔩(V) is nilpotent [KL]. It is immediate that all nonzero eigenvalues of a nil endomorphism must have positive valuation. Since a regular semisimple endomorphism with characteristic polynomial as above has eigenvalues of the form xπ^{q} for some x ∈ k, we obtain the following corollary:

Corollary 2.5. Let 𝒞 be a nil, regular semisimple conjugacy class in 𝔤𝔩(V_{F}). Then there is a representative N ∈ 𝒞 almost commuting with a Q′-action if and only if the characteristic polynomial char_{𝒞} (μ) is a homogeneous polynomial of degree n in μ and π^{q} for some positive rational number q ∈ \frac{1}{n−1}Z ∪ \frac{1}{n}Z.

3. k^{∗}-actions on fixed point varieties of affine flag manifolds: The special linear algebras

We will now apply our results on k^{∗}-actions to the study of fixed point varieties on affine flag manifolds. In this section, we will consider the affine flag manifolds obtained for the root systems A_{n}.

Let V be a k-vector space of dimension n ≥ 2. The special linear group SL(V) is the group of linear automorphisms of V which induce the identity on the highest exterior power of V, i.e. the volume-preserving linear automorphisms. Its Lie algebra is denoted 𝔰𝔩(V). Fixing a basis e_{1}, …, e_{n} identifies SL(V) with SL_{n}(k), the semisimple, simply connected, algebraic group over k of type A_{n−1}.

Let 𝔟 ⊂ 𝔰𝔩_{n}(k) be the standard Borel subalgebra determined by the fixed basis \underline{e}, namely the upper triangular matrices in 𝔰𝔩_{n}(k). The standard Iwahori algebra 𝔟̂ is the pullback of 𝔟 to 𝔰𝔩_{n}(A). We now define algebras 𝔮̂_{i} ⊂ 𝔰𝔩_{n}(F) for i ∈ [0, n − 1] by

i n − i
𝔮̂_{i} = 𝔰𝔩_{n}(F) ∩ (\begin{matrix}A & π^{−1}A \\ πA & A \end{matrix}) \begin{matrix}i \\ n − i. \end{matrix}

The notation means that 𝔮̂_{i} consists of those matrices in 𝔰𝔩_{n}(F) of the above form, i.e. the upper right block is an i × i matrix with entries in A and similarly for the other blocks. Blocks with zero as a dimension are ignored, so 𝔮̂_{0} = 𝔰𝔩_{n}(A). These are the maximal (proper) parahoric subalgebras containing 𝔟̂. The stabilizer of 𝔮̂_{i} is the group Q̂_{i} of matrices in SL_{n}(F) with the same block structure. We can now obtain all parahorics containing 𝔟̂ in terms of the maximal ones: 𝔭̂_{I} = ⋂_{i∉I} 𝔮̂_{i} with stabilizer P̂_{I} = ⋂_{i∉I} Q̂_{i}. It is possible to choose simple reflections for the affine Weyl group Ŵ, indexed by the set [0, n − 1], such that this notation agrees with the notation of section 2 [S].

The affine flag manifolds of SL(V_{F}) may be interpreted as varieties of chains of A-lattices in V_{F}. Let ℒ denote the space of all A-lattices in V_{F}. This set has a natural ind-variety structure. To see this, note that for any lattice L, there exists a natural number m such that π^{m}V_{A} ⊆ L ⊆ π^{−m}V_{A}. The natural map L ↦ L/π^{m}V_{A} from the set of lattices satisfying this condition for a fixed m to the variety of subspaces of π^{−m}V_{A}/π^{m}V_{A} is an embedding, identifying the domain as a projective variety. Taking the union over m makes ℒ into an infinite-dimensional variety. (For classical groups, affine flag manifolds can be interpreted in terms of lattices, and similar arguments show that they are ind-varieties.)

If L is a lattice with basis f_{1}, …, f_{n}, then f_{1} ∧ ⋯ ∧ f_{n} = αe_{1} ∧ … e_{n} for some α ∈ F^{∗}. The valuation of α is independent of the choice of bases for L and V; we call it the valuation of L and denote it by v(L). The space of lattices with fixed valuation m is denoted ℒ_{m}. The group GL(V_{F}) acts transitively on ℒ, and it is obvious that T ∈ GL(V_{F}) maps ℒ_{m} to ℒ_{m+v(det T)}. The SL(V_{F})-orbits of ℒ are just the varieties ℒ_{m} for m ∈ Z.

The actions of SL(V_{F}) and F^{∗} on ℒ commute, so we see that ℒ_{m} ≅ ℒ_{m′} as an SL(V_{F})-variety when m ≡ m′ (mod n). (All the ℒ_{m}’s are isomorphic as varieties, but the SL(V_{F})-structures are different.) As a result, for each i ∈ [0, n−1], SL(V_{F}) acts on the spaces X_{i} = {⋯ ⊃ π^{−1}L ⊃ L ⊃ πL ⊃ ⋯ | L ∈ ℒ_{i}}. The set X_{i} is just the variety of lattices modulo homothety with valuation congruent to i modulo n. This alternate point of view shows that the ℒ_{m}’s are just special cases of the following more general class of varieties:

Definition. Let I be a nonempty subset of [0, n − 1]. We call

X_{I} = {(L_{i}) | i ∈ I (mod n), L_{i} ∈ ℒ_{i}, L_{i} ⊃ L_{j} ⟺ i < j, and L_{i+n} = πL_{i}}

the space of (partial) lattice chains of type I.

The varieties X_{I} are in fact nothing more than the affine flag manifolds for SL(V_{F}). Note that if I ⊆ [0, n−1] contains 0, then for any L ∈ ℒ_{0}, the fiber over L of the projection X_{I} → X_{{0}} = ℒ_{0} is isomorphic as an SL(L/πL)-variety to ℱ_{I\{0}}, the variety of flags of type I \ {0} in L/πL. Using this observation, we can show that SL(V_{F}) acts transitively on each X_{I}. The automorphism of Z/nZ given by c ↦ −c defines an involution α on the set of coset representatives [0, n − 1]. The map I ↦ α(I) is thus a bijective self-map on the set of subsets of [0, n − 1]. We can find a lattice chain in X_{I} whose stabilizer is P̂_{α(I)^{c}}. This proves the following theorem (for more details, see [S, theorem 3.4]):

Theorem 3.1. The varieties X_{I} and 𝒫̂_{α(I)^{c}} are isomorphic for each nonempty subset I of [0, n − 1].

In order to apply the results of section 2 on almost commuting pairs to the present context, we will relate regular semisimple conjugacy classes in 𝔰𝔩(V_{F}) and 𝔤𝔩(V_{F}). Fix an algebraic closure \overline{F} of F, and let F_{s} be the separable closure of F in \overline{F}. Recall that for the simple root systems, the torsion primes are 2 for B_{n}, D_{n}, and G_{2}, 2 and 3 for E_{6}, E_{7}, and F_{4}, and 2, 3, and 5 for E_{8} [SS, I.4.4].

Proposition 3.2. Let G be a simple, simply connected, algebraic group over k with Lie algebra 𝔤, and assume that the characteristic of k is not a torsion prime for G. If N is a regular semisimple element of 𝔤_{F}, then the G(F)-conjugacy class of N in 𝔤_{F} is the intersection of 𝔤_{F} with the G(\overline{F})-conjugacy class of N in 𝔤\overline{F}.

Proof. Let N′ ∈ 𝔤_{F} be a G(\overline{F})-conjugate of N. We claim that N and N′ are G(F_{s})-conjugate. Let Z = Z(\overline{F}) and Z′ = Z′(\overline{F}) be the centralizers of N and N′ respectively in G(\overline{F}). Since the endomorphisms are semisimple, the corresponding Lie algebras 𝔷 and 𝔷′ are the centralizers of N and N′ in 𝔤\overline{F} [Bo, p. 128]. The group G is simply connected and the characteristic of F is not a torsion prime, so by a theorem of Springer and Steinberg, Z and Z′ are connected [SS, II.3.19]. Since N and N′ are regular semisimple, they are maximal tori defined over F.

The tori Z and Z′ split over F_{s} [Bo, III.8.11], hence are conjugate over G(F_{s}). Choose h ∈ G(F_{s}) such that hZh^{−1} = Z′, implying that h𝔷h^{−1} = 𝔷′. But G(\overline{F})- conjugacy in the Cartan subalgebra 𝔷′ is equivalent to W -conjugacy, where W is the Weyl group. Since W = N(Z′(F_{s}))/Z′(F_{s}), there exists h′ ∈ N(Z′(F_{s})) such that h′hNh^{−1}h′^{−1} = N′ as desired.

Set g = h′h. For each γ ∈ Gal(F_{s}/F), we have γ(g)Nγ(g^{−1}) = N′, so that g^{−1}γ(g) is in Z(F_{s}). The map γ ↦ γ^{−1}γ(g) is a cocycle with coefficients in the torus Z(F_{s}); we show that it is a coboundary. The field F is complete with respect to a discrete valuation, and its residue field is algebraically closed. Therefore, the Brauer group of every finite separable extension of F is zero, and this implies that H^{1}(Gal(F_{s}/F), T (F_{s})) = 0 for any torus T defined over F [S1, p. 170]. In particular, H^{1}(Gal(F_{s}/F), Z(F_{s})) = 0. Thus, there exists z ∈ Z(F_{s}) such that z^{−1}γ(z) = g^{−1}γ(g). Note that γ(gz^{−1}) = gz^{−1} for all γ ∈ Gal(F_{s}/F), so gz^{−1} ∈ G(F). The result follows, since gz^{−1}Nzg^{−1} = N′.

Remark. If we assume that char k = 0, then the statement holds for arbitrary semisimple N. In this case, the field F is perfect; moreover, it has cohomological dimension 1 because it is complete with respect to a discrete valuation and has algebraically closed residue field [S2, p. 97]. This implies that H^{1}(Gal(\overline{F} /F), L) = 0 for any connected linear group over F [St, I.9]. The theorem of Springer and Steinberg again implies that the centralizer Z is connected, and the rest of the proof goes through as above.

A semisimple (resp. regular semisimple) element of 𝔰𝔩(V_{F}) is just a (semisimple (resp. regular semisimple) element of 𝔤𝔩(V_{F}) with trace zero. Also, two elements of 𝔰𝔩(V\overline{F}) are SL(V\overline{F})-conjugate if and only if they are GL(V\overline{F})-conjugate. Proposition 3.2 now shows that the regular semisimple conjugacy classes of 𝔰𝔩(V_{F}) under the actions of SL(V_{F}) and GL(V_{F}) coincide. We thus obtain as an immediate corollary of theorem 2.4 and corollary 2.5 the following classification of regular semisimple conjugacy classes 𝒞 in 𝔰𝔩(V_{F}) coming from almost commuting pairs (N, f) with f a Q′_{SL(V)}-action:

Theorem 3.3. Let 𝒞 be a regular semisimple (resp. nil regular semisimple) conjugacy class in 𝔰𝔩(V_{F}). Then there is a representative N ∈ 𝒞 almost commuting with a Q′_{SL(V)}-action if and only if the characteristic polynomial char_{𝒞} (μ) is a homogeneous polynomial of degree n in μ and π^{q} for some nonzero (resp. positive) rational number q ∈ \frac{1}{n−1}Z ∪ \frac{1}{n}Z.

Note that c_{1} = tr 𝒞 = 0 in char_{𝒞} (μ) = ∑_{i=0}^{n} c_{i}π^{qi}μ^{n−i}, so the standard representative for a conjugacy class given in the proof of theorem 2.4 lies in 𝔰𝔩(V_{F}).

We are primarily interested in defining k^{∗}-actions on the fixed point varieties 𝒫̂_{N} when they are algebraic varieties. Recall that a torus T over F is called elliptic if no nontrivial cocharacter of T is defined over F. We say that a regular semisimple element N ∈ 𝔰𝔩(V_{F}) is elliptic if its centralizer Z_{SL(V_{F})}(N) in SL(V_{F}) is an elliptic torus. Suppose that N ∈ 𝔰𝔩(V_{F}) is nil. By a theorem of Kazhdan and Lusztig [KL], dim 𝒫̂_{N} < ∞ if and only if N is regular semisimple. Furthermore, 𝒫̂_{N} is an algebraic variety, i.e. has a finite number of irreducible components precisely for N elliptic. (For positive characteristic, we note that the fixed point subvarieties considered below are algebraic varieties for an arbitrary algebraically closed field; this follows from [LS].)

It is known that a regular semisimple element N of 𝔰𝔩(V_{F}) is elliptic if and only if char_{N}(μ) ∈ F [μ] is an irreducible, separable polynomial [S, proposition 3.10]. If F has characteristic zero, then for each positive integer m, the field F has a unique extension of degree m in \overline{F}; it is generated by an mth root of π. If we call this cyclic extension F (π^{1/m}), we have \overline{F} = ⋃_{m=1}^{∞} F (π^{1/m}). In characteristic p, the structure of \overline{F} and the separable closure F_{s} is much more complicated. However, the previous theorem guarantees that any elliptic, regular semisimple N almost commuting with a Q′_{SL(V)}-action has eigenvalues lying in ⋃_{p∤m} F (π^{1/m}) ⊂ F_{s}. Therefore, if N is nil-elliptic, the splitting field of char_{N} is F (π^{1/n}) and n can not be a multiple of the characteristic. By corollary 2.3, each eigenvalue is homogeneous of valuation q = s/n > 0 where s is relatively prime to n. Since π^{qi} ∉ F for i ∈ [1, n−1], we have char_{N}(μ) = μ^{n} − bπ^{s}. Let a ∈ k be some nth root of b. Then the roots of char_{N} over \overline{F} are \{aζπ^{q} : ζ an nth root of unity in k\}. These are distinct if and only if a ≠ 0 and the characteristic of k is either 0 or relatively prime to n. This can be rewritten more simply as (char′ k, n) = 1, where char′ k denotes the characteristic exponent of k. (Recall that the characteristic exponent of a field is 1 if the field has characteristic 0 and equals the characteristic otherwise ([B2, 5.1.5]). Summing up, we have:

Proposition 3.4. The nil-elliptic conjugacy classes in 𝔰𝔩(V_{F}) obtained from almost commuting pairs (N, f) with f a Q′_{SL(V)}-action are precisely those with char_{𝒞} (μ) = μ^{n} − bπ^{s} where b ∈ k^{∗}, s > 0, (s, n) = 1, and (char′ k, n) = 1. Fixing a basis \underline{e} for V, we have the standard matrix representative

N_{bs} = (\begin{matrix}0 & 1 & 0 & 0 & … & 0 \\ 0 & 0 & 1 & 0 & … & 0 \\ ⋮ & ⋮ & ... & ... & & ⋮ \\ ⋮ & ⋮ & ... & ... & 0 & 1 \\ bπ^{s} & 0 & 0 & … & 0 & 0 \end{matrix}) .

Each N_{bs} almost commutes with exponent s with the Q′-action f, diagonal with respect to \underline{e}, given by ν(m, i) = nm − is.

Note that the field automorphism of F given by π ↦ b^{1/s}π (where b^{1/s} is a fixed sth root of b) induces an automorphism of 𝔤_{F} sending N_{1s} to N_{bs}. Accordingly, 𝒫̂_{N_{bs}} is isomorphic to 𝒫̂_{N_{1s}}, and we need only consider the case b = 1. For simplicity of notation, we set N_{s} =^{def} N_{1s}.

Remark. Lusztig and Smelt used different representatives for these nil-elliptic conjugacy classes and almost commuting actions [LS].

For the remainder of the section, we assume that (n, char′ k) = 1 and fix a basis \underline{e} of V. By the Iwasawa decomposition for GL_{n}(F), any lattice L has a basis \underline{z} such that z\underline{e} is upper triangular. Indeed, L has a unique basis of the form z_{j} = π^{r_{j}}e_{j} + ∑_{i<j} z_{ij}e_{i} for j ∈ [1, n], where r_{j} ∈ Z and z_{ij} ∈ F is a terminating Laurent series with all nonzero terms of degree strictly smaller than r_{i}. The diagonal lattices (with respect to \underline{e}) are the lattices Λ_{r} = Aπ^{r_{1}}e_{1}+⋯+Aπ^{r_{n}}e_{n} for r ∈ Z^{n}; they form a countable closed subset 𝒟 of ℒ. Note that 𝒟_{m} =^{def} 𝒟 ∩ ℒ_{m} = {Λ_{r} | ∑_{i=1}^{n} r_{i} = m}. We call the set of lattices chains in X_{I} composed of diagonal lattices 𝒟_{I}.

It is obvious that any Q′-action diagonal with respect to \underline{e} fixes every lattice in 𝒟. The converse is not true in general, but it does hold for the Q′-actions defined in the above theorem.

Lemma 3.5. Let f be the Q′-action almost commuting with N_{s} in theorem 3.4. The fixed point set of f on ℒ is 𝒟.

Proof. Let L be a lattice fixed by f, and choose a standard basis \underline{z} for L as described above. We will show that the off-diagonal components of z_{j} are zero by induction on j. The claim is vacuous for j = 1. Suppose that j > 1. Since f(λ)z_{j} ∈ L, we have

f(λ)z_{j} = ∑_{i′=1}^{n} a_{i′}z_{i′} (3-1)

for some elements a_{i′} ∈ A. It is immediate that a_{i′} = 0 for i′ > j; if i > j is the largest index such that a_{i} ≠ 0, then equating the coefficients of e_{i} in this expression gives a_{i} = 0, a contradiction. By comparing the e_{j} terms, we now get λ^{ν(r_{j},j)}π^{r_{j}} = a_{j}π^{r_{j}}, i.e. a_{j} = λ^{ν(r_{j},j)}. If i < j, then the e_{i} component of z_{i′} vanishes unless i′ = i or i′ = j by inductive hypothesis. Writing z_{ij} = ∑_{m≫−∞}^{r_{i}−1} z_{ij}^{m}π^{m} and equating the e_{i} terms on each side of (3-1), we have

∑_{m≫−∞}^{r_{i}−1} λ^{ν(m,i)}z_{ij}^{m}π^{m} = a_{i}π^{r_{i}} + λ^{ν(r_{j},j)} ∑_{m≫−∞}^{r_{i}−1} z_{ij}^{m}π^{m}.

Therefore,

a_{i} = π^{−r_{i}} ∑_{m≫−∞}^{r_{i}−1}(λ^{ν(m,i)} − λ^{ν(r_{j},j)})z_{ij}^{m}π^{m} ∈ A.

Since m − r_{i} < 0, (λ^{ν(m,i)} − λ^{ν(r_{j},j)})z_{ij}^{m} = 0 for each m. If λ^{ν(m,i)} − λ^{ν(r_{j},j)} = 0, then n(r_{j} − m) = (j − i)s, a contradiction since (s, n) = 1 and n does not divide j − i ∈ [1, n − 1]. Thus, z_{ij}^{m} = 0 for all m, i.e. z_{j} = π^{r_{j}}e_{j}.

Remarks. 1. Using a similar argument, it is possible to show that if f is a diagonal Q′-action almost commuting with a nonnilpotent element N ∈ 𝔤𝔩_{n}(F) and if (N, f) has loop index s/n with (s, n) = 1, then the fixed point set of f on ℒ is 𝒟 [S, proposition 3.12].

2. It is an immediate consequence that 𝒟 is closed in ℒ.

For any nil-elliptic N ∈ 𝔰𝔩(V_{F}), 𝒟 ∩ ℒ_{m,N} is an algebraic variety and hence has a finite number of components. Since 𝒟 is discrete, the proposition then shows that the fixed point sets (𝒫̂_{N_{s}})^{k^{∗}} for the k^{∗}-action f of proposition 3.4 are all finite. A theorem of Bialynicki-Birula [B-B1] implies that χ(𝒫̂_{N_{s}}) = χ((𝒫̂_{N_{s}})^{k^{∗}}), so the calculation of these Euler characteristics has been reduced to a problem in combinatorics.

We can decompose the computation into two complementary parts. Take I ⊊ [0, n − 1] and let J = [0, n − 1] \ {m} be a maximal proper subset of [0, n − 1] containing I. The fiber bundle 𝒫̂_{I} →^{ρ} 𝒫̂_{J} with fiber 𝒫 induces a map 𝒫̂_{I,N} → 𝒫̂_{J,N}.

Upon passing to points fixed by f, we see that

χ(𝒫̂_{I,N}) = χ((𝒫̂_{I,N})^{k^{∗}}) = ∑_{z∈(𝒫̂_{J,N} )^{k^{∗}}} |(ρ^{−1}(z))^{k^{∗}}|. (3-2)

Suppose z ∈ (𝒫̂_{J,N})^{k^{∗}} corresponds to the lattice L ∈ ℒ_{m}. Let N̄ ∈ 𝔰𝔩(L/πL) be the nilpotent endomorphism induced by N. The summand may then be interpreted as the Euler characteristic of 𝒫N̄, a fixed point variety on a classical flag manifold. We can thus approach the problem by examining the fixed points in the base and fibers of ρ separately.

We will first determine the fixed point sets (ℒ_{m,N_{s}})^{k^{∗}}. Let Λ be a lattice of valuation m fixed by f and N_{s}. Then Λ must equal Λ_{r} for some r ∈ Z^{n} with ∑_{i=1}^{n} r_{i} = m. The inclusion N_{s}Λ ⊂ Λ implies that π^{r_{i}}e_{i−1} ∈ Aπ^{r_{i−1}}e_{i−1} for i ∈ [2, n] and π^{r_{1}+s} ∈ Aπ^{n}. Thus, (ℒ_{m,N_{s}})^{k^{∗}} = {Λ_{r} | r ∈ R_{sm}} where

R_{sm} = {r ∈ Z^{n} | ∑_{i=1}^{n} r_{i} = m and r_{1} ≤ r_{2} ≤ ⋯ ≤ r_{n} ≤ r_{1} + s}. (3-3)

To determine the size of R_{sm}, we will use an argument due to Lusztig and Smelt [LS]. For i ∈ [1, n], set ζ_{i} = δ_{i1}s and define a function Θ_{s} : Z → Z by Θ_{s}(l) = ∑_{i=1}^{n}(l + i)(r_{i} − r_{i−1} + ζ_{i}) (where we view r as a periodic sequence indexed by Z). We have

\begin{aligned}Θ_{s}(l) = Θ_{s}(0) + l ∑_{i=0}^{n−1}(r_{i+1} − r_{i} + ζ_{i}) & = nr_{n} − ∑_{i=1}^{n} r_{i} + s + ls \\ & = −m + (l + 1)s + nr_{n} \\ & ≡ −m + (l + 1)s (mod n). \end{aligned} (3-4)

Hence, Θ_{s} induces a map Z/nZ → Z/nZ, and it is in fact a bijection because s and n are relatively prime. Set C_{s}^{n} = {c ∈ N^{n} | ∑_{i=1}^{n} c_{i} = s} and C_{sm}^{n} = {c ∈ C_{s}^{n} | ∑_{i=1}^{n}(l+i)c_{i} ≡ −m+(l+1)s (mod n)}. Usually the superscript will be suppressed from the notation. Just as in (3-4), the congruence will hold for each integer if it holds for a single l. The group Z/nZ acts on C_{s} via cyclic permutations of the coordinates; let C̃_{s} denote the set of orbits.

Lemma 3.6. There are bijections R_{sm} →^{ϕ} C_{sm} →^{ψ} C̃_{s}.

Proof. Set ϕ(r)_{i} = r_{i} − r_{i−1} + ζ_{i}. If ϕ(r) = ϕ(r′), then equality of the last n − 1 components implies that r′ = r + t1 where t ∈ Z and 1 is the constant n-tuple (1, …, 1). The sums of the coefficients are equal, so t = 0. Thus, ϕ is injective. Now take c ∈ C_{sm}. Set

r_{n} = \frac{m − s + ∑_{i=1}^{n} ic_{i}}{n} and r_{j} = r_{n} − (c_{j+1} + ⋯ + c_{n}) for j ∈ [1, n − 1].

These are integers by definition of C_{sm}. We have

∑_{i=1}^{n} r_{i} = nr_{n} − ∑_{i=1}^{n}(i − 1)c_{i} = m − s + ∑_{i=1}^{n} c_{i} = m.

This proves surjectivity.

Let ψ be the restriction of the orbit map to C_{sm}. The n-cycle τ = (1, 2, …, n) generates the Z/nZ action. Note that for any c ∈ C_{s}, ∑_{i=1}^{n} i(τ · c)_{i} = ∑_{i=1}^{n}(1 + i)c_{i} − nc_{n} ≡ s + ∑_{i=1}^{n} ic_{i} (mod n). This means that within the orbit of c, the sum runs through the residue classes mod n, i.e. precisely one representative lies in C_{sm}. Therefore, ψ is bijective.

Corollary 3.7. For any m ∈ Z,

χ(ℒ_{m,N_{s}}) = \frac{(n + s − 1)!}{n!s!}.

Proof. The proof of the lemma shows that Z/nZ acts freely on C_{s}. Consequently, χ(ℒ_{m,N_{s}}) = |R_{sm}| = |C̃_{s}| = |C_{s}|/n. It now suffices to recall the well known fact that |C_{s}| = \binom{n+s−1}{n−1} = \frac{(n+s−1)!}{(n−1)!s!}; it is just the number of ways of putting s indistinguishable balls into n boxes.

We have just shown that |C̃_{s}^{n}| = \frac{(n+s−1)!}{n!s!}. This expression is symmetrical in n and s, so we have a bijection C̃_{s} ˜→C̃_{n}^{s}. We can describe this bijection explicitly. The orbit c ∈ C̃_{s}^{n} may be interpreted as an arrangement of s indistinguishable balls in n boxes placed around a circle. Viewing a box as two boundary walls, we see that this is just a configuration of s balls and n walls up to cyclic permutation of the walls. However, because n and s are relatively prime, this is equivalent to an arrangement of the n + s objects up to cyclic permutation. Let a cell be a “dual box”, where the boundaries are now balls and the contents of the cell are walls. The analogous argument now gives a way of placing n walls in s cells around a circle and hence defines the desired bijection. More concretely, number the boxes and walls clockwise around the circle so that the (t − 1)st and tth walls bound the tth box. Label the balls and cells similarly, starting with the first ball counterclockwise of wall 1. Letting B_{k} denote the set of walls contained in the kth cell, we have b_{k} = |B_{k}|. The cell numbering guarantees that for walls t, t′ within a given cell, t < t′ if and only if the path from t to t′ staying in the cell runs clockwise.

We now turn to lattice chains of type J′ where J′ is a nonempty subset of [0, n−1]. Let l be the number of elements in J′. Choose m ∈ J′ and let J = {j_{1} < ⋯ < j_{l} = n} ⊆ [1, n] be the unique set such that J′ = m + ((J \ {n}) ∪ {j_{0} = 0}) (mod n). Set p_{i} = j_{i} − j_{i−1}. A chain in X_{J′} is uniquely determined by its component lattices in ℒ_{m+j} for j ∈ J. Thus, (X_{J′,N_{s}})^{k^{∗}} = {(Λ_{r^{j}})_{j∈J} | (r^{0}, …, r^{j_{l}}) ∈ Y_{Jsm}} where

Y_{Jsm} = {(r^{0}, …, r^{j_{l}}) | r^{j_{i}} ∈ R_{s,m+j_{i}} and r^{0} ≤ r^{j_{1}} ≤ ⋯ ≤ r^{j_{l−1}} ≤ r^{n} = r^{0} + 1}.

The partial order on vectors is defined componentwise.

Fix r ∈ R_{sm} and let Y_{Jsm}(r) denote those chains in Y_{Jsm} with r^{0} = r. Given any chain r̂ in this fiber, r_{t} ≤ r_{t}^{j_{1}} ≤ ⋯ ≤ r_{t}^{j_{l}} = r_{t} + 1 for each t ∈ [1, n]. Hence, there exists a unique σ(t) = j_{i} ∈ J such that r_{t}^{j_{i−1}} = r_{t} and r_{t}^{j_{i}} = r_{t} + 1. The map σ : [1, n] → J partitions [1, n] into subsets σ^{−1}(j_{i}) of size p_{i} and uniquely determines r̂. Let b ∈ C_{n}^{s} be a representative of the element of C̃_{n}^{s} corresponding to r through the bijections R_{sm} → C̃_{s} → C̃_{n}^{s}. We define a set of functions

Q′_{Js}(b) = {σ : [1, n] → J | |σ^{−1}(j)| = p_{j} and σ is nonincreasing on each B_{k}}.

Note that although this set depends on a choice of labeling of the cells B_{k}, any other labeling (not just those obtained through cyclic permutations) or even any decomposition of [1, n] into s disjoint subsets of sizes b_{k} gives a set of the same size.

Proposition 3.8. The map r̂ ↦ σ defines a bijection Y_{Jsm}(r) ˜→Q′_{Js}(b).

Proof. Take r̂ ∈ Y_{Jsm}. Note that t − 1 and t are in B_{k} if and only if c_{t} = 0; c_{t} ≠ 0 means that box t contains a ball, separating walls t − 1 and t into different cells. Suppose that t − 1 and t are in B_{k}, but σ(t) > σ(t − 1) = j_{i}. Then r_{t}^{j_{i}} = r_{t} and r_{t−1}^{j_{i}} = r_{t−1} + 1, implying that c_{t}^{j_{i}} = r_{t} − (r_{t−1} + 1) = c_{t} − 1 = −1, a contradiction. Moreover, for each j ∈ J, ∑_{t=1}^{n} r_{t}^{j} = m + j, so r_{t}^{j_{i−1}} < r_{t}^{j} for exactly p_{i} elements of [1, n]. Thus, the map sends Y_{Jsm} into Q′_{Js}(b).

The map is clearly injective. Take σ ∈ Q′_{Js}(b) and define r^{j_{i}} recursively by

r_{t}^{j_{i}} = \begin{cases}r_{t}^{j_{i−1}} + 1 & if t ∈ σ^{−1}(j_{i}) \\ r_{t}^{j_{i−1}} & if t ∉ σ^{−1}(j_{i}). \end{cases}

We will show that r^{j_{i}} ∈ R_{s,m+j_{i}} by induction on i. This is trivial for i = 0. Now suppose that i > 0 and that the claim is true for indices smaller than i. In particular, c^{j_{i′}} = ϕ(r^{j_{i′}}) is well-defined for all i′ < i. The coordinates of the n-tuple r^{j_{i}} sum to m+j_{i} because ∑_{t=1}^{n} r_{t}^{j_{i}} = ∑_{t=1}^{n} r_{t}^{j_{i−1}} +|σ^{−1}(j_{i})| = m+j_{i−1}+p_{i} = j_{i}. Also, observe that if r_{t}^{j_{i}} + ζ_{t} < r_{t−1}^{j_{i}}, then r_{t}^{j_{i−1}} + ζ_{t} = r_{t−1}^{j_{i−1}}, i.e. c_{t}^{j_{i−1}} = 0. Thus, σ(t − 1) = j_{i}, and since c_{t}^{j_{i′}} only decreases where r_{t−1}^{j_{i′}} increases, namely at the ith step, c_{t}^{j_{i′}} = 0 for each i′ < i. In particular, c_{t} = 0. But this means that t and t − 1 are in the same cell, so j_{i} = σ(t − 1) ≥ σ(t). This is a contradiction; if σ(t − 1) = σ(t), then 0 = r_{t}^{j_{i−1}} − r_{t−1}^{j_{i−1}} + ζ_{t} = r_{t}^{j_{i}} − r_{t−1}^{j_{i}} + ζ_{t} < 0 and if σ(t − 1) > σ(t) = j_{i′}, then c_{t}^{j_{i′}} = 1 for some i′ < i. This proves surjectivity.

Definition. Let d = (d_{0}, d_{1}, …, d_{l}) be a nondecreasing sequence and t a positive integer. We call a matrix (q_{ik}) ∈ M_{lt}(N) a (d, t)-intersection matrix if ∑_{k=1}^{t} q_{ik} = d_{i} − d_{i−1} for i ∈ [1, l]. We denote the set of (d, t)-intersection matrices by Q_{dt}. If d is strictly increasing, then the d_{i} − d_{i−1}’s are determined by D = {d_{0}, …, d_{l}}. In this case, Q_{Dt} =^{def} Q_{dt} is called the set of (D, t)-intersection matrices. For any b ∈ C_{d_{l}−d_{0}}^{t}, let Q_{dt}(b) ⊆ Q_{dt} be the set of (d, t)-intersection matrices with ∑_{i=1}^{l} q_{ik} = b_{k} for each k ∈ [1, t]. The group Z/tZ acts freely on Q_{dt} by cyclic permutation of the columns; we call the set of orbits Q̃_{dt}.

Lemma 3.9.

|Q_{dt}| = ∏_{i=1}^{l} \binom{s + d_{i} − d_{i−1} − 1}{d_{i} − d_{i−1}}.

Proof. The ith row of a (d, t)-intersection matrix is just an element of C_{d_{i}−d_{i−1}}^{t}, so we have a bijection Q_{dt} → ∏_{i=1}^{l} C_{d_{i}−d_{i−1}}^{t}. The result follows, since |C_{d_{i}−d_{i−1}}^{t}| = \binom{t+d_{i}−d_{i−1}−1}{d_{i}−d_{i−1}} by the proof of corollary 3.7.

We will describe the Euler characteristics of X_{J,N_{s}} in terms of (J, s)-intersection matrices. Define a map Q′_{Js}(b) → Q_{Js}(b) by σ ↦ (|σ^{−1}(j_{i}) ∩ B_{k}|). (This is the motivation for the terminology.)

Proposition 3.10. The map Q′_{Js}(b) ˜→Q_{Js}(b) is a bijection.

Proof. Fix an intersection matrix Q = (q_{ik}) ∈ Q_{Js}(b), and suppose σ is mapped to Q. The function σ is nonincreasing on B_{k}, so the top q_{1k} elements of B_{k} are in σ^{−1}(1), the next highest q_{2k} are in σ^{−1}(2), and so on. Thus, σ is completely determined by its intersection matrix. Conversely, this procedure clearly defines σ ∈ Q′_{Js}(b) mapping to Q.

We can apply this proposition to obtain a combinatorial description of the Euler characteristic of 𝒫_{N} for any classical partial flag variety 𝒫 and any nilpotent N ∈ 𝔰𝔩_{n}(k). Nilpotent conjugacy classes in 𝔰𝔩_{n}(k) are determined by the Jordan blocks of N, in other words, by a partition of n, say n = n_{1} + ⋯ + n_{t} where each n_{i} is positive.

Theorem 3.11. Assume that (char′ k, n) = 1. Let N ∈ 𝔰𝔩_{n}(k) be a nilpotent element in the conjugacy class corresponding to the partition n = n_{1} + ⋯ + n_{t}. Let s be any integer greater than t and relatively prime to n. Define b ∈ C_{n}^{s} by b_{i} = n_{i} for i ∈ [1, t] and 0 otherwise. Then for each I ⊆ [1, n−1], χ(ℱ_{I,N}) = |Q_{I∪{n},s}(b)|.

In particular, χ(ℬ_{N}) = \frac{n!}{n_{1}! ⋯ n_{t}!}.

Proof. Let r ∈ R_{s0} correspond to the orbit of b. Since the fiber of the fixed point set over r corresponds bijectively to Y_{I∪{n},s0}(r) ˜→Q_{I∪{n},s}(b), it will suffice to show that the nilpotent map N̄_{s} induced on W = Λ_{r}/πΛ_{r} has Jordan block structure given by the partition of n.

The images w_{i} of π^{r_{i}}e_{i} form a basis for W, and the matrix for N̄_{s} with respect to this basis has the same form as does N_{s}. Thus, only the (n, 1) and (i, i + 1) coefficients can possibly be nonzero. The image of w_{i} under N̄_{s} is zero if and only if r_{i} − r_{i−1} + ζ_{i} ≥ 1, i.e. exactly when c_{i} > 0. Suppose the ith and i′th columns are nonzero with vanishing intervening columns (in increasing order with 1 following n). Then c_{i} and c_{i′} are positive, and there is a cell containing precisely the walls i through i′ − 1. But a Jordan block of size l is given by a string of l − 1 consecutive nonzero columns surrounded by two zero columns, implying that Jordan blocks correspond to cells. The b_{k}’s are the cell sizes, so the claim is proved.

Since ℬ = ℱ_{[1,n−1]}, we have χ(ℬ_{N}) = |Q_{[1,n],s}(b)|. Each row of a ([1, n], s)-in-tersection matrix sums to 1, implying that the only nonzero entry in any row is a 1. Such a matrix is in the fiber over b if and only if there are exactly b_{k} 1’s in the kth column. Thus, an element of Q_{[1,n],s}(b) is determined by arbitrarily selecting subsets of sizes b_{1}, …, b_{k} from the n rows. This can be done in \frac{n!}{b_{1}!···b_{s}!} = \frac{n!}{n_{1}!···n_{t}!} ways.

Remark. The smallest integer which can be used for s in the above theorem for every nilpotent conjugacy class is n + 1.

The bijection between the fiber Y_{Jsm}(r) and Q_{Js}(b) extends to a one-to-one correspondence between the entire fixed point set and all orbits of (J, s)-intersection matrices.

Proposition 3.12. There is a bijection Y_{Jsm} →^{α} Q̃_{Js}.

Proof. We define the function α on Y_{Jsm}(r) to be the composition of the maps Y_{Jsm}(r) → Q′_{Js}(b) → Q_{Js}(b) → Q̃_{Js}. The s-tuple b ∈ C_{n}^{s} is only defined up to cyclic permutation, but this ambiguity disappears upon passing to orbits.

Suppose α(r̂) = α(r̂′). Note that Q_{Js}(b) and Q_{Js}(b′) have disjoint images in the orbit space unless b and b′ are cyclic permutations of each other. Hence, r̂ and r̂′ must be in the same fiber Q_{Js}(b). Propositions 3.8 and 3.10 now imply that r̂ = r̂′, so α is injective.

Given Q = (q_{ik}) ∈ Q_{Js}, define b by b_{k} = ∑_{i=1}^{l} q_{ik}. These are nonnegative integers, satisfying ∑_{k=1}^{s} b_{k} = ∑_{k=1}^{s} ∑_{i=1}^{l} q_{ik} = ∑_{i=1}^{l} ∑_{k=1}^{s} q_{ik} = ∑_{i=1}^{l} p_{i} = n.

This shows that b ∈ C_{n}^{s} and Q ∈ Q_{Js}(b). It is now clear that the corresponding orbit is in the image of α, proving surjectivity.

We obtain the following theorem on Euler characteristics of fixed point varieties on affine flag manifolds:

Theorem 3.13. Assume that (n, char′ k) = 1, and let s be a positive integer relatively prime to n. Let I = {j_{1} < ⋯ < j_{l}} ⊆ [0, n − 1] be a nonempty set of size l with p_{i} = j_{i+1} − j_{i} (where j_{l+1} = j_{1} + n). Then

χ(X_{I,N_{s}}) = s^{−1} ∏_{i=1}^{l} \binom{s + p_{i} − 1}{p_{i}}.

In particular, χ(ℬ̂_{N_{s}}) = s^{n−1} and if 𝒦 is any variety of maximal parahoric subal-

gebras, χ(𝒦̂_{N_{s}}) = \frac{(n + s − 1)!}{n!s!}.

Proof. Let J ⊆ [1, n] be the unique set such that I = j_{1} + ((J \ {n}) ∪ {0}) (mod n). Note that J = {j_{2} − j_{1}…. , j_{l} − j_{1}, n}, so the definition of the p_{i}’s given in the statement coincides with the earlier definition. Since there is a bijection Y_{Jsj_{1}} ˜→Q̃_{Js} by proposition 3.12, we have

χ(X_{I,N_{s}}) = χ((X_{I,N_{s}})^{k^{∗}}) = |Y_{Jsj_{1}}| = |Q̃_{Js}| = \frac{|Q_{Js}|}{s}.

The formula now follows from lemma 3.9.

The varieties of maximal parahorics correspond to lattice chains of type I with I a singleton. In this case, l = 1 and p_{1} = n, and the Euler characteristic is

\frac{(n + s − 1)}{n!(s − 1)!s} = \frac{(n + s − 1)!}{n!s!}.

We thus recover corollary 3.7. The full affine flag variety ℬ̂ is isomorphic to X_{[0,n−1]}.

This gives l = n and p_{i} = 1 for each i, so χ(ℬ̂_{N_{s}}) = s^{−1}\binom{s}{1}^{n} = s^{n−1}.

Remarks. 1. Let λ_{1}, …, λ_{n−1} be the roots of 𝔰𝔩_{n}(k) and λ_{0} the affine root, and consider the subset of the positive Weyl chamber in an apartment of the affine building given by the conditions λ_{i} ≥ 0 for i ∈ [1, n − 1] and λ_{0} ≤ s. The Euler characteristic of 𝒫̂_{N_{s}} is then just the number of simplices of type corresponding to the parahoric class in this region.

2. For s = 1, the theorem implies that the Euler characteristic is 1 for each parahoric while if n is odd and s = 2, the Euler characteristics are given by

χ(X_{I,N_{b2}}) = \frac{1}{2} ∏_{i=1}^{l}(p_{i} + 1).

3. Lusztig and Smelt obtained the formulas for χ(ℬ̂_{N_{1s}}) and χ(ℒ_{0,N_{1s}}) [LS].

Finally, we recall a theorem due to Lusztig and Smelt:

Theorem 3.14. The varieties 𝒫̂_{N_{s}} have stratifications with finitely many strata, each of which is isomorphic to an affine space over k.

Proof. Since N_{s} is conjugate to the nil-elliptic element T_{s} considered by Lusztig and Smelt, their result carries over immediately[LS]. (Their argument, given for Iwahori and maximal parahoric subalgebras applies to the other parahoric subalgebras as well.)

Remark. The particular k^{∗}-action considered by Lusztig and Smelt came from the Q′-action f(λ)π^{m}e_{i} = λ^{mn−i}π^{m}e_{i}. The cells of the Bruhat decomposition for 𝒫̂ for this choice of ordered basis are just the cells of the Bialynicki-Birula plus decomposition for f. For example, consider the Bruhat cells of ℒ_{m}. Set i, j ∈ [1, n] with i ≠ j, set < i, j >= 0 if i < j and 1 otherwise. The cells are indexed by r ∈ Z^{n} such that ∑_{i=1}^{n} r_{i} = m, and they consist of those lattices with an A-basis z_{1}, …, z_{n} of the form z_{j} = π^{r_{j}}(e_{j} + ∑_{i≠j}(∑_{l≥0} a_{ij}^{l}π^{l+<i,j>})e_{i}), where a_{ij}^{l} ∈ k for i ≠ j, l ≥ 0 and a_{ij}^{l} = 0 if l ≥ r_{i} − r_{j}− < i, j >. It is easy to see that f preserves the Bruhat cells and acts on the coordinates a_{ij}^{l} for a given cell by sending them to λ^{nl+n<i,j>+j−i}a_{ij}^{l}. The coefficient of λ is always positive, so the limit as λ goes to zero of f(λ)L for any lattice in this cell is the diagonal lattice with A-basis π^{r_{1}}e_{1}, … π^{r_{n}}e_{n}. Since the cells of Lusztig and Smelt’s stratification of 𝒫̂_{T_{s}} are just the intersections of the fixed point variety with the Bruhat cells, we see that the stratification coincides with the plus decomposition.

Corollary 3.15. The étale cohomology of the varieties 𝒫̂_{N_{s}} vanishes in odd degrees.

4. k^{∗}-actions on fixed point varieties of affine flag manifolds: The symplectic algebras

We now consider the simple root systems C_{n}. Let V be a 2n-dimensional vector space over k endowed with a nondegenerate, alternating, bilinear form ⟨ , ⟩. The symplectic group Sp(V) is the group of linear automorphisms of V preserving the form. The corresponding Lie algebra is denoted 𝔰𝔭(V). A basis \underline{e} = (e_{1}, …, e_{2n}) of V is called symplectic if the subspaces ke_{1} ⊕ ⋯ ⊕ ke_{n} and ke_{n+1} ⊕ ⋯ ⊕ ke_{2n} are isotropic, i.e. the restriction of the form to these subspaces is identically 0, and ⟨e_{i}, e_{n+j}⟩ = δ_{ij} for i, j ∈ [1, n]. It is well known that V admits such a basis. Define a matrix

J = (\begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix}) .

A choice of symplectic basis identifies Sp(V) with Sp_{2n}(k) = {g ∈ GL_{2n}(k) | g^{t}Jg = J}, the semisimple, simply connected, algebraic group of type C_{n}. If n = 1, this is just SL_{2}(k). The symplectic Lie algebra with respect to this basis 𝔰𝔭_{2n}(V) is the space of 2n × 2n matrices M satisfying M^{t}J + JM = 0; equivalently,

M = (\begin{matrix}M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) (4-1)

where M_{22} = −M_{11}^{t} and the off-diagonal n × n blocks are symmetric.

The symplectic basis \underline{e} determines a standard Borel subalgebra 𝔟 consisting of the matrices M ∈ 𝔰𝔭_{2n}(k) such that M_{21} = 0 and M_{11} is upper triangular, using the notation of (4-1). (The subalgebra of upper triangular matrices in 𝔰𝔭_{2n}(k) is not a Borel subalgebra.) Let 𝔟̂ ⊂ 𝔰𝔭_{2n}(A) be the corresponding standard Iwahori subalgebra. We define the algebras 𝔮̂_{i} ⊂ 𝔰𝔭_{2n}(F) for i ∈ [0, n] by

i n − i i n − i

𝔮̂_{i} = 𝔰𝔭_{2n}(F) ∩ (\begin{matrix}A & A & π^{−1}A & A \\ πA & A & A & A \\ πA & πA & A & πA \\ πA & A & A & A \end{matrix}) \begin{matrix}i \\ n − i \\ i \\ n − i \end{matrix}

These are the maximal parahoric subalgebras 𝔮̂_{i} containing our fixed Iwahori subalgebra. Their stabilizers Q̂_{i} ⊆ Sp_{2n}(F) have the same block form. The parahoric subalgebras containing 𝔟̂ are now given by 𝔭̂_{I} = ∩_{i∉I} 𝔮̂_{i} with stabilizer P̂_{I} = ∩_{i∉I} Q̂_{i} for each proper subset I of [0, n]; this notation is compatible with notation of section 2 for a suitable choice of simple reflections for Ŵ, indexed by [0, n]. The affine flag manifolds 𝒫̂_{I} for Sp(V_{F}) are identified with the quotient spaces Sp(V_{F})/P̂_{I}.

We will again interpret the spaces of parahoric subalgebras as chains of A-lattices in V_{F}. Let L ⊂ V_{F} be a lattice. The set L^{∗} = {v ∈ V_{F} | ⟨v, L⟩ ⊆ A} is called the dual lattice to L; it is a lattice with valuation −v(L).

Definition. A lattice L is called symplectic if the set of lattices generated from L by homothety and duality (or equivalently, the set of lattices homothetic to L or L^{∗}) is totally ordered by inclusion, i.e. a chain. The variety of symplectic lattices is denoted ℒ^{Sp}.

The symplectic group Sp(V_{F}) acts on ℒ^{Sp}, and the orbits of this action are the subvarieties ℒ_{m}^{Sp} =^{def} ℒ^{Sp} ∩ ℒ_{m}. We can apply the theory of alternating forms on free modules over a principal ideal domain (see theorem IX.5.1 in [B1]) to obtain a more explicit characterization of these orbits. For m ≥ 0, write m = qn + r with r ∈ [0, n − 1]. The variety ℒ_{m}^{Sp} is the set of lattices contained in their duals and admitting a basis \underline{z} such that ⨁_{i=1}^{n} Az_{i} and ⨁_{i=n+1}^{2n} Az_{i} are isotropic submodules and ⟨z_{i}, z_{n+j}⟩ equals δ_{ij}π^{q} for i ∈ [1, n − r] and δ_{ij}π^{q+1} for i > n − r. For m ≤ 0, ℒ_{m}^{Sp} is the set of lattices whose duals are in ℒ_{|m|}^{Sp}. In particular, ℒ_{0}^{Sp} is the space of self-dual lattices.

As in the case of SL(V_{F}), if m ≡ m′ (mod 2n), then ℒ_{m}^{Sp} ≅ ℒ_{m′}^{Sp} as an Sp(V_{F})- variety. In addition, the correspondence L ↦ L^{∗} defines an Sp(V_{F})-isomorphism ℒ_{m}^{Sp} → ℒ_{−m}^{Sp}. Thus, by identifying a symplectic lattice L with the totally ordered set {π^{q}L, π^{q′}L^{∗} | q, q′ ∈ Z}, we can view ℒ_{m}^{Sp} as a variety of chains of symplectic lattices. More generally, we make the following definition:

Definition. Let I be a nonempty subset of [0, n]. The space of (partial) symplectic lattice chains of type I is the set

X_{I}^{Sp} = {(L_{i}) | i ∈ I ∪ −I (mod 2n), L_{i} ∈ ℒ_{i}^{Sp},

L_{i} ⊃ L_{j} ⟺ i < j, L_{i+2n} = πL_{i}, and L_{−i} = L_{i}^{∗}}.

The Sp(V_{F})-action on the ℒ_{m}^{Sp}’s extends to an action on the X_{I}^{Sp}’s. Note that X_{I}^{Sp} is a subvariety of X_{I∪(2n−I)\{2n}}, the set of lattice chains of type I ∪(2n−I)\{2n}.

An argument similar to that sketched in section 3 shows that the varieties X_{I}^{Sp} are the affine flag manifolds for Sp(V_{F}) (see proposition 4.6 and theorem 4.8 in [S]):

Theorem 4.1. For each proper subset of [0, n], 𝒫̂_{I} ≅ X_{I^{c}}^{Sp} as an Sp(V_{F})-variety. Moreover, if 0 ∈ I, then the fiber over the self-dual lattice L of the projection X_{I}^{Sp} → X_{{0}}^{Sp} = ℒ_{0}^{Sp} is isomorphic as an Sp(L/πL)-variety to ℱ_{I\{0}}^{Sp}(L/πL), the variety of isotropic flags of type I \ {0} in L/πL.

Let Q̃′_{Sp} = GSp(V) × k^{∗}, where GSp(V) is the group of symplectic similitudes of V. We call a Q′-action f a Q̃′_{Sp}-action if f(k^{∗}) ⊆ Q̃′_{Sp}; these are precisely the Q′-actions which preserve the space of symplectic lattices. Such a homomorphism induces a Q′_{Sp}-action via the projection GSp(V) → P GSp(V) as described in section 2.

We now assume that char k ≠ 2. If N ∈ 𝔰𝔭(V_{F}) is semisimple, its eigenvalues must be of the form ±a_{1}, …, ±a_{n} where a_{i} ∈ \overline{F}. The characteristic polynomial char_{N}(μ) = ∏_{i=1}^{n}(μ^{2} − a_{i}^{2}) is an element of F [μ^{2}], and we define a polynomial h_{N} ∈ F [μ] by h(μ^{2}) = char_{N}(μ). The eigenvalues of ad N : 𝔤 → 𝔤 are ±a_{i} ± a_{j} for i < j, ±2a_{i}, and 0 with multiplicity n. The endomorphism N is regular semisimple if the kernel of N has dimension n; this occurs when the a_{i}^{2}’s are distinct and nonzero. It is well-known that the characteristic polynomial, or equivalently h_{N}, determines the Sp(V\overline{F})-conjugacy classes of regular semisimple elements of 𝔰𝔭(V\overline{F}). Proposition 3.2 shows that the same is true for regular semisimple classes in 𝔰𝔭(V_{F}).

Let 𝒞 be a regular semisimple conjugacy class in 𝔰𝔭(V_{F}), and suppose that there exists N ∈ 𝒞 and a Q′_{Sp}-action f almost commuting with N. The endomorphism N is also regular semisimple viewed as an element of 𝔤𝔩(V_{F}), so by theorem 2.4, char_{N} is a homogeneous polynomial in μ and π^{q′} for some nonzero q′ ∈ \frac{1}{2n−1}Z ∪ \frac{1}{2n}Z. Moreover, since det N ≠ 0, the μ^{0}π^{2nq′} term does not vanish, implying that 2nq′ ∈ Z. It is immediate that h_{N} is homogeneous in μ and π^{q} where q = 2q′ ∈ \frac{1}{n}Z. The converse can also be proven by an argument similar to the proof of theorem 2.4.

Theorem 4.2. Let 𝒞 be a regular semisimple conjugacy class in 𝔰𝔭(V_{F}). Then there is a representative N ∈ 𝒞 almost commuting with a Q′_{Sp}-action if and only if h_{𝒞} (μ) is a homogeneous polynomial in μ and π^{q} for some nonzero q ∈ \frac{1}{n}Z. 𝒞 is nil if and only if q > 0.

If we write h_{𝒞} (μ) = ∑_{i=0}^{n} c_{i}π^{qi}μ^{n−i} and identify 𝔰𝔭(V_{F}) with 𝔰𝔭_{2n}(F) by choosing a symplectic basis for V, we can define a representative N = (a_{ij}) for 𝒞 by:

a_{ij} = \begin{cases}1 & if i = j + 1, j ∈ [1, n − 1] or (i, j) = (1, n + 1) \\ −1 & if j = i + 1, i ∈ [n + 1, 2n − 1] \\ (−1)^{i}c_{i}π^{qi} & if i = j + n, j ∈ [1, n] \\ 0 & otherwise. \end{cases}

Corollary 4.3. The regular semisimple conjugacy classes in 𝔰𝔭(V_{F}) obtained from almost commuting pairs (N, f) with f a Q′_{Sp}-action are precisely those with h_{𝒞} (μ) = ∏_{i=1}^{r}(μ^{d} − b_{i}π^{s}) where rd = n, (s, d) = 1, (char′ k, d) = 1, and the b_{i}’s are distinct elements of k^{∗}.

Proof. If 𝒞 is such a class, then h_{𝒞} is a homogeneous polynomial in μ and π^{q}. Let q = s/d with (s, d) = 1 and d > 0. All the eigenvalues are nonzero, so by corollary 2.3, each is homogeneous of degree q. If aπ^{q} is an eigenvalue, then it satisfies the irreducible polynomial μ^{d} − a^{d}π^{s}. Consequently, every irreducible factor of h_{𝒞} has degree d, implying that n = rd for some positive integer r. Since these r factors must be distinct, we obtain the r different b_{i}’s. Finally, the factors are separable polynomials if and only if (char′ k, d) = 1.

We now find an explicit criterion for ellipticity of regular semisimple conjugacy classes in 𝔰𝔭(V_{F}). Write h_{N} = h_{1} … h_{l} where the h_{i}’s are distinct irreducible polynomials. If we set h′_{i}(μ) = h_{i}(μ^{2}), then char_{N} = h′_{1} … h′_{l}. Let E_{i} = F [μ]/(h_{i}); it is a field extension of degree n_{i} = deg h_{i} over F. By the Chinese remainder theorem, E = F [μ]/(h_{N}) ≅ ⊕_{i=1}^{l}E_{i}. Similarly, if we set E′_{i} = F [μ]/(h′_{i}), E′ = F [μ]/(char_{N}) ≅ ⊕_{i=1}^{l}E′_{i}. The F -algebra injection E ↪ E′ given by μ ↦ μ^{2} induces injections E_{i} ↪ E′_{i} for each i, making E′_{i} into a 2-dimensional E_{i}-algebra. If h′_{i} is irreducible, then E′_{i} is a field. Otherwise, let g_{1}, …, g_{t} be the monic, irreducible factors of h′_{i}. If g_{j}(μ)|h′_{i}, then so does g_{j}(−μ). Moreover, if the ideals generated by g_{j}(μ) and g_{j}(−μ) were equal, then g_{j} would be a polynomial in μ^{2} and would give rise to a proper irreducible factor of h_{i}. We can thus assume that g_{2}(μ) = (−1)^{n}g_{1}(−μ); the coefficient guarantees that g_{1} and g_{2} are simultaneously monic. But now g_{1}g_{2} is a polynomial in μ^{2}, and this contradicts irreducibility of h_{i} unless h′_{i} = g_{1}g_{2}. Since g_{1} and g_{2} are distinct, E′_{i} ≅ F [μ]/(g_{1})⊕F [μ]/(g_{2}) ≅ E_{i}⊕E_{i}. The last isomorphism follows because the homomorphisms E_{i} → F [μ]/(g_{j}) for j ∈ [1, 2] are nonzero maps between fields which have the same dimension as F - vector spaces.

Proposition 4.4. Let N be a regular semisimple element of 𝔰𝔭(V_{F}). Then N is elliptic if and only if F [μ]/(h_{N}) and F [μ]/(char_{N}) decompose into direct sums of the same number of fields. Equivalently, for each irreducible factor h_{i} of h_{N}, h′_{i}(μ) = h_{i}(μ^{2}) is also irreducible.

Proof. We construct an element N_{i} ∈ 𝔰𝔭_{2n_{i}}(F) with h_{N_{i}} = h_{i} by the procedure of theorem 4.2. Using the injection ∏_{i=1}^{l} 𝔰𝔭_{2n_{i}}(F) ↪ 𝔰𝔭_{2n}(F), we obtain a ‘blockdiagonal’ matrix N′ ∈ 𝔰𝔭_{2n}(F) with h_{N′} = h_{N}. (The restriction of N′ to W_{1} = ⊕_{j=1}^{n_{1}}(F e_{j} ⊕ F e_{n+j}) is N_{1} and so on.) Since N is regular semisimple, N and N′ are conjugate, and there is a symplectic basis \underline{w} of V_{F} in which N is given by N′.

Suppose some h′_{i} is reducible. Since V_{F} = W_{1} ⊥ ⋯ ⊥ W_{l}, it suffices to exhibit an F -cocharacter F^{∗} → Z_{Sp(W_{i})}(N_{i}). We thus assume without loss of generality that h_{N} is irreducible, but char_{N} is not. We then have char_{N} = g_{1}g_{2} with g_{2}(μ) = (−1)^{n}g_{1}(−μ). Construct a matrix M ∈ 𝔤𝔩_{n}(F) with characteristic polynomial g_{1} as in the proof of theorem 2.4. It has n distinct eigenvalues a_{1}, …, a_{n} ∈ \overline{F} and g_{1}(μ) = ∏_{i=1}^{n}(μ − a_{i}). The transpose M^{t} has the same characteristic polynomial and eigenvalues, so char_{−M^{t}}(μ) = ∏_{i=1}^{n}(μ + a_{i}) = (−1)^{n}g_{1}(−μ) = g_{2}(μ). The block-diagonal matrix M′ = diag(M, −M^{t}) ∈ 𝔰𝔭_{2n}(F) has the same characteristic polynomial as N, and again invoking regularity, N and M′ are Sp_{2n}(F)-conjugate. Since the F -cocharacter F^{∗} → Z_{Sp_{2n}(F)}(M′) given by x ↦ diag(xI_{n}, x^{−1}I_{n}) is nontrivial, N is not elliptic.

Conversely, suppose N is not elliptic, so that there exists a nontrivial cocharacter F^{∗} →^{ϕ} Z_{Sp_{2n}(F)}(N). Using the above orthogonal decomposition of V_{F}, we obtain

cocharacters F^{∗} →^{ϕ_{i}} Z_{Sp(W_{i})}(N_{i}). Since at least one ϕ_{i} is nontrivial, we can assume without loss of generality that h_{N} is irreducible. If char_{N} is also irreducible, then there cannot even be a nontrivial cocharacter F^{∗} → Z_{SL_{2n}(F)}(N). Thus, char_{N} is reducible as desired.

Suppose that N is nil-elliptic and almost commutes with a Q′_{Sp}-action. The irreducible factors of h_{N} are all of the form h_{i}(μ) = μ^{d} − b_{i}π^{s} with b_{i} ∈ F^{∗}, (d, s) = 1, and s > 0. The polynomial h′_{i}(μ) = μ^{2d} − b_{i}π^{s} is irreducible if and only if (2d, s) = 1, i.e. when (d, s) = 1 and s is odd. Applying the previous result, we have:

Proposition 4.5. The nil-elliptic conjugacy classes in 𝔰𝔭(V_{F}) obtained from almost commuting pairs (N, f) with f a Q′_{Sp}-action are precisely those with h_{𝒞} (μ) = ∏_{i=1}^{r}(μ^{d} − b_{i}π^{s}) where s > 0, rd = n, (s, 2d) = 1, (char′ k, d) = 1, and the b_{i}’s are distinct elements of k^{∗}. If N is the standard matrix representative of theorem 4.2, f almost commutes with exponent s with the diagonal Q′_{Sp}-action f given by ν(m, i) = 2dm + is and ν(m, n + i) = 2dm + (1 − i)s for i ∈ [1, n].

The varieties (𝒫̂_{N})^{k^{∗}} are in general much simpler than the 𝒫̂_{N}’s, but, because the k^{∗}-actions of the above proposition do not always have discrete fixed point sets, they are not necessarily finite. It is possible to identify the fixed lattices precisely; the off-diagonal entries which may be nonzero as well as the terms present in these Laurent series are fully determined by r.

We now restrict ourselves to the case r = 1, d = n, and (char′ k, 2n) = (s, 2n) = 1. We let N_{bs} denote the standard representative of the class with h_{𝒞} (μ) = μ^{n} −bπ^{s} given in theorem 4.2. (This does not seriously conflict with our previous notation in section 3 because the element of 𝔰𝔩_{2n}(F) given by the two definitions are in the same SL_{2n}(F)-conjugacy class. Also, note that the fixed point varieties for N_{bs} viewed as an element of 𝔰𝔭_{2n}(F) are closed subvarieties of the analogous fixed point varieties with N_{bs} viewed as an element of 𝔰𝔩_{2n}(F); in particular, they are algebraic varieties even in positive characteristic.) Again, we can assume b = 1, and we set N_{s} =^{def} N_{1s}. By the remark after lemma 3.4, the Q′_{Sp}-action f fixes only the diagonal lattices. The set of symplectic lattice chains of type J composed of diagonal lattices will be denoted by 𝒟_{J}^{Sp}; in particular, we set 𝒟_{m}^{Sp} = 𝒟_{m} ∩ℒ_{m}^{Sp}. Given I ⊆ [1, n], let δ_{i}^{I} = 1 if i ∈ I and δ_{i}^{I} = 0 otherwise. It is immediate that for 0 ≤ m ≤ n, Λ_{r} ∈ 𝒟_{m}^{Sp} if and only if there exists I ⊆ [1, n] such that |I| = m and r_{n+i} = −r_{i} + δ_{i}^{I}.

We will again apply the theorem of Bialynicki-Birula to give explicit combinatorial interpretations for the Euler characteristics χ(𝒫̂_{N_{s}}). Because of the considerable loss of symmetry involved in the passage to the symplectic algebras, the resulting counting problems are more complicated than in the special linear case.

We begin by determining the fixed point sets (ℒ_{m,N_{s}}^{Sp})^{k^{∗}}. Suppose that Λ = Λ_{r} ∈ 𝒟_{m}^{Sp} is fixed by N_{s}. We must have π^{r_{i}}e_{i+1} ∈ Aπ^{r_{i+1}}e_{i+1} and −π^{r_{n+i+1}}e_{n+i} ∈ Aπ^{r_{n+i}}e_{n+i} for i ∈ [1, n − 1], π^{r_{n+1}}e_{1} ∈ Aπ^{r_{1}}e_{1}, and (−1)^{n+1}π^{r_{n}+s}e_{2n} ∈ Aπ^{r_{2n}}e_{2n}. As a result, (ℒ_{m,N_{s}}^{Sp})^{k^{∗}} = {Λ_{r} | r ∈ R_{sm}^{Sp}} where R_{sm}^{Sp} is the union over I ⊆ [1, n] of size m of the disjoint sets

R_{sI}^{Sp} = {r ∈ Z^{2n} | r_{n} ≤ r_{n−1} ≤ ⋯ ≤ r_{1} ≤ r_{n+1} ≤ r_{n+2} ≤ ⋯ ≤ r_{2n} ≤ r_{n} + s

and r_{n+i} = −r_{i} + δ_{i}^{I}}.

Let R_{sm} be the set of 2n-tuples such that (ℒ_{m,N_{s}})^{k^{∗}} = {Λ_{r} | r ∈ R_{sm}}. This is the same definition as in (3-4) up to the numbering of the entries. Thus, we have a bijection R_{sm} →^{ϕ} C_{sm} = C_{sm}^{2n} as in lemma 3.6. The indexing c = (c_{1}, …, c_{2n}) is not convenient here, and we will write instead

(c_{1}, …, c_{2n}) = (q_{n}, q_{n−1}, …, q_{1}, q_{0}, q_{−1}, …, q_{−(n−1)}).

The function ϕ is given by q_{0} = r_{n+1} − r_{1}, q_{n} = r_{n} − r_{2n} + s, q_{i} = r_{i} − r_{i+1}, and q_{−i} = r_{n+i+1} − r_{n+i} for i ∈ [1, n − 1].

We now determine ϕ(R_{sm}^{Sp}). Given a set I ⊆ [1, n] and i ∈ [1, n − 1], let ϵ_{i}^{I} = δ_{i+1}^{I} − δ_{i}^{I}. Define

C_{sI}^{Sp} = {c ∈ C_{s}^{2n} | q_{−i} = q_{i} + ϵ_{i}^{I} for i ∈ [1, n − 1], q_{n} is even ⟺ n ∈ I,

and q_{0} is odd ⟺ 1 ∈ I}

and

C_{sm}^{Sp} = ⋃_{\begin{matrix}I⊆[1,n] \\ |I|=m \end{matrix}} C_{sI}^{Sp}.

The union is clearly disjoint. Note that it is not a priori obvious that C_{sm}^{Sp} ⊆ C_{sm}. We will show that ϕ(R_{sm}^{Sp}) = C_{sm}^{Sp}. We need the following lemmas:

Lemma 4.6. For any n ≥ 1 and any I ⊆ [1, n] with |I| = m,

∑_{i=1}^{n−1} iϵ_{i}^{I} = \begin{cases}n − m & if n ∈ I \\ −m & if n ∉ I. \end{cases}

Proof. We have

∑_{i=1}^{n−1} iϵ_{i}^{I} = ∑_{i=1}^{n−1} i(δ_{i+1}^{I} − δ_{i}^{I}) = nδ_{n}^{I} − ∑_{i=1}^{n} δ_{i}^{I} = nδ_{n}^{I} − m

as desired.

Lemma 4.7. C_{sm}^{Sp} ⊆ C_{sm}.

Proof. We must show that ∑_{i=1}^{2n}(l + i)c_{i} ≡ −m + (l + 1)s (mod 2n) for some l ∈ Z. Set l = −(n+1). The sum is then −nq_{n}+0q_{0}+∑_{i=1}^{n−1} i(q_{−i}−q_{i}) = −nq_{n}+∑_{i=1}^{n−1} iϵ_{i}^{I}. By lemma 4.6, this is congruent to n − m regardless of the parity of q_{n}. Since s is odd, −m − ns ≡ −m + n (mod 2n), and the result follows.

We are now ready to prove:

Proposition 4.8. The map ϕ induces bijections R_{sI}^{Sp} ˜→C_{sI}^{Sp} for each I ⊆ [1, n] and hence bijections R_{sm}^{Sp} ˜→C_{sm}^{Sp} for all m ∈ [0, n].

Proof. It is clear from lemma 4.7 that ϕ(R_{sI}^{Sp}) ⊆ C_{sI}^{Sp} ⊆ C_{s|I|}^{Sp}, so it will suffice to

show that ϕ^{−1}(C_{sI}^{Sp}) ⊆ R_{sI}^{Sp}. Choose c ∈ C_{sI}^{Sp}, and let r = ϕ^{−1}(c). We will show by downward induction on i that r_{i} + r_{n+i} = δ_{i}^{I}.

Using the formula for ϕ^{−1} given in lemma 3.6, we have

r_{2n} = \frac{m − s + ∑_{j=−n+1}^{n}(n + 1 − j)q_{j}}{2n} and
r_{n} = \frac{m − s + q_{n} + ∑_{j=−n+1}^{n−1}(−n + 1 − j)q_{j}}{2n} .

Adding, we get

r_{n} + r_{2n} = \frac{m − s + q_{n} + ∑_{j=−n+1}^{n−1}(1 − j)q_{j}}{n}

= \frac{m − s + q_{n} + ∑_{j=−n+1}^{n−1} q_{j} − ∑_{j=−n+1}^{n−1} jq_{j}}{n}

= \frac{m − s + q_{n} + (s − q_{n}) + ∑_{j=1}^{n−1} jϵ_{j}^{I}}{n} = δ_{n}^{I},

where the last step is lemma 4.6.

Now suppose r_{i+1} + r_{n+i+1} = δ_{i+1}^{I}. By the formula for ϕ^{−1}, r_{i} + r_{n+i} = r_{i+1} + r_{n+i+1} + (q_{i} − q_{−i}). If i + 1 ∈ I, then the term in parentheses is −1 (resp. 0) for i ∉ I (resp. i ∈ I) as desired. The case i + 1 ∉ I is checked similarly.

Given a set I ⊆ [1, n], define Î = I ∪ {n + 1}. Let

G_{sI}^{Sp} = {p = (p_{0}, …, p_{n}) ∈ C_{s}^{n+1} | p_{i} is even ⟺ i, i + 1 ∈ Î or i, i + 1 ∉ Î},

and let G_{sm}^{Sp} denote the (disjoint) union of these sets over all I of size m. Define a

function C_{sI}^{Sp} →^{ψ} G_{sI}^{Sp} by q ↦ (q_{0}, q_{1} + q_{−1}, …, q_{n−1} + q_{−(n−1)}, q_{n}). We also have a

map G_{sI}^{Sp} → C_{sI}^{Sp} with q_{0} = p_{0}, q_{n} = p_{n}, and for i ∈ [1, n − 1],

(q_{i}, q_{−i}) = \begin{cases}([p_{i}/2], [p_{i}/2] + 1) & if i ∉ I, i + 1 ∈ I \\ ([p_{i}/2] + 1, [p_{i}/2]) & if i ∈ I, i + 1 ∉ I \\ ([p_{i}/2], [p_{i}/2]) & otherwise. \end{cases}

It is easy to see that these maps are inverses, so we have proven:

Lemma 4.9. The function ψ defines bijections C_{sI}^{Sp} ˜→G_{sI}^{Sp} and C_{sm}^{Sp} ˜→G_{sm}^{Sp} for all I ⊆ [1, n] and m ∈ [0, n].

For any I ⊆ [1, n] with |I| = m, we can interpret G_{sI}^{Sp} as the set of distinct arrangement of s balls in n + 1 boxes with the parity of the number of balls in each box determined by I. Suppose l of the boxes contain an odd number of balls. Since these boxes are nonempty and s = 2t + 1 is odd, l = 2a + 1 for some a ∈ [0, t]. Note that ∑_{i=0}^{n}[p_{i}/2] = (s − l)/2 = t − a, and any such arrangement is uniquely determined by the [p_{i}/2]’s. Thus, there is a bijection from G_{sI}^{Sp} to C_{t−a}^{n+1}, implying that |G_{sI}^{Sp}| = \binom{n+t−a}{n}.

Let h_{n,m,a} be the number of subsets I ⊆ [1, n] with |I| = m such that 2a + 1 of the p_{i}’s are odd. A succession in Î is a pair of numbers i, i + 1 ∈ Î. Let g_{n,m,j} be the number of I’s of size m such that Î has j successions. Write Î = {i_{1} < ⋯ < i_{m} < i_{m+1} = n + 1}, and partition it into maximal sequences of consecutive integers {i_{l_{j−1}+1} < ⋯ < i_{l_{j}}}. Suppose there are d sequences in the partition. Since 0 ∉ Î and n + 1 ∈ Î, l_{0} = 0 and l_{d} = m + 1. Each maximal sequence contributes l_{j} −l_{j−1}−1 successions to Î, so there are a total of l_{d}−l_{0}−d = m+1−d successions. The boxes containing an odd number of balls are those indexed by i_{l_{j−1}+1} − 1 for j ∈ [1, d] and i_{l_{j}} for j ∈ [1, d − 1]. Hence, there are 2(d − 1) + 1 such boxes, i.e. a = d − 1, implying that h_{n,m,a} = g_{n,m,m−a}. We must have 0 ≤ a ≤ m, since the number of successions is evidently between 0 and m.

Lemma 4.10. For each j ∈ [0, m],

g_{n,m,j} = \binom{m}{j}\binom{n − m}{m − j}.

Proof. Let f_{n,m,j} be the number of size m subsets I ⊆ [1, n] having j successions. It is well known (see [R, p. 12]) that

f_{n,m,j} = \binom{m − 1}{j}\binom{n − m + 1}{m − j}.

We see that g_{n,m,j} is the number of subsets of [1, n + 1] of size m + 1 which contain the element n+1 and have j successions. This collection of subsets consists of those size m + 1 subsets with j successions which are not already subsets of [1, n]. Thus,

\begin{aligned}g_{n,m,j} = f_{n+1,m+1,j} − f_{n,m+1,j} & = \binom{m}{j}\binom{n − m + 1}{m + 1 − j} − \binom{m}{j}\binom{n − m}{m + 1 − j} \\ & = \binom{m}{j}\binom{n − m}{m − j}. \end{aligned}

We can finally compute |G_{sm}^{Sp}|.

Proposition 4.11.

|G_{sm}^{Sp}| = \binom{[s/2] + m}{m}\binom{[s/2] + n − m}{n − m}.

Proof. We have

|G_{sm}^{Sp}| = ∑_{\begin{matrix}I⊆[1,n] \\ |I|=m \end{matrix}} |G_{sI}^{Sp}| = ∑_{a=0}^{m} h_{n,m,a}\binom{n + t − a}{n} = ∑_{a=0}^{m} g_{n,m,m−a}\binom{n + t − a}{n}

= ∑_{a=0}^{m} \binom{m}{m − a}\binom{n − m}{a}\binom{t + n − a}{n}

= ∑_{a=0}^{m} \binom{m}{a}\binom{n − m}{a}\binom{t + n − a}{n}

= \binom{t + m}{m}\binom{t + n − m}{n − m}.

The last equality is proven in [R, p. 17]; it is an application of the Vandermonde convolution formula. The result follows since t = [s/2]. Note that the summation from 0 to m is correct even when t is smaller than m because \binom{n+t−a}{n} = 0 for a > t.

Combining the fact that χ(ℒ_{m,N_{s}}^{Sp}) = |R_{sm}^{Sp}| with lemma 4.9 and propositions 4.8 and 4.11, we obtain:

Corollary 4.12. For any m ∈ [0, n],

χ(ℒ_{m,N_{s}}^{Sp}) = \binom{[s/2] + m}{m}\binom{[s/2] + n − m}{n − m}.

In particular, χ(ℒ_{m,N_{s}}^{Sp}) = χ(ℒ_{n−m,N_{s}}^{Sp}) and χ(ℒ_{0,N_{s}}^{Sp}) = χ(ℒ_{n,N_{s}}^{Sp}) = \binom{[s/2]+n}{n}.

Remarks. 1. The variety (ℒ_{m,N_{s}}^{Sp}) is isomorphic to the varieties (ℒ_{m+2ni,N_{s}}^{Sp}) and (ℒ_{−m+2ni,N_{s}}^{Sp}) for any i ∈ Z via the maps L ↦ π^{i}L and L ↦ π^{i}L^{∗} respectively. Thus, the result is true for all m ∈ Z if m is replaced by m′ ∈ [0, n] such that m ≡ ±m′ (mod 2n).

2. The formula for self-dual lattices can be proven in a much simpler way. Since r_{n+i} = −r_{i} for all i, a self-dual lattice Λ_{r} is fixed by N_{s} when r_{n} ≤ r_{n−1} ≤ ⋯ ≤ r_{1} ≤ −r_{1} ≤ ⋯ ≤ −r_{n} ≤ r_{n} + s. Setting d_{i} = −r_{i}, we have 0 = d_{0} ≤ d_{1} ≤ ⋯ ≤ d_{n} ≤ d_{n+1} = [s/2]. This is just an arrangement of [s/2] balls in n + 1 boxes (the

ith box contains d_{i} − d_{i−1} balls), and there are \binom{[s/2] + n}{n} such arrangements.

We now study the fixed symplectic lattice chains of type J for J a nonempty subset of [0, n]. Let J = {m_{1} < ⋯ < m_{l}}. It is clear from the definition that a lattice chain (L_{m}) ∈ X_{J}^{Sp} is determined by the L_{m_{i}}’s. Conversely, any sequence L_{m_{1}} ⊃ ⋯ ⊃ L_{m_{l}} with L_{m_{i}} ∈ ℒ_{m_{i}}^{Sp} extends uniquely to a chain in X_{J}^{Sp}. To see this, note that if L ⊂ L′, then L^{∗} ⊃ L′^{∗}. Thus, if L and L′ are both symplectic, the totally ordered set of lattices associated to L and L′ interlock to form a chain of lattices. This shows that (X_{J,N_{s}}^{Sp})^{k^{∗}} = {(Λ_{r^{m}}) | (r^{m_{1}}, …, r^{m_{l}}) ∈ Y_{Js}^{Sp}} where

Y_{Js}^{Sp} = {(r^{m_{1}}, …, r^{m_{l}}) | r^{m_{i}} ∈ R_{sm_{i}}^{Sp} and r^{m_{1}} ≤ ⋯ ≤ r^{m_{l}}}.

For r ∈ R_{sm_{1}}^{Sp}, let Y_{Js}^{Sp}(r) denote the chains in Y_{Js}^{Sp} with r^{m_{1}} = r.

These sets can be given a uniform combinatorial description in terms of a certain directed graph Δ. The vertex set of Δ is C_{s}^{n+1}, the distinct arrangements of s balls in n + 1 boxes. We index the boxes from 0 to n, and call the number of balls in the ith box p_{i}. Arrange the boxes in a line with the index increasing to the right. Viewing a box as two boundary walls, a vertex is just a string of s balls and n walls, both numbered starting from 1; the first and last walls are redundant because they do not separate adjacent boxes. Such a string also uniquely determines an arrangement of n walls in s + 1 cells, where a cell is determined by boundary balls. Let b_{j} be the number of walls in the jth cell for j ∈ [0, s], with the cell index also increasing to the right.

Two vertices v and v′ are connected by an edge if one configuration is obtained from the other by moving a ball between two adjacent boxes, i.e. by reversing the order of a wall and ball next to each other. Given a vertex v, define I_{v} ⊆ [1, n] recursively by prescribing that for i ∈ [0, n−1], the size of {i, i+1}∩I_{v} is congruent to p_{i} modulo 2. The recursion can begin because 0 ∉ I_{v}. If v and v′ are adjacent, then there exists i ∈ [0, n − 1] such that p_{j} and p′_{j} have the same parity if and only if j is neither i nor i + 1. Thus, I_{v} and I_{v′} differ only at the single element i + 1. We assign a direction to the edge from v to v′ by having it point towards the vertex with the larger associated set, namely, the set containing i + 1.

The sets G_{sI}^{Sp} are disjoint subsets of the vertex set and in fact define a partition of V (Δ).

Proposition 4.13. The vertex set V (Δ) = C_{s}^{n+1} is the disjoint union of the G_{sI}^{Sp}’s. Furthermore, \overrightarrow{vv′} is an edge if and only if there exists m ∈ [0, n − 1] such that ϕ^{−1}ψ^{−1}(v) ∈ R_{sm}^{Sp}, ϕ^{−1}ψ^{−1}(v′) ∈ R_{s,m+1}^{Sp}, and ϕ^{−1}ψ^{−1}(v) ≤ ϕ^{−1}ψ^{−1}(v′).

Proof. Since the G_{sI}^{Sp}’s are disjoint, in order to prove the first claim, it suffices to show that v ∈ G_{sI_{v}}^{Sp}. It is obvious from the construction of I_{v} that p_{i} has the parity required of an element of G_{sI_{v}}^{Sp} for i ∈ [0, n − 1]. It only remains to check that p_{n} is even if and only if n ∈ I_{v}. Since ∑_{i=0}^{n} p_{n} = s is odd, this is a consequence of the following lemma.

Lemma 4.14. Suppose ∑_{i=0}^{j} p_{i} is odd (resp. even). Then j ∉ I_{v} if and only if p_{j} is odd (resp. even).

Proof. We use induction on j. Since 0 ∉ I_{v}, the statement is vacuous for j = 0. Assume it is true for j − 1. Suppose ∑_{i=0}^{j} p_{i} is odd and p_{j} is odd (resp. even). Then ∑_{i=0}^{j−1} p_{i} is even (resp. odd). If p_{j−1} is even, then j − 1 ∉ I_{v} (resp. j − 1 ∈ I_{v}) by induction. Since p_{j−1} even also implies that j − 1 and j are simultaneously in or out of I_{v}, we have j ∉ I_{v} (resp. j ∈ I_{v}). If p_{j−1} is odd, then j − 1 ∈ I_{v} (resp. j − 1 ∉ I_{v}), again giving the desired result. The proof is similar for ∑_{i=0}^{j} p_{i} even.

Suppose that r ∈ R_{sm}^{Sp}, r′ ∈ R_{s,m+1}^{Sp}, and r ≤ r′. Then there exist subsets I and I′ of [1, n] such that r ∈ R_{sI}^{Sp}, r′ ∈ R_{sI′}^{Sp}, |I| = m, |I′| = m + 1, and I ⊂ I′. If i is the unique element of I′ \ I, then r′_{j} = r_{j} except at one of the two indices i and n + i where it is increased by 1. Let p = ψϕ(r) ∈ G_{sI}^{Sp} and p′ = ψϕ(r′) ∈ G_{sI′}^{Sp}. It is immediate that p′ is obtained from p by shifting a ball between the (i − 1)th and ith boxes. Since I ⊂ I′, \overrightarrow{pp′} is an edge. The argument is reversible, so the proposition is proved.

Set m_{0} = 0 and m_{l+1} = n. Let E_{Js}^{n} denote the set of sequences of vertices (v_{1}, …, v_{l}) such that v_{i} ∈ G_{sm_{i}}^{Sp} and there exists a (directed) path in Δ going through all of the v_{i}’s. In particular, if J = {m, m + 1, …, m′}, then E_{Js}^{n} is the set of paths from G_{sm}^{Sp} to G_{sm′}^{Sp}. For each v ∈ G_{sm_{i}}^{Sp}, we let E_{Js}^{n}(v) be the subset of E_{Js}^{n} consisting of sequences starting at v. Proposition 4.13 implies that there is a bijection Y_{Js}^{Sp} ˜→E_{Js}^{n} which restricts to bijections Y_{Js}^{Sp}(r) ˜→E_{Js}^{n}(ψϕ(r)). We thus get a concrete combinatorial interpretation of the Euler characteristics of the fixed point varieties with respect to N_{s} on each affine flag manifold:

Proposition 4.15. Suppose that char k ≠ 2 and (char′ k, n) = 1. Let s be an odd integer relatively prime to n and J a nonempty subset of [0, n]. Then χ(X_{J,N_{s}}^{Sp}) = |E_{Js}^{n}|.

We define a set

Z_{Js}^{n} = {(v, σ) | σ : [1, n] → [0, l], σ^{−1}(0) = I_{v}, and |σ^{−1}(i)| = j_{i} =^{def} m_{i+1} − m_{i}}.

Let θ : E_{Js}^{n} → Z_{Js}^{n} be the function (v_{1}, v_{2}, …, v_{l}) ↦ (v_{1}, σ) where σ^{−1}(0) = I_{v_{1}}, σ^{−1}(l) = I_{v_{l}}^{c}, and σ^{−1}(i) = I_{v_{i+1}} − I_{v_{i}} for i ∈ [1, l − 1]. (Note that σ^{−1}(0) and σ^{−1}(l) can be empty.) Consequently,

|E_{Js}^{n}| = ∑_{(v,σ)∈Z_{Js}^{n}} θ^{−1}((v, σ)). (4-2)

This observation provides an algorithm for computing the Euler characteristics in terms of a game. If v is a vertex viewed as a string of balls and walls, then we obtain all v′ adjacent to v by replacing some substring |• by •| or vice versa, where • denotes a ball and | a wall. The edge \overrightarrow{vv′} is contained in Δ if and only if the wall involved in the switch is not in I_{v}. The elements of θ^{−1}((v, σ)) may be found by the following procedure. Find all possible configurations of balls and walls obtained from v by one move for each wall in σ^{−1}(1). For each such v_{2}, repeat the process with the walls in σ^{−1}(2). We continue until a ball has been moved over each wall in σ^{−1}([1, l − 1]), thereby getting all l-tuples of vertices (v, v_{2}, …, v_{l}) ∈ θ^{−1}((v, σ)).

In order to evaluate the sum (4-2), we will pass from Z_{Js}^{n} to a more familiar index set consisting of intersection matrices. As a first step, we define a map η : V (Δ) → C_{n}^{t+1} where t = [s/2]. Given a vertex v, let b ∈ C_{n}^{s+1} be the corresponding arrangement of n walls in s + 1 cells, and set η(v) = a with a_{k} = b_{2k} + b_{2k+1}. This function simply pools the walls in cells 2k and 2k + 1 into one larger cell. We let η_{m} denote the restriction of η to G_{sm}^{Sp}. We also set C_{m}^{t+1}(a) = {(c_{0}, …, c_{t}) | c_{k} ∈ [0, a_{k}] and ∑_{k=0}^{t} c_{k} = m}.

Proposition 4.16. There is a bijection between η_{m}^{−1}(a) and C_{m}^{t+1}(a); in particular, η_{0} is a bijection. Moreover, identifying G_{s0}^{Sp} and C_{n}^{t+1} via η_{0}, η(v) is the unique element of G_{s0}^{Sp} from which v can be obtained using only moves of the form |• ⇝ •|.

Proof. Suppose that p ∈ G_{s0}^{Sp}, so that each box contains an even number of balls except for the last one. The ith wall has p_{0} + ⋯ + p_{i−1} balls to its left, and since this is an even number, it is in a cell with an even index. Thus, b_{j} = 0 for j odd. In this case, a_{k} = b_{2k} and η_{0} is injective because b is uniquely determined by p. Surjectivity follows since |G_{s0}^{Sp}| = |C_{n}^{t+1}| by proposition 4.11.

Consider a game starting at p with only moves of the form |• ⇝ •| allowed. The only balls which can take part in these switches are the leftmost balls in boxes 1, …, n. To see this, note that walls can only move to the right. Consequently, the balls in box 0 play no role. Also, if a ball becomes the first ball in its box during the course of the game without starting out in this position, then the wall left of it must already have moved. Since each wall can only move once, the ball is not involved in the game. It follows that no even-numbered balls can move. Thus, b_{2k} + b_{2k+1}, the number of walls between balls 2k and 2(k + 1), never changes, i.e. η is constant on the set of configurations obtained as outcomes of this game. This proves the uniqueness part of the final statement.

We will show existence of such a path by induction on m. This is trivial for m = 0. Suppose m ≥ 1 and the claim is true for m − 1. Take v = p ∈ G_{sm}^{Sp}, and let i be the smallest integer such that p_{i} is odd. Equivalently, i + 1 is the smallest element of Î_{v}; in fact, i + 1 ∈ I_{v} because m ≠ 0. Since p_{i} ≠ 0, the (i + 1)st wall is immediately preceded by a ball. We obtain a vertex v′ adjacent to v by moving this wall to the left, and \overrightarrow{v′v} is an edge since i + 1 ∈ I_{v}. This implies that, v′ ∈ G_{s,m−1}^{Sp}, so there exists a path to v′ of the correct form by inductive hypothesis. Concatenating \overrightarrow{v′v} with this path proves the result.

Finally, given c ∈ C_{m}^{t+1}(a), we obtain a vertex τ(c) in G_{sm}^{Sp} by moving ball 2k + 1 past the last c_{k} walls in the 2kth cell; such a ball exists because the last cell has odd index. These moves are of the specified type, so η(τ(c)) = a. The map C_{nm}^{t+1}(a) →^{τ} η_{m}^{−1}(a) is surjective by the previous paragraph. It is injective because the (2k + 1)st cell of τ(c) contains c_{k} walls.

Let m_{J} be the sequence (0 = m_{0}, m_{1}, …, m_{l}, m_{l+1} = n). We define a map ξ from Z_{Js}^{n} to the set of (m_{J}, t + 1)-intersection matrices Q_{m_{J} ,t+1}. (Both row and column indexing start at 0.) Let a = η(v), and let A_{k} be the set of walls in the 2kth cell of η_{0}^{−1}(a). Set ξ(v, σ) = (|σ^{−1}(i) ∩ A_{k}|) and ζ = ξ ∘ θ. We obtain

|E_{Js}^{n}| = ∑_{Q∈Q_{m_{J} ,t+1}} |ζ^{−1}(Q)|. (4-3)

Fix an intersection matrix Q. Notice that the only balls which play a role in moves involving the walls in A_{k} are the boundaries of the 2kth cell. Since a ball bounds only one even-numbered cell, this shows that the manipulations of the game take place independently within each A_{k}. Consequently, it will suffice to understand the game for the special cases of a string of walls with a boundary ball on both sides or only on the right. These correspond to the game restricted to the 2kth cell for k > 0 and k = 0 respectively.

Assume the cell contains d walls and suppose z_{0}, …, z_{l} are nonnegative integers such that ∑_{i=1}^{l} z_{i} = d. Consider a game with z_{i} moves at the ith step for i ∈ [0, l − 1], where only moves of the form |• ⇝ •| are allowed at the 0th step. Call the number of outcomes of the game α_{z}^{d} if the cell is only bounded on the right and β_{z}^{d} if it is bounded on both sides.

Lemma 4.17.

α_{z}^{d} = 1 and β_{z}^{d} = ∏_{i=1}^{l−1}(z_{i} + 1).

Proof. Suppose there is only a boundary ball on the right. Then there is only one outcome of the game; indeed, we have already seen in the proof of proposition 4.16 that each move consists of the ball shifting one box to the left. Now suppose the cell has a ball on each side. The 0th move is uniquely determined, and setting z′_{0} = 0 and z′_{i} = z_{i} for i ∈ [1, l], we have β_{z}^{d} = β_{z′}^{d−z_{0}}. We may thus assume that z_{0} = 0. The left ball can only move to the right and vice versa because otherwise a ball would jump over the same wall twice. Hence, for each c ∈ [0, d], there are c + 1 possible configurations after c moves corresponding to c′ ∈ [0, c] jumps of the left ball and c − c′ jumps of the right ball. The resulting strings have the form of the following diagram:

\overbrace{|| ⋯ ||}^{c′} • \overbrace{|| ⋯ ||}^{d−c} • \overbrace{|| ⋯ ||}^{c−c′} .

The outer walls have all been used, so the c moves reduce the game to the game of the same form with d − c walls. The lemma follows from these observations by induction.

Corollary 4.18.

|ζ^{−1}(Q)| = ∏_{i=1}^{l−1} ∏_{k=1}^{t}(q_{ik} + 1)

Proof. Let q^{k} be the kth column of Q. Then |ζ^{−1}(Q)| = α_{q^{0}}^{a_{0}} ∏_{k=1}^{t} β_{q^{k}}^{a_{k}}, and lemma 4.17 completes the proof.

Let γ_{d}^{t} = ∑_{y∈C_{d}^{t}} ∏_{k=1}^{t}(y_{k} + 1). We need the following lemma:

Lemma 4.19. For any d ≥ 0 and t > 0,

γ_{d}^{t} = \binom{2t + d − 1}{d}.

Proof. We use induction on t. The claim is true for t = 1, since γ_{d}^{1} = d + 1 = \binom{d+1}{d}. Now take t > 1 and assume that the result holds for t − 1. Then

γ_{d}^{t} = ∑_{y∈C_{d}^{t}} ∏_{k=1}^{t}(y_{k} + 1) = ∑_{y_{d}=0}^{d}(y_{d} + 1) ∑_{x∈C_{d−y_{d}}^{t−1}} ∏_{k=1}^{t−1}(x_{k} + 1)

= ∑_{y_{d}=0}^{d}(y_{d} + 1)\binom{2(t − 1) + d − y_{d} − 1}{d − y_{d}}

= ∑_{y_{d}=0}^{d} \binom{y_{d} + 1}{y_{d}}\binom{2(t − 1) + d − y_{d} − 1}{d − y_{d}}

= \binom{2t + d − 1}{d},

where the last step is a version of the Vandermonde convolution formula [R, p. 10].

Corollary 4.20. For any d ≥ 0 and t > 0,

∑_{z∈C_{d}^{t+1}} ∏_{k=1}^{t}(z_{k} + 1) = \binom{2t + d}{d}.

Proof. We have

∑_{z∈C_{d}^{t+1}} ∏_{k=1}^{t}(z_{k} + 1) = ∑_{z_{d}=0}^{d} γ_{d−z_{d}}^{t} = ∑_{z_{d}=0}^{d} \binom{2t + d − z_{d} − 1}{d − z_{d}} = \binom{2t + d}{d};

the last equality is a standard binomial identity [R, p. 7].

We can now calculate |E_{Js}^{n}|.

Proposition 4.21.

|E_{Js}^{n}| = \binom{t + j_{0}}{t}\binom{t + j_{l}}{t} ∏_{i=1}^{l−1} \binom{s + j_{i} − 1}{j_{i}}.

Proof. Let q_{i} be the ith row of Q. Recall from lemma 3.9 that the map Q ↦ (q_{0}, …, q_{l}) is a bijection of Q_{m_{J} ,s} onto ∏_{i=0}^{l} C_{j_{i}}^{t+1}. From (4-3) and corollaries 4.18 and 4.20, we have

|E_{Js}^{n}| = ∑_{Q∈Q_{m_{J} ,t+1}} |ζ^{−1}(Q)| = ∑_{\begin{matrix}(q_{0},...,q_{l})∈ \\ ∏_{i=0}^{l} C_{j_{i}}^{t+1} \end{matrix}} ∏_{i=1}^{l−1} ∏_{k=1}^{t}(q_{ik} + 1)

= |C_{j_{0}}^{t+1}||C_{j_{l}}^{t+1}| ∏_{i=1}^{l−1} ∑_{q_{i}∈C_{j_{i}}^{t+1}} ∏_{k=1}^{t}(q_{ik} + 1) = \binom{t + j_{0}}{t}\binom{t + j_{l}}{t} ∏_{i=1}^{l−1} γ_{j_{i}}^{t+1}

= \binom{t + j_{0}}{t}\binom{t + j_{l}}{t} ∏_{i=1}^{l−1} \binom{2t + j_{i}}{j_{i}} = \binom{t + j_{0}}{t}\binom{t + j_{l}}{t} ∏_{i=1}^{l−1} \binom{s + j_{i} − 1}{j_{i}}.

Combining this result with proposition 4.15 gives the following theorem:

Theorem 4.22. Suppose that char k ≠ 2 and (char′ k, n) = 1. Let s be an odd integer relatively prime to n with t = [s/2] and J = {m_{1} < ⋯ < m_{l}} a nonempty subset of [0, n]. Let j_{i} = m_{i+1} − m_{i} where m_{0} = 0 and m_{l+1} = n. Then

χ(X_{J,N_{s}}^{Sp}) = \binom{t + j_{0}}{t}\binom{t + j_{l}}{t} ∏_{i=1}^{l−1} \binom{s + j_{i} − 1}{j_{i}}.

In particular, χ(ℬ̂_{N_{s}}) = s^{n} and χ(ℒ_{m,N_{s}}^{Sp}) = \binom{t + j_{0}}{t}\binom{t + j_{l}}{t}.

Proof. It only remains to check the final statement. The symplectic lattices of type m correspond to J = {m}, so l = 1, j_{0} = m, and j_{1} = n − m. Substituting, we obtain another proof of corollary 4.12. The full affine flag manifold is isomorphic to X_{[0,n]}^{Sp}. In this case, l = n + 1, j_{0} = j_{n+1} = 0, and j_{i} = 1 for i ∈ [1, n], and the result follows.

Remark. As in the case of the root system A_{n}, the Euler characteristic is just the number of simplices of the type corresponding to the parahoric class within the subset of the positive Weyl chamber given by λ_{0} ≤ s, where λ_{0} is the affine root.

Corollary 4.23. Under the hypotheses of the previous theorem,

χ(X_{J,N_{s}}^{Sp}) ≤ χ(X_{J∪(2n−J)\{2n},N_{s}}),

with equality for n = 1 or s = 1 and strict inequality otherwise.

Proof. The inequality follows immediately from the fact that (X_{J,N_{s}}^{Sp})^{k^{∗}} is a subset of the finite set (X_{J∪(2n−J)\{2n},N_{s}})^{k^{∗}}. These sets are equal when n = 1 because Sp_{2}(k) = SL_{2}(k); thus the formulas for the Euler characteristics of the affine flag varieties of Sp_{2}(F) given in theorem 4.22 agree with those given in theorem 3.13. Also, if s = 1, then the Euler characteristics are 1 for both the symplectic and special linear groups.

Now suppose that n and s are both greater than 1. We have

χ(X_{J∪(2n−J)\\{2n\},N_{s}}) = s^{−1}\binom{2t + 2j_{0}}{2t}\binom{2t + 2j_{l}}{2t} ∏_{i=1}^{l−1} \binom{s + j_{i} − 1}{j_{i}}^{2}.

Note that \binom{2t+2q}{2t} = \binom{t+q}{t} for q = 0 and

\binom{2t + 2q}{2t} = \binom{t + q}{t} ∏_{p=1}^{q} \frac{2t + 2p − 1}{2p − 1} ≥ s\binom{t + q}{t}

for q ≥ 1. Since t ≥ 1, the inequality is strict when q ≥ 2. Furthermore, \binom{s+q−1}{q}^{2} ≥ s\binom{s+q−1}{q} for q ≥ 1, with strict equality for q ≥ 2. This proves the claim if j_{i} ≥ 2

for any i ∈ [0, l]. Otherwise, ∑_{i=0}^{l} j_{i} = n ≥ 2 implies that there exist two indices i_{1} and i_{2} such that j_{i_{1}} = j_{i_{2}} = 1. The result follows because s^{2} > s.

We can also obtain a combinatorial description of the Euler characteristics of the fixed point varieties in the classical symplectic flag manifolds for certain nilpotent conjugacy classes of 𝔰𝔭_{2n}(k). The nilpotent conjugacy classes of 𝔰𝔭_{2n}(k) are characterized by symplectic partitions of 2n, i.e. partitions whose odd parts come in pairs [Sp]. These partitions are the dimensions of the Jordan blocks of the nilpotent element of 𝔰𝔭_{2n}(k). Let 𝒞 be a nilpotent conjugacy class corresponding to a partition with at most one part appearing with odd multiplicity. Such a class is determined by n_{0} ∈ [0, n] and a partition n_{1}, …, n_{l} of n − n_{0}. Let M_{0} and M_{i} be matrices with a single Jordan block in 𝔰𝔭_{2n_{0}}(k) and 𝔤𝔩_{n_{i}}(k) respectively. Recall that 𝔤𝔩_{n}(k) injects into 𝔰𝔭_{2n}(k) via M ↦ diag(M, −M^{t}). Then the image of (M_{0}, M_{1}, …, M_{l}) under the injections 𝔰𝔭_{2n_{0}}(k) × ∏_{i=1}^{t} 𝔤𝔩_{n_{i}}(k) ↪ ∏_{i=0}^{l} 𝔰𝔭_{2n_{i}}(k) ↪ 𝔰𝔭_{2n}(k) is an element in this class; it has two Jordan blocks of size n_{i} and one of size 2n_{0}.

Theorem 4.24. Assume that char k ≠ 2 and (n, char′ k) = 1. Let N ∈ 𝔰𝔭_{2n}(k) be an element of a nilpotent conjugacy class with at most one Jordan block size having odd multiplicity, and let 2n = 2n_{0} + 2 ∑_{i=1}^{l} n_{i} be the corresponding symplectic partition of 2n. Let s = 2t + 1 be any odd integer greater than 2l and relatively prime to n. Define a ∈ C_{n}^{t+1} by a_{k} = n_{k} for k ∈ [0, l] and a_{k} = 0 for k > l. Then for each nonempty J ⊆ [1, n],

χ(ℱ_{J,N}^{Sp}) = |E_{J∪{0},s}^{n}(η^{−1}(a))| = ∑_{Q∈Q_{m_{J} ,t+1}(a)} ∏_{i=1}^{l−1} ∏_{k=1}^{t}(q_{ik} + 1).

In particular,

χ(ℬ_{N}^{Sp}) = \frac{n! 2^{n_{1}+···+n_{l}}}{n_{0}! ⋯ n_{l}!} .

Proof. Let r ∈ R_{s0}^{Sp} correspond to a, so that setting q = ϕ(r) and p = ψ(q), we have η_{0}(p) = a. We show that the nilpotent map N̄_{s} induced on W = Λ_{r}/πΛ_{r} has Jordan block structure given by n_{0}, …, n_{l}. It is clear from the definition that η_{0}^{−1}(a) is the string of s + 1 balls and n walls with n_{k} walls in the 2kth cell for k ∈ [0, l] and the other cells empty. Thus, the nonempty boxes are precisely those numbered ∑_{i=0}^{d} n_{i} for d ∈ [0, l]. The nonzero q_{i}’s then have indices n and ± ∑_{i=0}^{d} n_{i} for d ∈ [0, l − 1]. As shown in the proof of theorem 3.11, a Jordan block of size d corresponds to j ∈ [−(n − 1), n] such that q_{j} = q_{j+d} = 0 and q_{j+d′} ≥ 1 for 0 < d′ < d. (It is not necessary to understand the addition of the indices modulo 2n because q_{n} ≠ 0.) It is immediate that there is one block of size 2n_{0} and two of size n_{i} for each i ∈ [1, l] and that the block structure comes from an injection of the form 𝔰𝔭_{2n_{0}}(k) × ∏_{i=1}^{t} 𝔤𝔩_{n_{i}}(k) ↪ 𝔰𝔭_{2n}(k) as described above. The first statement follows, since

\begin{aligned}χ(ℱ_{J,N}^{Sp}) = |E_{J∪\{0\},s}^{n}(η^{−1}(a))| & = ∑_{Q∈Q_{m_{J} ,t+1}(a)} |ζ^{−1}(Q)| \\ & = ∑_{Q∈Q_{m_{J} ,t+1}(a)} ∏_{i=1}^{l−1} ∏_{k=1}^{t}(q_{ik} + 1). \end{aligned}

Now suppose J = [1, n]. A (m_{[0,n]}, s)-intersection matrix has a single 1 in each row, so there are \frac{n!}{a_{0}!···a_{t}!} elements Q ∈ Q_{m_{[0,n]},s}(a). Also, |ζ^{−1}(Q)| = 2^{a_{1}+···+a_{t}}, and

since this is constant on the fiber over a, χ(ℬ_{N}^{Sp}) = \frac{n! 2^{a_{1}+···+a_{t}}}{a_{0}! ⋯ a_{t}!} = \frac{n! 2^{n_{1}+···+n_{l}}}{n_{0}! ⋯ n_{l}!} .

Remarks. 1. Since \frac{n! 2^{a_{1}+...a_{t}}}{a_{0}! ⋯ a_{t}!} is the term corresponding to a in the multinomial

expansion of s^{n} = (1 + 2 + ⋯ + 2)^{n}, we obtain another proof of the formula for the Euler characteristic of the full affine flag manifold.

2. The Euler characteristic of each 𝒫_{N} can be obtained from the game with 2n+1 balls.

Corollary 4.25. Assume char k ≠ 2 and (n, char′ k) = 1. Let N ∈ 𝔰𝔭_{2n}(k) be an element of a nilpotent conjugacy class with at most one Jordan block size having odd multiplicity. Then for each nonempty J ⊆ [1, n], χ(ℱ_{J,N}^{Sp}) ≤ χ(ℱ_{J∪(2n−J)}).

Proof. The proof is similar to the first part of corollary 4.23.

Remark. The formulas given in theorems 4.24 and 3.11 coincide for dim V = 2.

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Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

   E-mail address: sage@math.utah.edu