Journal of Lie Theory Volume 6 (1996) 53–68 ◯^{C} 1996 Heldermann Verlag

The Weyl group as fixed point set of smooth involutions

Filippo De Mari

Communicated by G. Mauceri

Abstract. We show that the Weyl group W = M′/M of a noncompact semisimple Lie group is obtained by taking fixed point sets of smooth involutions in K/M. More precisely, one considers first the fixed point set X of the involutions defined on K/M by the elements of order 2 in exp i𝔞. The Weyl group is either X, or the fixed point set of the involutions defined on X by special elements of order 4 in exp i𝔞.

1. Introduction.

The primary motivation for studying the problem at hand comes from the observation ([2]) that if the Weyl group W can be obtained from K_{0}/M_{0} by successively taking fixed point sets of smooth involutions preserving Hessenberg manifolds, then one can also reverse Morse inequalities for real Hessenberg manifolds using Floyd’s theorem [4] (see Section 1 for notations, and the second remark at the end of Section 2 for the Hessenberg–preserving property). The result proved in this paper, however, is of some independent interest, and we think that it might be useful in a variety of contexts. If one removes the smoothness assumption, it is easy to obtain W as the fixed point set of a single discontinuous involution: just view K_{0}/M_{0} as the adjoint K_{0}–orbit of a suitable regular element in 𝔞_{0} and “flip” across 𝔞_{0}. In order to explain our ideas, we now briefly discuss an example. Let the group M_{0} = {diag (ε_{1}, ε_{2}, ε_{3}) | ε_{j} = ±1, ∏ ε_{j} = 1} act by conjugation on K_{0} = SO(3, ℝ). Clearly, each m ∈ M_{0} induces a smooth involution on the flag manifold K_{0}/M_{0}. The points in K_{0}/M_{0} simultaneously fixed by all three non– trivial involutions are in correspondence to those k ∈ K_{0} for which the following property holds: for all m ∈ M_{0} there exists m′ ∈ M_{0} such that mkm = km′. It is immediate to see that such a k normalizes – by conjugation – the set 𝔞_{0} of trace zero diagonal matrices. Indeed, if H ∈ 𝔞_{0} and Y = kHk^{−1}, then Y is left fixed by all m ∈ M_{0}. On the other hand, this last condition is expressed by the equalities Y_{ij} = ε_{i}ε_{j}Y_{ij}, so that Y is itself diagonal. Thus, modulo M_{0}, k is a permutation, i.e. an element of the Weyl group of SL(3, ℝ).

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag If we attempt to obtain the same result for SL(2, ℝ) using as set of involutions the centralizer of 𝔞_{0} in K_{0} = SO(2, ℝ), that is M_{0} = {± id}, we get as fixed point set in the projective space K_{0}/M_{0} the identity coset alone. But if we use the involution defined by diag (i, −i), which still centralizes 𝔞_{0}, we achieve the target. These considerations suggest on the one hand that the right involutions should centralize 𝔞_{0}, and on the other hand that the problem should be analyzed inside some “complexification” of the semisimple Lie group G_{0}, that is, inside the adjoint group G of its complexified Lie algebra. The appropriate set of involutions turns out to be F_{2} = {f ∈ exp i𝔞_{0} | f^{2} = e}, where 𝔞_{0} denotes as usual a maximal abelian subspace of the symmetric part of the Lie algebra 𝔤_{0} of G_{0}. When acting as group of smooth maps on X_{0} := K_{0}/M_{0}, however, F_{2} singles out the Weyl group W as fixed point set only if the (restricted) root system associated with (𝔤_{0}, 𝔞_{0}) is reduced. Otherwise X_{1} := Fix(F_{2}, X_{0}) contains properly W and one has to consider a special set F_{4} of elements of order 4 in exp i𝔞_{0}. At this stage one gets equality, namely Fix(F_{4}, X_{1}) = W. The elements of F_{4} take into precise account the non–reduced roots, in a sense that will be made clear in Section 4. The nature of Fix(F_{4}, X_{1}), in particular the fact that the action of F_{4} on X_{1} is well–defined, is a slightly delicate matter, and is best understood via the Bruhat decomposition. The key step (Theorem 10) is proved by using basic properties of the Bruhat decomposition (Crollary 5.3) and SU(2, 1)–reduction (Lemma 5.6).

2. Preliminaries and notation.

Let G_{0} be a semisimple, connected, non–compact Lie group with finite center, 𝔤_{0} its Lie algebra, and 𝔤 = 𝔤_{0}^{c} its complexification viewed as real Lie algebra. Thus 𝔤 is semisimple. Denote by σ the automorphism of 𝔤 corresponding to conjugation with respect to 𝔤_{0}, i.e. σ : X + iY ↦ X − iY for X, Y ∈ 𝔤_{0}. Let ad denote the adjoint representation of 𝔤. We then have Lie algebra inclusions ad 𝔤_{0} ⊂ ad 𝔤 ⊂ 𝔤𝔩(𝔤) = End(𝔤). Let G = Int(𝔤) be the adjoint group of 𝔤, i.e. the connected Lie subgroup of GL(𝔤) = Aut(𝔤) correspoding to ad 𝔤. If G_{∗} denotes the connected Lie subgroup of Int(𝔤) corresponding to ad 𝔤_{0}, then G_{∗} is a closed Lie subgroup of G diffeomorphic to Int(𝔤_{0}), the adjoint group of 𝔤_{0} ([6], Lemma 6.2, Ch. III, p.181). The adjoint representation of G_{0} maps G_{0} onto G_{∗} with kernel Z_{0}, the center of G_{0}. Thus G_{∗} ≃ G_{0}/Z_{0}. Let now θ be a Cartan involution of 𝔤_{0} and 𝔤_{0} = 𝔨_{0} + 𝔭_{0} the resulting Cartan decomposition. Let 𝔞_{0} be a maximal abelian subspace of 𝔭_{0} of dimension say l. Call a linear functional α ∈ 𝔞_{0}^{∗} a restricted root if

𝔤_{0α} = {X ∈ 𝔤_{0} | [H, X] = α(H)X ∀ H ∈ 𝔞_{0}} ≠ 0.

The set of non–zero restricted roots (resp. positive, simple) will be denoted by Σ (resp. Σ^{+}, Δ). The simultaneous diagonalization of all the X ↦ [H, X] ∈ End(𝔤_{0}) with H ∈ 𝔞_{0} leads to the root–space decomposition of 𝔤_{0}

𝔤_{0} = 𝔤_{00} + ∑_{α∈Σ} 𝔤_{0α}.

In turn,

𝔤_{00} = 𝔞_{0} + 𝔪_{0},

where 𝔪_{0} = {X ∈ 𝔨_{0} | [X, H] = 0 ∀H ∈ 𝔞_{0}} is the centralizer of 𝔞_{0} in 𝔨. Put

𝔫_{0} = ∑_{α∈Σ^{+}} 𝔤_{0α},

so that the Iwasawa decomposition of 𝔤_{0} reads:

𝔤_{0} = 𝔫_{0} + 𝔞_{0} + 𝔨_{0}.

Let now N_{0}, A_{0} and K_{0} denote the Lie subgroup of G_{0} corresponding to 𝔨_{0}, 𝔞_{0} and 𝔫_{0}. Thus G_{0} = N_{0}A_{0}K_{0} is the Iwasawa decomposition of G_{0}. Here K_{0} is a maximal compact subgroup of G_{0}, A_{0} is abelian and N_{0} is nilpotent. Moreover, there exists an involutive automorphism Θ of G_{0} with dΘ = θ, such that K_{0} is the set of points fixed by Θ. Let M_{0} and M′_{0} denote respectively the centralizer and normalizer of 𝔞_{0} in K_{0}, i.e. M_{0} = {m ∈ K_{0} | Ad m(H) = H, ∀H ∈ 𝔞_{0}} and M′_{0} = {m ∈ K_{0} | Ad m(H) ∈ 𝔞_{0}, ∀H ∈ 𝔞_{0}}. Here Ad, stands for the adjoint representation of G_{0}. The Lie algebras of M_{0} and M′_{0} coincide and are equal to 𝔪_{0}. The finite group W = M′_{0}/M_{0} is the Weyl group of G_{0} associated to the previous data. Clearly, W sits inside the boundary K_{0}/M_{0}.

3. The case of reduced root systems.

The Cartan involution θ extends in a unique fashion to an involution of 𝔤, also denoted by θ ([7], Ch. III, p. 368), and there exists an involutive automorphism Θ of G such that dΘ = θ. Let K_{Θ} denote the Lie subgroup of G of fixed points of Θ and by K its identity component. Put

F_{2} = {f ∈ exp i𝔞_{0} | f^{2} = e} ⊂ G.

It is easy to give an explicit description of F_{2} ([7], ex. 7 p. 384). Indeed, if {H_{1}, …, H_{l}} is the basis of 𝔞_{0} dual to Δ, then:

F_{2} = {exp iπ ∑_{j=1}^{l} ν_{j}H_{j} | ν_{j} = 0, 1} .

Thus card F_{2} = 2^{l}. Observe also that if f = exp iA ∈ F_{2}, then

Θf = Θ exp iA = exp(θiA) = exp(−iA) = f^{−1} = f,

so that F_{2} ⊂ K_{Θ}. More precisely, F_{2} is the group of components of K_{Θ}, i.e. ([8])

K_{Θ} = K · F_{2}.

The first step of our construction consists in showing that F_{2} acts on X_{0} = K_{0}/M_{0} by smooth involutions. The fixed point set X_{1} = Fix(F_{2}, X_{0}) is in many cases W, but not always. If the root system Σ is not reduced, then X_{1} is an “intermediate” manifold between K_{0}/M_{0} and W. ¿From now until the end of this section, we will use the subscript ∗ to indicate images under the adjoint representation Ad of G_{0}.

Proposition 3.1. If f ∈ F_{2}, , then fK_{∗}f = K_{∗}.

Proof. Let fK_{∗}f = K_{f}, and let 𝔨_{f} be its Lie algebra in 𝔤. It is clear that 𝔨_{f} = Ad_{G} f𝔨_{0}. Therefore, 𝔨_{f} is θ–invariant. If f = exp iA, and T_{0} ∈ 𝔨_{0}, then:

σ(Ad_{G}fT_{0}) = σ(Ad_{G}(exp iA)T_{0})
= Ad_{G}(exp σ(iA))(σT_{0})
= Ad_{G}(exp −iA)T_{0}
= Ad_{G}fT_{0}.

Thus Ad_{G} fT_{0} ∈ 𝔤_{0} ∩𝔨 = 𝔨_{0}. This shows 𝔨_{f} = 𝔨_{0}, thereby proving the Proposition, since K_{∗} is the connected subgroup of G corresponding to the Lie subalgebra 𝔨_{0} of 𝔤.

Proposition 3.2. F_{2} acts on K_{0}/M_{0} by:

f · ⟨k⟩ = ⟨Ad^{−1}(fk_{∗}f)⟩,  (1)

⟨·⟩ denoting the class mod M_{0} and Ad the adjoint representation of G_{0}. Moreover, as maps, all the f ’s commute with each other and f^{2} = id.

Proof. First of all notice that Ad^{−1} e = Z_{0}, the center of G_{0}. But Z_{0} is contained in M_{0}, so that all the elements in Ad^{−1} x belong to the same M_{0} coset. Since F_{2} centralizes M_{∗}, for k ∈ K_{0} and m ∈ M_{0}, we have:

f(km)_{∗}f = fk_{∗}m_{∗}f
= (fk_{∗}f)(fm_{∗}f)
= (k′)_{∗}m_{∗}
= (k′m)_{∗},

where k′_{∗} = fk_{∗}f ∈ K_{∗} because of the previous Proposition. Thus:

Ad^{−1}(f(km)_{∗}f) = k′mZ_{0}

and ⟨k′mZ_{0}⟩ = ⟨k′⟩ depends only on ⟨k⟩. The remaining assertions are clear, since F_{2} is abelian in G.

Denote by Fix(F_{2}, X_{0}) the set of points in X_{0} = K_{0}/M_{0} which are simultaneously fixed by all f ∈ F_{2}. It is clear that Fix(F_{2}, X_{0}) is a smooth manifold. Indeed, if F_{2} = {f_{1}, …, f_{N}}, N = 2^{l}, let X_{0}^{1} = Fix(f_{1}, X_{0}) and for 1 ≤ j ≤ N put X_{0}^{j} = Fix(f_{j}, X_{0}^{j−1}), (here X_{0}^{0} = X_{0}). Then X_{0}^{1} is a smooth manifold because it is the image under π : K_{0} → K_{0}/M_{0} of K_{0}^{1} = {k ∈ K_{0} | k^{−1}f_{1}kf_{1} ∈ M_{0}}. Similarly, X_{0}^{j} is a smooth manifold because it is the image under π : K_{0} → K_{0}/M_{0} of K_{0}^{j} = {k ∈ K_{0}^{j−1} | k^{−1}f_{j}kf_{j} ∈ M_{0}}. But X_{0}^{N} = Fix(F_{2}, X_{0}) because all the f ’s commute as maps.

Theorem 3.3. If Σ is reduced, then Fix(F_{2}, X_{0}) = W.

Proof. Let \overline{H} ∈ 𝔞_{0} be a regular element, and let ⟨k⟩ ∈ Fix(F_{2}, X_{0}). This means that given f ∈ F_{2} there exists m ∈ M_{0} such that:

fk_{∗}f = k_{∗}m_{∗}.

Then Y := Ad k_{∗}\overline{H} = Ad k\overline{H} ∈ 𝔭_{0} depends only on ⟨k⟩ and we may write:

Y = Y_{0} + ∑_{α∈Σ} Y_{α},

where Y_{0} ∈ 𝔪_{0} and Y_{α} ∈ 𝔤_{α}. Since f centralizes 𝔞_{0},

Ad fY = Ad fk_{∗}f\overline{H} = Ad k_{∗}m_{∗}\overline{H} = Ad k_{∗}\overline{H} = Y,

so that Y is fixed by all f ∈ F_{2}. Therefore, if f = exp iA_{0},

Y_{α} = (Ad exp iA_{0}Y)_{α} = (e^{ad iA_{0}}Y)_{α} = e^{iα(A_{0})}Y_{α}.

Fix now α and write

α = ∑_{δ∈Δ} ν_{δ}(α)δ.

If Σ is reduced, there are no roots α for which ν_{δ}(α) is even for all δ ([1]). Thus ν\overline{δ}(α) is odd for at least one simple restricted root \overline{δ}. Select A_{0} = πH\overline{δ}, so that α(A_{0}) is an odd multiple of π. It follows that Y_{α} = −Y_{α} = 0. This shows that Y = Y_{0} ∈ 𝔞_{0}, namely that k ∈ M′_{0}. Thus ⟨k⟩ ∈ W and Fix(F_{2}, X_{0}) ⊂ W. The reverse inclusion is obvious.

Remarks i) It is clear from the proof of 3.3 that if the root system Σ is not reduced, ⟨k⟩ ∈ Fix(F_{2}, X_{0}) and \overline{H} ∈ 𝔞_{0} is a regular element, then Y = Ad k\overline{H} = Y_{0}+∑_{α∈E} Y_{α}, where E is the set of even roots, namely those roots α = ∑_{δ∈Δ} ν_{δ}(α)δ for which ν_{δ}(α) ∈ 2ℤ for all δ ∈ Δ.

ii) The involutions defined by F_{2} have the additional property of preserving Hessenberg manifolds. We recall ([2], [3]) that for a fixed regular element \overline{H} ∈ 𝔞_{0}, a (real) Hessenberg manifold Hess_{R}(\overline{H}) is defined for any subset R of the set Σ^{−} of negative roots having the following Hessenberg property:

α ∈ R, β ∈ Σ^{+}, α + β ∈ Σ^{−} ⟹ α + β ∈ R.

One then defines the Hessenberg subspaces 𝔭_{0}(R) of 𝔭_{0} by summing vectors of the form X_{α} − θX_{α} with α ∈ R. More precisely, set

𝔭_{0}(R) = 𝔞_{0} + ∑_{α∈R}((𝔤_{0,α} + 𝔤_{0,−α}) ∩ 𝔭_{0}).

Finally,

Hess_{R}(\overline{H}) = \{⟨k⟩ ∈ K_{0}/M_{0} | Ad k^{−1}\overline{H} ∈ 𝔭_{0}(R)\} .

It is clear that since F_{2} centralizes 𝔞_{0}, it maps Hess_{R}(\overline{H}) into itself. A similar situation occurs for the action of F_{4} which will be presented in Section 5.

We will simplify the notation thinking of all the subgroups of G_{0} as subgroups of G under the adjoint representation. In this process, the center becomes trivial. Thus we suppress the subscripts ∗ and write, for example, K_{0} in place of K_{∗} whenever no confusion arises.

4. The Weyl group of SU(2, 1).

As we will see in the next section, the general case is handled by reducing the problem to an SU(2, 1) computation, via the “Bruhat decomposition” of K_{0}. We therefore analyze this group in full detail. Recall that G_{0} = SU(2, 1) consists of those elements in GL(3, ℂ) having determinant equal to one and satisfying g^{∗}I_{2,1}g = I_{2,1}, where

I_{2,1} = [ \begin{matrix}−I_{2} & 0 \\ 0 & 1 \end{matrix} ] .

The Lie algebra 𝔤_{0} of G_{0} is therefore

𝔰𝔲(2, 1) = {X ∈ 𝔤𝔩(3, ℂ) | X^{∗}I_{2,1} + I_{2,1}X = 0, tr X = 0}
= \{[ \begin{matrix}A & B \\ B^{∗} & − tr A \end{matrix} ] | A ∈ 𝔲(2), B ∈ M_{2,1}(ℂ)\} ,

where evidently 𝔲(2) is the Lie algebra of 2 × 2 skew–hermitian matrices. Let θ and Θ denote the Cartan involutions on 𝔤_{0} and G_{0} respectively, so that θ = dΘ. Then:

θX = I_{2,1}XI_{2,1}, Θg = I_{2,1}gI_{2,1}.

Consequently,

𝔨_{0} = \{[ \begin{matrix}A & 0 \\ 0 & − tr A \end{matrix} ] | A ∈ 𝔲(2)\} ,

𝔭_{0} = \{[ \begin{matrix}0 & ^{t}ζ \\ \overline{ζ} & 0 \end{matrix} ] | ζ ∈ ℂ^{2}\} .

A maximal abelian suspace of 𝔭_{0} is:

𝔞_{0} = \{tH_{0} | H_{0} = [ \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix} ] , t ∈ ℝ\} .

Next, a maximal compact subgroup of G_{0} corresponding to 𝔨_{0} is:

K_{0} = S(U_{2} × U_{1}) = \{[ \begin{matrix}A & 0 \\ 0 & e^{it} \end{matrix} ] | A ∈ U(2), det A = e^{−it}, t ∈ ℝ\} .

The centralizer M_{0} of 𝔞_{0} in K_{0} is

M_{0} = \{[ \begin{matrix}e^{is} & 0 & 0 \\ 0 & e^{−2is} & 0 \\ 0 & 0 & e^{is} \end{matrix} ] | s ∈ ℝ\} ,

whereas the normalizer M′_{0} of 𝔞_{0} in K_{0} is

M′_{0} = \{[ \begin{matrix}εe^{is} & 0 & 0 \\ 0 & εe^{−2is} & 0 \\ 0 & 0 & e^{is} \end{matrix} ] | s ∈ ℝ, ε = ±1\} .

It follows that the Weyl group W = M′_{0}/M_{0} is (isomorphic to) the two–element group. The root–space structure of SU(2, 1) is easily written. Indeed, 𝔤_{00} = 𝔞_{0} +𝔪_{0} with

𝔪_{0} = \{tT_{0} | T_{0} = [ \begin{matrix}i & 0 & 0 \\ 0 & −2i & 0 \\ 0 & 0 & i \end{matrix} ] , t ∈ ℝ\} ,

and if α : 𝔞_{0} → ℝ is the linear functional defined by tH_{0} ↦ t, then:

𝔤_{0,α} = \{[ \begin{matrix}0 & z & 0 \\ −\overline{z} & 0 & \overline{z} \\ 0 & z & 0 \end{matrix} ] , z ∈ ℂ\} , 𝔤_{0,−α} = \{[ \begin{matrix}0 & z & 0 \\ −\overline{z} & 0 & −\overline{z} \\ 0 & −z & 0 \end{matrix} ] , z ∈ ℂ\} ,

𝔤_{0,2α} = \{[ \begin{matrix}it & 0 & −it \\ 0 & 0 & 0 \\ it & 0 & −it \end{matrix} ] , t ∈ ℝ\} , 𝔤_{0,−2α} = \{[ \begin{matrix}it & 0 & it \\ 0 & 0 & 0 \\ −it & 0 & −it \end{matrix} ] , t ∈ ℝ\} .

Taking exponentials in GL(3, ℂ), we have

exp i𝔞_{0} = \{[ \begin{matrix}cos t & 0 & i sin t \\ 0 & 1 & 0 \\ i sin t & 0 & cos t \end{matrix} ] | t ∈ ℝ\} ,

so that

F_{2} = \{f ∈ exp i𝔞_{0} | f^{2} = 1\} = \{[ \begin{matrix}ε & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ε \end{matrix} ] | ε = ±1\} .

Observe that:

exp iπH_{0} = [ \begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ] .

We will also need the following set of elements of order 4 in exp i𝔞_{0}: F_{4} = {f ∈ exp i𝔞_{0} | f^{2} = exp iπH_{0}}. Clearly:

F_{4} = \{[ \begin{matrix}0 & 0 & iε \\ 0 & 1 & 0 \\ iε & 0 & 0 \end{matrix} ] | ε = ±1\} .

Next we analyze X_{1} := Fix(F_{2}, K_{0}/M_{0}). To this end, let ⟨k⟩ ∈ X_{1}. If k =

[ \begin{matrix}A & 0 \\ 0 & e^{it} \end{matrix} ] with A = [ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ U(2), and f = f_{ε} ∈ F_{2}, then there exists

m = m_{s}(ε) ∈ M_{0} such that fkf = km. Since

fkf = [ \begin{matrix}a & εb & 0 \\ εc & d & 0 \\ 0 & 0 & e^{it} \end{matrix} ] , and km = [ \begin{matrix}ae^{is} & be^{−2is} & 0 \\ ce^{is} & de^{−2is} & 0 \\ 0 & 0 & e^{it}e^{is} \end{matrix} ] ,

we immediately obtain e^{is} = 1, i.e. m_{s}(ε) = id = e independently of ε. Choosing ε = −1, we see that A must be diagonal, i.e.,

k = [ \begin{matrix}e^{iu} & 0 & 0 \\ 0 & e^{iv} & 0 \\ 0 & 0 & e^{it} \end{matrix} ] u + v + t ∈ 2πℤ.

We stress that we have proved that if ⟨k⟩ ∈ X_{1}, then fkf = k for all f ∈ F_{2}, a very special situation which will be used in the proof of 5.6. Nonetheless, X_{1} ≠ W. In order to obtain W as the fixed point set of (smooth) involutions, we need to consider the action of F_{4} on Fix(F_{2}, K_{0}/M_{0}). This fact illustrates a general situation. Observe that if ⟨k⟩ ∈ X_{1} and f ∈ F_{4}, then:

fkf^{−1} = [ \begin{matrix}0 & 0 & iε \\ 0 & 1 & 0 \\ iε & 0 & 0 \end{matrix} ] [ \begin{matrix}e^{iu} & 0 & 0 \\ 0 & e^{iv} & 0 \\ 0 & 0 & e^{it} \end{matrix} ] [ \begin{matrix}0 & 0 & −iε \\ 0 & 1 & 0 \\ −iε & 0 & 0 \end{matrix} ] = [ \begin{matrix}e^{it} & 0 & 0 \\ 0 & e^{iv} & 0 \\ 0 & 0 & e^{iu} \end{matrix} ] ,

so that F_{4} sends fixed points into fixed points. Now, the requirement that ⟨k⟩ ∈ Fix(F_{4}, X_{1}) is equivalent to asking that, given ε, there exists m ∈ M_{0} such that fkf^{−1} = km. Since

km = [ \begin{matrix}e^{i(u+s)} & 0 & 0 \\ 0 & e^{i(v−2s)} & 0 \\ 0 & 0 & e^{i(t+s)} \end{matrix} ] ,

we obtain the system

\{ \begin{matrix}u + s = t + 2n_{1}π \\ v − 2s = v + 2n_{2}π \\ t + s = u + 2n_{3}π \end{matrix}

for some integers n_{1}, n_{2} and n_{3}, that is:

s = nπ, u = t + n′π,

for some integers n and n′. If n is odd, then m = [ \begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ], and fkf = km

implies e^{it} = −e^{iu}, whence v = (2n′′ + 1)π − 2u, and so:

k = [ \begin{matrix}−e^{it} & 0 & 0 \\ 0 & −e^{−2it} & 0 \\ 0 & 0 & e^{it} \end{matrix} ] ∈ M′_{0}.

Finally, if n is even, then m = id and fkf = km forces e^{it} = e^{iu}, whence e^{iv} = −e^{−2it}, i.e.

k = [ \begin{matrix}e^{it} & 0 & 0 \\ 0 & e^{−2it} & 0 \\ 0 & 0 & e^{it} \end{matrix} ] ∈ M_{0} ⊂ M′_{0}.

This shows that Fix(F_{4}, X_{1}) = W.

For the reader’s convenience, we conclude this section by briefly recalling the general structure of a root system of type (BC)_{l} because it will be used in Section 5. The root system associated to SU(2, 1) is of course of type (BC)_{1}. Let e_{j} denote the j^{th} standard basis element in ℝ^{l}. The rank l root system of type (BC)_{l} is (isomorphic to):

(BC)_{l} = {±e_{i} ± e_{j}, 1 ≤ i < j ≤ l, ±e_{j}, 1 ≤ j ≤ l, ±2e_{j}, 1 ≤ j ≤ l} .

Slightly modifying the standard notation ([6], Theorem 3.25, Ch. X, p. 475), a basis Δ = {δ_{0}, δ_{1}, … δ_{l−1}} of simple roots is given by:

δ_{0} = e_{l}, δ_{j} = e_{j} − e_{j+1}, 1 ≤ j ≤ l − 1.

Finally we list the expression of the positive roots in terms of the basis elements; here 1 ≤ i < j ≤ l, and if j = l the sum δ_{j} + … + δ_{l−1} is zero:

e_{i} − e_{j} = δ_{i} + … + δ_{j−1};
e_{j} = δ_{0} + (δ_{j} + … + δ_{l−1});
2e_{j} = 2δ_{0} + 2(δ_{j} + … + δ_{l−1});
e_{i} + e_{j} = (δ_{i} + … + δ_{j−1}) + 2δ_{0} + 2(δ_{j} + … + δ_{l−1}).

Using the terminology introduced in the first remark following Theorem 3, the only even roots in (BC)_{l} are 2e_{j}, for j = 1, …, l.

5. Bruhat decomposition and SU(2, 1) reduction. The general case.

We recall that if B_{0} = N_{0}A_{0}M_{0}, then the Bruhat decomposition of G_{0} is the disjoint union:

G_{0} = ∐_{w∈W} B_{0}wB_{0}.

A better parametrization of the double cosets B_{0}wB_{0} may be achieved as follows (see [5]). For w ∈ W, let Σ_{w}^{+} = {α ∈ Σ^{+} | −wα ∈ Σ^{+}}, and set

\overline{𝔫}_{w} = ∑_{α∈Σ_{w}^{+}} 𝔤_{wα}, N_{w}^{−} = exp \overline{𝔫}_{w}.

Then the Bruhat decomposition may be rewritten as

G_{0} = ∐_{w∈W}(N_{0}A_{0})N_{w}^{−}wM_{0},

giving rise to a “Bruhat decomposition” of K_{0}:

K_{0} = ∐_{w∈W} k(N_{w}^{−})wM_{0},

where k(·) refers to the K_{0}–coordinate function in the Iwasawa decomposition G_{0} = N_{0}A_{0}K_{0}. When we write k = k(\overline{n})wm, we think of w as a fixed representative in M′_{0}. We will write B(w) = B_{0}wB_{0} = (N_{0}A_{0})N_{w}^{−}wM_{0} and C(w) = k(N_{w}^{−})wM_{0}, and refer to B(w) and C(w) as the Bruhat cells of G_{0} and K_{0}, respectively. Observe that:

C = k(B).

Next, we recall ([1]) a few crucial properties enjoied by the cells B(w), and show that they hold for the cells C(w) as well. Let S be a set of generators of W, with e ∉ S. Let s ∈ S and w ∈ W, then

B(s)B(w) = \{ \begin{matrix}B(sw) & if B(w) ⊄ B(s)B(w) \\ B(w) ∪ B(sw) & if B(w) ⊂ B(s)B(w) \end{matrix}  (2)

Let w = s_{1} · . . . · s_{p} be a reduced expression in terms of generators, and let v ∈ W; then

B(s_{1} · . . . · s_{p})B(v) ⊂ ∐_{1≤i_{1}≤...≤i_{t}≤p} B(s_{i_{1}} · . . . · s_{i_{t}}v),  (3)

where (i_{1}, …, i_{t}) ranges over all – possibly empty – t-uples of increasing integers in the interval [1, p]. Finally, let l_{S}(·) denote the length function on W with respect to the set S of generators of W. Let w_{1}, …, w_{p} ∈ W and suppose that l_{S}(w) = l_{S}(w_{1}) + … + l_{S}(w_{p}), then B(w) = B(w_{1}) · . . . · B(w_{p}). In particular, if w = s_{1} · . . . · s_{p} is a reduced expression of w in terms of generators, then:

B(w) = B(s_{1}) · . . . · B(s_{p}).  (4)

Our first concern will be to show that 2, 3 and 4 hold for C(w) as well.

Lemma 5.1. Let u, w ∈ W. Then k(B(u)) · k(B(w)) = k(B(u) · B(w)).

Proof. Let x ∈ B(u) and y ∈ B(w). Write x = b_{1}ub_{2}, b_{i} ∈ B_{0}, and y = na · k(y). Then xy = b_{1}ub_{2}na · k(y) = (b_{1}ub′_{2}) · k(y). Since b_{1}ub′_{2} ∈ B(u), b_{1}ub′_{2} = n′a′k′ with k′ ∈ k(B(u)). Thus xy = n′a′k′ · k(y), so that k(xy) = k′ · k(y) ∈ k(B(u)) · k(B(w)), thereby showing k(B(u) · B(w)) ⊂ k(B(u)) · k(B(w)). On the other hand, k(B(u)) · k(B(w)) = k(B(u) · k(B(w))) ⊂ k(B(u) · B(w)), and the lemma is proved.

Lemma 5.2. k(B) ⊂ B and N_{0}A_{0} · k(B) = B.

Proof. As for the first statement, let nak ∈ B = B_{0}wB_{0} and put b = (na)^{−1} ∈ N_{0}A_{0}M_{0} = B_{0}. Then k = b(nak) ∈ B_{0}B = B. As for the second, N_{0}A_{0} · k(B) ⊂ B_{0} · k(B) ⊂ B_{0}B = B, and B ⊂ n(B) · a(B) · k(B) ⊂ N_{0}A_{0} · k(B).

Corollary 5.3. Formulae (2), (3) and (4) hold for the Bruhat cells in K_{0}.

Proof. Suppose C(w) ⊂ C(s)C(w), i.e. k(B(w)) ⊂ k(B(s))k(B(w)). Then N_{0}A_{0} · k(B(w)) ⊂ N_{0}A_{0} · k(B(s))N_{0}A_{0} · k(B(w)), that is, by Lemma 5, B(w) ⊂ B(s)B(w). Thus (2) gives B(s)B(w) = B(w) ∪ B(sw), so that, applying Lemma 4 and observing that k(A ∪ B) = k(A) ∪ k(B), we obtain C(s)C(w) = C(w) ∪ C(sw). If instead C(w) ⊄ C(s)C(w), then necessarily B(w) ⊄ B(s)B(w), otherwise taking k(·) and applying Lemma 5.1 we would obtain the inclusion which negates our assumption. But then, by (2), B(s)B(w) = B(sw), which yields C(s)C(w) = C(sw). This proves that 2 holds for the cells in K_{0}. As for 3 and 4, simply apply Lemma 5.1.

Next we analyze the action of F_{2} on the Bruhat cells C(w). We simply write fkf in place of (1), and keep in mind the convention established at the end of Section 2.

Lemma 5.4. F_{2} leaves each Bruhat cell C(w) invariant. In particular, if f ∈ F_{2} and k = k(\overline{n}_{w})wm ∈ C(w):

fkf = k(f\overline{n}_{w}f)w^{f}m,

where f\overline{n}_{w}f ∈ N_{w}^{−}, w^{f} = w · m(f, w) ∈ M′_{0}, and m(f, w) ∈ M_{0}.

Proof. Let \overline{n}_{w} = nau be the Iwasawa decomposition of \overline{n}_{w}, so that u = k(\overline{n}_{w}). Then f\overline{n}_{w}f = (fnf)(faf)(fuf) = n′a(fuf), where clearly

fnf = exp ∑_{α∈Σ^{+}} Ad fX_{α} = ∑_{α∈Σ^{+}} e^{iπα(A)}X_{α} ∈ N_{0}.

Therefore k(f\overline{n}_{w}f) = fk(\overline{n}_{w})f. Moreover, if f = exp iπA, and \overline{n}_{w} = exp ∑_{α∈Σ_{w}^{+}} X_{wα}, then

f\overline{n}_{w}f = exp ∑_{α∈Σ_{w}^{+}} Ad fX_{wα} = exp ∑_{α∈Σ_{w}^{+}} e^{iπwα(A)}X_{α} ∈ N_{w}^{−}.

so that f\overline{n}_{w}f ∈ N_{w}^{−}. Next, it is clear that since f centralizes M_{0}, w^{f} = fwf normalizes 𝔞_{0}:

Ad w^{f}A = Ad fwfA = Ad fwA = Ad wA ∈ 𝔞_{0},

which also shows that w^{f} ∈ M′_{0} coincides with w modulo M_{0}, i.e. w^{f} = wm(f, w). The result follows, since fkf = (fk(\overline{n}_{w})f)(fwf)(fmf) and fmf = m.

Let now S = {s_{1}, …, s_{l}} be the set of generators of W consisting of all the reflections associated with the simple roots Δ = {δ_{1}, …, δ_{l}}. Recall that {H_{1}, …, H_{l}} is the dual basis of Δ. Thus to each s ∈ S there corresponds a unique H of the basis, and viceversa.

Lemma 5.5. Let s ≠ s_{0}, with s, s_{0} ∈ S. Let f_{0} = exp iπH_{0} ∈ F_{2}, where H_{0} is the element corresponding to s_{0}. Then f_{0} leaves C(s) pointwise fixed.

Proof. Let k ∈ C(s). Thanks to Lemma (5.4), if k = k(\overline{n}_{s})sm, then f_{0}kf_{0} = k(f_{0}\overline{n}_{s}f_{0})s^{f_{0}}m. Now, Σ_{s}^{+} consists of the integral multiples of δ in Σ^{+}. Indeed, s permutes the positive roots which are not multiples of δ, and sends δ (resp. 2δ) to −δ (resp. −2δ). It follows that \overline{n}_{s} = exp(X_{−δ} + X_{−2δ}), where we agree that X_{−2δ} = 0 if 2δ is not a root. Therefore:

f_{0}\overline{n}_{s}f_{0} = f_{0} exp(X_{−δ} + X_{−2δ})f_{0}
= exp Ad f_{0}(X_{−δ} + X_{−2δ})
= exp Ad(exp iπH_{0})(X_{−δ} + X_{−2δ})
= exp e^{ad iπH_{0}}(X_{−δ} + X_{−2δ})
= exp(e^{−iπδ(H_{0})}X_{−δ} + e^{−iπ2δ(H_{0})}X_{−2δ})
= exp(X_{−δ} + X_{−2δ}).
= \overline{n}_{s}

Moreover, Ad sH_{0} = ∑_{j=1}^{l} ν_{j}H_{j} for some coefficients ν_{1}, …, ν_{l}. But if δ corresponds to s, from the fact that δ(H_{0}) = 0 it follows:

ν_{j} = δ_{j}(Ad sH_{0}) = s · δ_{j}(H_{0}) = (δ_{j} − c_{δ_{j},δ}δ)H_{0} = δ_{j}(H_{0}),

namely Ad sH_{0} = H_{0}. Thus

sf_{0}s = exp iπ(Ad sH_{0}) = exp iπH_{0} = f_{0},

that is s^{f_{0}} = s.

Let now Σ = ∪_{j}Σ^{j} be the decomposition of Σ into irreducible root systems, and assume Σ non–reduced. Then at least one of the Σ^{j} is non–reduced, hence of type (BC)_{l} ([6], Theorem 3.25, Ch. X, p. 475). Let Σ_{0} = ∪_{j}Σ_{0}^{j} denote the collection of all such subsystems, let δ_{0}^{j} be the only simple root in Σ_{0}^{j} such that 2δ_{0}^{j} is a root, and let H_{0}^{j} be associated with δ_{0}^{j}. These data enable us to select special elements in F_{2}, namely

f_{0}^{j} = exp iπH_{0}^{j}.

Denote by F_{2}^{0} the set of all such elements and define a new set of elements of order 4 :

F_{4} = {f ∈ exp i𝔞_{0} | f^{2} ∈ F_{2}^{0}} .

For simplicity, we may assume that Σ_{0} consists of a single system, so that F_{2}^{0} reduces to {f_{0}}. All the results that follow hold in full generality, but in order to avoid cumbersome notation, they will be stated and proved under this simplifying assumption. The necessary modifications are obvious.

Lemma 5.6. Let k ∈ C(s_{0}) be such that f_{0}kf_{0} = km for some m ∈ M_{0}. Then m = e.

Proof. Let k = k(\overline{n}_{0})s_{0}μ. Then \overline{n}_{0} = exp(X_{0}^{1} + X_{0}^{2}), with X_{0}^{1} (resp. X_{0}^{2}) in the root space corresponding to −δ_{0} (resp. −2δ_{0}). Select now two non-zero vectors in these spaces, coinciding with X_{0}^{1} and X_{0}^{2} if neither one vanishes, and denote them again X_{0}^{1} and X_{0}^{2}. Then since δ_{0} is indivisible, the Lie algebra 𝔤_{0}^{∗} generated by X_{0}^{1}, X_{0}^{2}, θX_{0}^{1} and θX_{0}^{2} is isomorphic to 𝔰𝔲(2, 1) ([6], Theorem 3.1, Ch. IX, p. 409). Give now to all general Lie algebra concepts connected with 𝔤_{0}^{∗} the superscript ^{∗}. In particular, let G_{0}^{∗}, K_{0}^{∗} A_{0}^{∗},N_{0}^{∗} and \overline{N}_{0}^{∗} be the analytic subgroups corresponding to 𝔤_{0}^{∗}, 𝔨_{0}^{∗}, 𝔞_{0}^{∗}, 𝔫_{0}^{∗} and \overline{𝔫}_{0}^{∗}. In particular \overline{n_{0}} ∈ exp \overline{𝔫}_{0}^{∗}. Next, consider the elements Y ∈ 𝔭_{0}^{∗} and Z ∈ 𝔨_{0}^{∗} corresponding to

\overline{Y} = [ \begin{matrix}0 & 0 & 0 \\ 0 & 0 & −i \\ 0 & i & 0 \end{matrix} ] ∈ 𝔰𝔲(2, 1), \overline{Z} = [ \begin{matrix}0 & i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ] ∈ 𝔰𝔲(2, 1),

respectively. It is immediate to check that for all H ∈ 𝔞_{0}^{∗}:

[H, Y] = δ_{0}(H)Z, [H, Z] = δ_{0}(H)Y.

Thus, ([6], Lemma 2.4, Ch. VII, p. 286), s_{0} may be realized as

s_{0} = exp(\frac{π}{⟨δ_{0}, δ_{0}⟩^{1/2}} Z) ∈ G^{∗} ∩ K_{0} = K_{0}^{∗}.

Observe also that k(\overline{n}_{0}) ∈ K_{0}^{∗}. It follows that the equality f_{0}(k(\overline{n}_{0})s_{0}μ)f_{0} = (k(\overline{n}_{0})s_{0}μ)m in K_{0} yields the equality:

f_{0}(k(\overline{n}_{0})s_{0})f_{0} = (k(\overline{n}_{0})s_{0})μmμ^{−1}

in K_{0}^{∗}, that is in S(U_{2} × U_{1}) ⊂ SU(2, 1). This gives μmμ^{−1} = e, namely m = e.

Remark. The previous lemma should be thought of as an “SU(2, 1)–reduction”. It also explains the first reason for choosing δ_{0} as we did; the second reason will become clear in the proof of Theorem 12.

Theorem 5.7. Let k be such that ⟨k⟩ ∈ Fix(F_{2}, K_{0}/M_{0}). Then f_{0}kf_{0} = k.

Proof. Let k ∈ C(w) be as in the statement. We now show by induction on l_{S}(w) that f_{0}kf_{0} = k. If l_{S}(w) = 1 the results follows from Lemma 5.5 and Lemma 5.6. Suppose the statement true for l_{S}(w) ≤ p and let w = ss_{1}·, . . . · s_{p} be a reduced expression. Two cases arise: either s ≠ s_{0} or s = s_{0}. Suppose first that s ≠ s_{0}. By hypothesis, f_{0}kf_{0} = km for a suitable m ∈ M_{0}. Because of (4), C(w) = C(s)C(s_{1} · . . . · s_{p}) and writing σ = s_{1}·, . . . · s_{p}, we have k = k_{s}k_{σ}, where evidently k_{s} ∈ C(s) and k_{σ} ∈ C(σ). But s ≠ s_{0}, so that Lemma (5.5) gives f_{0}k_{s}f_{0} = k_{s}. Therefore

f_{0}(k_{s}k_{σ})f_{0} = (f_{0}k_{s}f_{0})(f_{0}k_{σ}f_{0}) = k_{s}(f_{0}k_{σ}f_{0}),

and this is equal to k_{s}k_{σ}m. It follows that f_{0}k_{σ}f_{0} = k_{σ}m. By induction, m = e. Suppose next that s = s_{0}. Put again σ = s_{1}·, . . . · s_{p} and k = k_{s}k_{σ}. From (f_{0}k_{s}f_{0})(f_{0}k_{σ}f_{0}) = k_{s}k_{σ}m we obtain

f_{0}k_{s}f_{0} = k_{s}(k_{σ}mf_{o}k_{σ}^{−1}f_{0}).

Now k_{s} ∈ C(s), and let k_{σ}mf_{o}k_{σ}^{−1}f_{0} ∈ C(u), so that:

f_{0}k_{s}f_{0} ∈ C(s) ∩ C(s)C(u).

Using now (2), we infer that

a) either f_{0}k_{s}f_{0} ∈ C(s) ∩ C(su).

b) or f_{0}k_{s}f_{0} ∈ C(s) ∩ C(u).

In case a), u must be equal to e, which yields k_{σ}mf_{o}k_{σ}^{−1}f_{0} ∈ C(e) = M_{0}, that is f_{0}k_{s}f_{0} = k_{s}μ for suitable μ ∈ M_{0} which is forced to be e by induction. Thus k_{σ}m = f_{0}k_{σ}f_{0} and again by induction m = e. Next, we see that case b) cannot occur. Indeed, in this circumstance u = s. Let us be more precise about u. The defining condition is that

ρ := (k_{σ}m)(f_{0}k_{σ}^{−1}f_{0}) ∈ C(u).

But k_{σ}m ∈ C(σ) and f_{0}k_{σ}^{−1}f_{0} ∈ C(σ^{−1}). By (3)

C(s_{1} · . . . · s_{p})C(σ^{−1}) ⊂ ∐_{1≤i_{1}≤...≤i_{t}≤p} C(s_{i_{1}} · . . . · s_{i_{t}}σ^{−1}),

so that ρ ∈ C(s_{i_{1}} · . . . · s_{i_{t}}σ^{−1}) for a suitable choice of indices. On the other hand, s_{i_{1}} · . . . · s_{i_{t}}σ^{−1} = u = s, i.e. s_{i_{1}} · . . . · s_{i_{t}} = sσ = w. This is a contradiction because l_{S}(s_{i_{1}} · . . . · s_{i_{t}}) = t ≤ p < p + 1 = l_{S}(w).

Recall that the set F_{4} of order 4 elements is now assumed for simplicity to be:

F_{4} = {f ∈ exp i𝔞_{0} | f^{2} = f_{0}} .

Proposition 5.8. F_{4} acts on X_{1} = Fix(F_{2}, K_{0}/M_{0}).

Proof. With the notation of Section 2, the action of F_{4} on X_{1} is given by f · ⟨k⟩ = ⟨Ad^{−1}(fk_{∗}f^{−1})⟩. As agreed earlier, however, whenever appropriate we will think of all the supgroups of G_{0} as subgroups of G under Ad (whereby the center has become trivial), and remove the subscript ∗. Let K_{1} = {k ∈ K_{0} | f_{0}kf_{0} = k}, a closed, hence compact, Lie subgroup of K_{0}. Let 𝔨_{1} be its Lie algebra. Our first concern is to show that:

Ad f_{0}T = T,  ∀T ∈ 𝔨_{1}  (5)

Indeed, for all t ∈ ℝ:

exp(t Ad f_{0}T) = f_{0}(exp tT)f_{0} = (exp tT) ∈ K_{1},

thereby showing Ad f_{0}T ∈ 𝔨_{1}. Next, since Ad f_{0} is an involution on 𝔨_{1}, we have a vector space decomposition 𝔨_{1} = 𝔨_{1}^{+} + 𝔨_{1}^{−} corresponding to the ±1 eigenvalues of Ad f_{0}. Thus, if T = T^{+} + T^{−} is the resulting decomposition of T, then for all t ∈ ℝ:

exp(t(T^{+} − T^{−})) = exp(t Ad f_{0}T) = f_{0}(exp tT)f_{0} = exp tT = exp(t(T^{+} + T^{−})).

For t sufficiently small, this yields T^{−} = 0, which is equivalent to saying 𝔨_{1}^{−} = 0. This proves (5). Now we prove that if k ∈ K_{1}, then fkf^{−1} is fixed by Θ. In fact since f^{2} = f_{0}, we have

f = f^{−1}f_{0} = f_{0}f^{−1}, f^{−1} = ff_{0} = f_{0}f.

On the other hand, since Θ(f) = Θ(exp iA) = exp(iθA) = exp(−iA) = f^{−1} we have

Θ(fkf^{−1}) = f^{−1}kf = ff_{0}kf_{0}f^{−1} = fkf^{−1}.

Suppose now k = exp T ∈ K_{1}. We may take T ∈ 𝔨_{1}. Then fkf = exp(Ad fT) and if σ denotes conjugation in 𝔤 with respect to 𝔤_{0}, then by (5)

σ(Ad fT) = σ(Ad(exp iA)T)
= Ad(exp σ(iA))(σ(T))
= Ad(exp −iA)T
= Ad f^{−1}T
= Ad ff_{0}T
= Ad fT.

Thus, Ad fT ∈ 𝔨_{0}, i.e. fkf^{−1} ∈ K_{0}. Finally, let ⟨k⟩ ∈ X_{1}, so that, by Theorem 5.7, k ∈ K_{1} and by the above argument fkf^{−1} ∈ K_{0}. In order to see that F_{4} sends fixed points of F_{2} into fixed points of F_{2}, we must show that for all f ∈ F_{4} and for all g ∈ F_{2} there exists m ∈ M_{0} such that g(fkf^{−1})g = (fkf^{−1})m. But this is obvious because f and g commute, and there exists m ∈ M_{0} such that gkg = km.

Theorem 5.9. Fix(F_{4}, X_{1}) = W.

Proof. Since Σ_{0} is of type (BC)_{l}, all roots of the form α = ∑_{j=0}^{l−1} ν_{j}(α)δ_{j} with ν_{j}(α) ∈ 2ℤ (i.e. even roots) are such that ν_{0}(α) = ±2, as it is clear from the list at the end of Section 2. More precisely, all the non–vanishing coefficients of an even root must be equal to ±2, but for each fixed δ_{j} ≠ δ_{0} there is an even root α for which ν_{j}(α) = 0. This is the second reason for choosing δ_{0} as we did. Let now ⟨k⟩ ∈ Fix(F_{4}, X_{1}), \overline{H} ∈ 𝔞_{0} a regular element, and Y = Ad k\overline{H}. Then Y is fixed by all elements of F_{4}. According to the remark following Theorem 3.3, we may write:

Y = Y_{0} + ∑_{α∈E} Y_{α},

where E is the set of even roots in Σ_{0}. Therefore, if f = exp iA_{0}

Y_{α} = e^{iα(A_{0})}Y_{α}.

Take now A_{0} = \frac{π}{2}H_{0} (so that (exp iA_{0})^{2} = f_{0}) and observe that:

α(A_{0}) = ∑_{j=0}^{l−1} ν_{j}(α)δ_{j}(\frac{π}{2} H_{0}) = \frac{πν_{0}(α)}{2} = ±π.

Thus Y_{α} = −Y_{α} = 0 and Y ∈ 𝔞_{0}. Hence k ∈ M′_{0}, that is ⟨k⟩ ∈ W, which shows that Fix(F_{4}, X_{1}) ⊂ W. The reverse inclusion is obvious.

Acknowledgements. I would like to thank V. Baldoni Silva, C. Bartocci, M. Pedroni and R. Stanton for very useful conversations.

References

[1] Bourbaki, N., “Eléments de Mathématique. Groupes et algèbres de Lie,” Ch. IV, Hermann, Paris, 1960-75.

[2] De Mari, F. and M. Pedroni, Toda flows and real Hessenberg manifolds, Jour. Geom. An., to appear.

[3] De Mari, F., C. Procesi and M. A. Shayman, Hessenberg varieties, Trans. Amer. Math. Soc. 332 (1992), 529–534.

[4] Floyd, E. E., Periodic Maps via Smith theory, Seminar on Transformation Groups, A. Borel ed., 1960, Ann. Math. Std. n.46 Princeton U.P., Princeton N.J.

[5] Goodman, R. and N. R. Wallach, Classical and Quantum Mechanical Systems of Toda–Lattice Type II, Solutions of the Classical Flows, Commun. Math. Phys.94 (1984), 177–217.

[6] Helgason, S., “Differential Geometry, Lie Groups and Symmetric Spaces,” Academic Press, New York, 1978.

[7] Helgason, S., “Groups and Geometric Analysis,” Academic Press, New York, 1984.

[8] Kostant, B. and S. Rallis, Orbits and Lie group representations associated to symmetric spaces, Amer. J. Math. 93 (1971), 753–809.

Filippo De Mari Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy demari@dima.unige.it

Received January, 16 1996 and in final form April, 26 1996