Decompositions of Bruhat type for the Kac-Moody groups.

Yuly Billig Department of Mathematics, University of Alberta Edmonton, Alberta, Canada T6G 2G1

Matthew Dyer Department of Mathematics, University of Notre Dame

Mail Distribution Center, Notre Dame, Indiana 46556-5683

ABSTRACT. In this article we construct new Bruhat type decompositions for the Kac-Moody groups, and present canonical forms for the elements of a group with respect to these decompositions. We also give some partial results on closure and intersection patterns in the corresponding decompositions of the flag variety.

AMS Subject Classification numbers: 22E65, 17B67

Introduction. There are several versions of the decompositions of Bruhat type for the Kac-Moody groups; the most important of them are the Bruhat decomposition and the Birkhoff decomposition [2], [13], [18]. These decompositions are powerful tools for the study of the Kac-Moody groups from both algebraic and geometric points of view.

Here we construct a new family of Bruhat type decompositions, including classical Bruhat and Birkhoff decompositions as special cases, and study the closure patterns of the double cosets involved. The decompositions are associated with (non-standard) positive root systems.

Positive root systems were studied in [9] and [10] , where the highest weight representation theory with respect to a positive root system is developed. It is known that in the finite-dimensional case all positive root systems are conjugate under the action of the Weyl group, so all decompositions are equivalent to classical Bruhat decomposition. In the affine case all positive root systems are classified. There exist, up to conjugation, a finite number of them, so we get a finite number of corresponding non-equivalent decompositions. For the Kac-Moody groups of indefinite type, the number of non-equivalent decompositions may be infinite.

The idea of the proof of the generalized Bruhat decomposition is the same as in Kac-Peterson paper [13], where the Birkhoff decomposition is proven. Victor Kac wrote us that this generalization was known to him.

Finite and cofinite Schubert varieties in the flag variety of a Kac-Moody group have been studied extensively (see e.g. [14, 15]). The closure patterns are described by Chevalley (Bruhat) order. We show here that closure patterns for the decompositions of flag varieties arising from the generalized Bruhat decompositions are described by the “twisted orders”on the Weyl group studied in [6,7].

The main results of the paper are as follows. Let G be a Kac-Moody group with standard Borel subgroup B. In Theorem 1 we prove that G decomposes as the product of its subgroups:

G = BW Q,

where Q is an arbitrary subgroup of G that contains “one half of all the real roots”. Theorem 2 states that the natural map from W to the set of double cosets B\G/Q is a bijection. Theorem 3 provides a canonical form of such a presentation of elements of G. Theorem 4 describes the closure patterns of the double cosets QwB in the “Zariski” topology considered by Kac-Peterson [14], and Theorem 5 gives results similar to those of Curtis [4] in this context.

The first author would like to thank Professors Jacques Tits and Robert Moody for helpful discussions. The second author was partially supported by NSF grant DMS90-12836.

Definitions and notations. It will be convenient for us to use the version of the Kac-Moody groups studied in [14, 15] over fields of characteristic zero. However, most results not involving the “Zariski topology” defined there can be proved by essentially identical arguments for some similar versions of the groups (e.g. those constructed over arbitrary fields in [11] using one of Tits’ ℤ-forms for enveloping algebras of Kac-Moody Lie algebras).

Let A be a symmetrizable generalized Cartan matrix and let (𝔥, Π, Π^{∨}) be a realization of A over a field 𝔽 of characteristic zero, (symmetrizability is required only for some of our results involving the Zariski topology). One then has the corresponding Kac-Moody Lie algebra 𝔤 = 𝔤(A) over 𝔽 generated by 𝔥 and symbols e_{α}, f_{α} for α ∈ Π with the usual relations [15, (1.1), (1.2)]. One has the root space decomposition 𝔤 = ⊕_{α∈h^{∗}}𝔤_{α}, and we denote the roots (resp. positive roots) by Δ (resp., Δ_{+}).

Let W denote the subgroup of Aut𝔽(𝔥) generated by the “simple reflections” v ↦ v − ⟨v, α^{∨}⟩α for α ∈ Π. The set { wα | w ∈ W, α ∈ Π } of real roots will be denoted Δ^{re}. Define positive and negative real roots as usual by Δ_{±}^{re} = Δ^{re}∩±Δ_{+}.

A 𝔤′-module V, or (V, π) where π: 𝔤′ → End𝔽(V), is said to be integrable if for all α ∈ Δ^{re} and v ∈ V, there exists N with π(𝔤_{α})^{N}(v) = 0. One associates a group G to 𝔤(A) as in [14, 15]; here we recall only that G is the quotient G = G^{∗}/N of the free product G^{∗} of the root subgroups 𝔤_{α} (for real roots α) by the largest normal subgroup N of G^{∗} acting trivially on all integrable 𝔤′-modules V (where 𝔤_{α} ⊂ G^{∗} acts on each such (V, π) by (e, v) ↦ exp(π(e))v). For real roots α, one has the canonical injection (also denoted exp) 𝔤_{α} → G^{∗} → G with image the one-parameter subgroup U_{α}. Then G is generated by the U_{α} for real roots α, and denoting the natural action of G on an integrable 𝔤′-module (V, π) also by π, one has π(exp e) = exp π(e) for e ∈ 𝔤_{α}.

We abbreviate x_{α}(t) := exp(te_{α}) and x_{−α}(t) := exp(tf_{α}) for α ∈ Π, t ∈ 𝔽. For α ∈ Π, there is a unique homomorphism ϕ_{α} : SL_{2}(𝔽) → G with ϕ_{α}\binom{1 t}{0 1} = x_{α}(t) and ϕ_{α}\binom{1 0}{t 1} = x_{−α}(t); ϕ_{α} is actually an isomorphism onto its image G_{α}. Let H_{α}(t) = ϕ_{α}(diag(t, t^{−1}) for t ∈ 𝔽^{∗} and s_{α} = ϕ_{α}\binom{0 1}{−1 0}, so one has

x_{α}(λ)x_{−α}(−λ^{−1})x_{α}(λ) = H_{α}(λ)s_{α}.  (1)

Let H be the (abelian) subgroup of G generated by the images of the H_{α} for α ∈ Π, and N be the sugroup generated by H and the s_{α} for α ∈ Π. Then N normalizes H, and one may identify W and the group N/H so that for α ∈ Π, the simple reflection v → v − ⟨v, α^{∨}⟩α identifies with the coset s_{α}H ∈ N/H (we often denote a coset representative for w ∈ W still by w). We may also identify W with the contragredient subgroup of Aut𝔽(𝔥^{∗}), where 𝔥^{∗} is the dual space. Then the bijection Π → Π^{∨} given by α ↦ α^{∨} extends to a W -invariant bijection Δ^{re} = W Π → W Π^{∨} (from the real roots to the real coroots) which we still denote by α ↦ α^{∨}. For α ∈ Δ^{re}, we have the corresponding reflection v ↦ v − ⟨v, α^{∨}⟩α, for v ∈ 𝔥, in W. We let S denote the set of simple reflections of W, and l : W → ℕ denote the usual length function of the Coxeter system (W, S).

The following relation holds for all w ∈ W, γ ∈ Δ^{re} :

wU_{γ}w^{−1} = U_{w(γ)}.  (2)

The subgroup of G generated by all U_{α} , where α ∈ Δ_{+}^{re} (resp. α ∈ Δ_{−}^{re} ), is denoted by U_{+} (resp. U_{−} ). The subgroup H normalizes each U_{α} for α ∈ Δ^{re}, hence also U_{+} and U_{−} . Define the Borel subgroup B (resp. B_{−}) as the product B := HU_{+} = U_{+}H (resp. B_{−} := HU_{−} = U_{−}H ).

The Bruhat decomposition for a Kac-Moody group is a presentation of G as a product of the subgroups:

G = BW B.  (3)

However, the presentation of g ∈ G in the form g = b_{1}wb_{2}; b_{1}, b_{2} ∈ B is not unique. The following version of Bruhat decomposition gives unique presentation:

G = ∪_{w∈W}BwU_{w} ,  (4)

where U_{w} = U_{+} ∩ (w^{−1}U_{−}w).

Denote by U′_{w} the subgroup U_{+} ∩ (w^{−1}U_{+}w). The group U_{+} is the product of these subgroups:

U_{+} = U_{w}U′_{w} = U′_{w}U_{w}.

The group U_{w} is nilpotent and may be presented as a product of those groups U_{α} that are contained in U_{w} . Precisely, let w = s_{n}s_{n−1} … s_{2}s_{1} be the presentation of w as a product of the elements from S of minimal length: l(w) = n. Then

U_{w} = U_{β_{n}} … U_{β_{2}}U_{β_{1}},  (5)

with uniqueness of expression, where β_{1} = α_{1} , β_{k} = s_{1} … s_{k−1}(α_{k}) , for k = 2, …, n .

Bruhat-type Decompositions. Theorem 1. Let Q be a subgroup of a Kac-Moody group G, such that

for all α ∈ Δ^{re} either U_{α} ⊂ Q or U_{−α} ⊂ Q. Then

G = BW Q.

Proof. It is sufficient to prove that

BwU_{w}W Q ⊂ BW Q.  (6)

Then, using the Bruhat decomposition (4) we get:

G = ∪_{w∈W}BwU_{w} ⊂ ∪_{w∈W}BwU_{w}W Q ⊂ BW Q.  (7)

We shall prove (6) by induction on l(w). If l(w) = 0 , then w = e and U_{w} = {e} , so

BwU_{w}W Q = BW Q.

Suppose l(w) = n and the induction assumption holds for all elements of the Weyl group of smaller length.

Let u ∈ U_{w} and w_{1} ∈ W. We shall prove that Bwuw_{1}Q ⊂ BW Q. We use (5) to present u as a product: u = u_{n} … u_{2}u_{1} , where u_{k} ∈ U_{β_{k}} . Now we will consider three cases:

1) u_{1} = e;

2) u_{1} ≠ e, U_{w_{1}^{−1}(α_{1})} ⊂ Q; 3) u_{1} ≠ e, U_{−w_{1}^{−1}(α_{1})} ⊂ Q.

Case 1. If u_{1} = e then u ∈ U_{β_{n}} … U_{β_{2}} .

Let w′ = ws_{1} = s_{n} … s_{2} ; l(w′) = n − 1.

Then

U_{β_{n}} … U_{β_{2}} = s_{1}U_{s_{1}(β_{n})} … U_{s_{1}(β_{2})}s_{1} = s_{1}U_{w′}s_{1}.

Hence,

Bwuw_{1}Q ⊂ Bws_{1}U_{w′}s_{1}w_{1}Q = Bw′U_{w′}s_{1}w_{1}Q.

As l(w′) < l(w) , we may apply the induction assumption.

Case 2. Let U_{w_{1}^{−1}(α_{1})} ⊂ Q. Then

u_{1}w_{1}Q ⊂ U_{α_{1}}w_{1}Q = w_{1}(w_{1}^{−1}U_{α_{1}}w_{1})Q = w_{1}U_{w_{1}^{−1}(α_{1})}Q = w_{1}Q.

Consequently,

Bwuw_{1}Q = Bwu_{n} … u_{2}u_{1}w_{1}Q ⊂ Bwu_{n} … u_{2}w_{1}Q.

So, this case reduces to Case 1.

Case 3. Let U_{−w_{1}^{−1}(α_{1})} ⊂ Q and u_{1} ≠ e. Then u_{1} = x_{α_{1}}(λ), for some λ ≠ 0.

Then (2) yields:

w_{1}^{−1}x_{−α_{1}}(−λ^{−1})w_{1} ∈ w_{1}^{−1}U_{−α_{1}}w_{1} = U_{−w_{1}^{−1}(α_{1})} ⊂ Q,

hence,

w_{1}Q = w_{1}(w_{1}^{−1}x_{−α_{1}}(−λ^{−1})w_{1})Q = x_{−α_{1}}(−λ^{−1})w_{1}Q.

Thus, applying (1) we get:

Bwuw_{1}Q =

= Bwu_{n} … u_{2}x_{α_{1}}(λ)x_{−α_{1}}(−λ^{−1})w_{1}Q =

= Bwu_{n} … u_{2}H_{α_{1}}(λ)s_{1}x_{α_{1}}(−λ)w_{1}Q ⊂

⊂ BHwU_{β_{n}} … U_{β_{2}}s_{1}U_{α_{1}}w_{1}Q = Bws_{1}s_{1}U_{β_{n}} … U_{β_{2}}s_{1}U_{α_{1}}w_{1}Q =

= Bw′U_{w′}U_{α_{1}}w_{1}Q ⊂ Bw′U_{+}w_{1}Q ⊂ Bw′U_{w′}w_{1}Q .

As l(w′) < l(w), we are able to apply the induction assumption. This completes the proof.

Corollary 1. (of the proof). If s = s_{α} ∈ S, where α ∈ Π, and w ∈ W , then

BsBwQ = \{ \begin{matrix}BwQ ∪ BswQ & if U_{−w^{−1}(α)} ⊂ Q \\ BswQ & if U_{w^{−1}(α)} ⊂ Q. \end{matrix}

Corollary 2. Let Q satisfy the condition of Theorem 1, and let P be a parabolic subgroup of G containing B . Suppose P ∩ W = W_{1} , Q ∩ W = W_{2}. Then

G = ∪_{w∈W_{1}\W/W_{2}}P wQ.

Examples.

1. If Q = B then the decomposition (6)

G = BW Q = BW B

is the Bruhat decomposition (3) for G .

2. If Q = B_{−} then (6) gives the Birkhoff decomposition for G :

G = BW B_{−} .

3. Let G be an affine Kac-Moody group. Consider the realization of the corresponding Kac-Moody algebra as a subalgebra in 𝔤^{0} ⊗ 𝔽[t, t^{−1}] ⊕ 𝔽c ⊕ 𝔽d , where 𝔤^{0} is a simple finite-dimensional Lie algebra. Define Ψ to be the set of real roots of the subalgebra 𝔫_{+}^{0} ⊗ 𝔽[t, t^{−1}] .

Consider in the corresponding group G the subgroup Q generated by all U_{α} such that α ∈ Ψ. The decomposition (6) we obtain in this way is known as the Iwasawa decomposition.

Now suppose that A is the extended Cartan matrix of SL_{2}. The affine Weyl group W has the corresponding finite Weyl group as parabolic subgroup in the usual way; taking P ⊃ B to be the corresponding parabolic subgroup of G, the decomposition of G/𝔽^{∗} ≅ SL_{2}(𝔽[t, t^{−1}]) from Corollary 2 is:

SL_{2}( 𝔽[t, t^{−1}] ) = ∪_{k∈ℤ} SL_{2}( 𝔽[t] ) ( \begin{matrix}t^{k} & 0 \\ 0 & t^{−k} \end{matrix} ) ( \begin{matrix}1 & 𝔽[t, t^{−1}] \\ 0 & 1 \end{matrix} ) =

= ∪_{k∈ℤ} SL_{2}( 𝔽[t] ) ( \begin{matrix}1 & t^{−1}𝔽[t^{−1}] \\ 0 & 1 \end{matrix} ) ( \begin{matrix}t^{k} & 0 \\ 0 & t^{−k} \end{matrix} ) .

Definition. A (non-standard) set of positive roots Ψ is defined as an arbitrary subset in Δ with the following properties:

a) Ψ ∪ −Ψ = Δ\{0}.

b) Ψ ∩ −Ψ = ∅.

c) Ψ is closed, i.e. if α, β ∈ Ψ and α + β ∈ Δ, then α + β ∈ Ψ .

Consider for some ordered basis over the reals of the real span of the roots, the set Ψ of all roots whose first non-zero component in that basis is positive (lexicographically positive roots with respect to this basis). Then Ψ is a set of positive roots. Let us prove that all positive root systems arise this way.

For the following results, it is convenient to identify the root lattice with ℤ^{n} so the simple roots coorespond to the standard coordinate vectors.

Proposition 1. If Ψ is a positive root system in Δ then there exists a basis B in ℝ^{n} such that Ψ is the set of roots that are lexicographically positive with respect to B.

To prove this proposition we need the following lemma.

Lemma 1. Let Ψ_{1} = {∑_{i=1}^{k} n_{i}μ_{i} | k ≥ 1 , 0 < n_{i} ∈ ℕ, μ_{i} ∈ Ψ} be the

semigroup in ℤ^{n} generated by Ψ. Then Ψ_{1} ∩ (−Ψ_{1}) = ∅.

Proof. If ∑_{i=1}^{k} n_{i}μ_{i} = ∑_{j=1}^{s} m_{j}β_{j}, where μ_{i} ∈ Ψ, β_{j} ∈ −Ψ , then ∑_{i=1}^{k} n_{i}μ_{i} +

∑_{j=1}^{s} m_{j}(−β_{j}) = 0, with (−β_{j}) ∈ Ψ. So, it is sufficient to consider the case when

∑_{i=1}^{k} n_{i}μ_{i} = 0 and we may assume that this sum has the minimal possible positive

∑_{i=1}^{k} n_{i}.

Suppose that one of the roots in this sum is real, let say μ_{1} ∈ Δ^{re}. Then

0 = ⟨∑_{i=1}^{k} n_{i}μ_{i} , μ_{1}^{∨}⟩ = n_{1}⟨μ_{1}, μ_{1}^{∨}⟩ + ∑_{i=2}^{k} n_{i}⟨μ_{i}, μ_{1}^{∨}⟩.

As ⟨μ_{1}, μ_{1}^{∨}⟩ > 0 then for some i we have ⟨μ_{i}, μ_{1}^{∨}⟩ < 0 , hence μ_{i} + μ_{1} is a root and μ_{i} + μ_{1} ∈ Ψ. This gives us a sum with lesser ∑_{i=1}^{k} n_{i}. Consequently, all μ_{i} should be imaginary and hence all n_{i} are equal to 1 as nμ_{i} ∈ Δ for μ_{i} ∈ Δ^{im}.

Now we may present our expression in the form:

∑_{i=1}^{k} β_{i} = ∑_{j=1}^{s} γ_{j}, where β_{i} ∈ Ψ ∩ Δ_{+}^{im}, γ_{j} ∈ (−Ψ) ∩ Δ_{+}^{im}  (1)

and we assume that k + s is the minimal possible.

Let us consider two cases:

I. Suppose that ∑_{i=1}^{k} β_{i} is not a root. Note that for all w ∈ W , w(∑_{i=1}^{k} β_{i}) ∈

ℕ^{n}. Choosing w so this element has minimal height, one has w(∑_{i=1}^{k} β_{i}) ∈ −C^{∨}.

As w(∑_{i=1}^{k} β_{i}) ∉ Δ , then the support of w(∑_{i=1}^{k} β_{i}) decomposes into several connected components. Choosing those β_{i} and γ_{j} for which the support of w(β_{i}) and w(γ_{j}) belongs to one of the connected components we get the equality of type (1) with lesser k + s , which is a contradiction.

II. Suppose that ∑_{i=1}^{k} β_{i} is a root. Then k = 1 or s = 1 because of the minimality of k + s. Without a loss of generality we can assume that

∑_{i=1}^{k} β_{i} = δ, β_{i} ∈ Ψ ∩ Δ_{+}^{im}, δ ∈ (−Ψ) ∩ Δ_{+}^{im}.

It follows from the axioms of the positive root system that k > 2. As k is the minimal possible then no subsum in ∑_{i=1}^{k} β_{i} with more than one summand may be equal to a root.

Following a similar argument as in (I) we conclude that there exists w ∈ W such that wβ_{2}, …, wβ_{k−1}, wβ_{k} belong to −C^{∨} and their supports are pairwise non-connected. Consequently, supp(wβ_{1}) connects all of them. In the same way there exists w_{1} ∈ W such that w_{1}β_{1}, w_{1}β_{2} ∈ −C^{∨} and their supports are nonconnected. Since C^{∨} is a fundamental domain for W on the (dual) Tits’ cone, wβ_{2} = w_{1}β_{2}, hence ww_{1}^{−1} ∈ Stab_{W} (wβ_{2}) which is generated by reflections in W that stabilize wβ_{2}. As wβ_{2} ∈ −C^{∨} then Stab_{W} (wβ_{2}) is generated by reflections with respect to the simple roots that are non-connected to the supp(wβ_{2}) or belong to the supp(wβ_{2}). Consequently, the support of wβ_{1} = ww_{1}^{−1}w_{1}β_{1} is non-connected to the support of wβ_{2}, this is a contradiction.

Corollary 3. Let Ψ_{2} = {∑_{i=1}^{k} q_{i}β_{i} | k ≥ 1, q_{i} ∈ ℚ_{+}, β_{i} ∈ Ψ}. Then

Ψ_{2} ∩ (−Ψ_{2}) = ∅.

Proof of the proposition. Let’s prove by induction on n that for any semigroup Ψ_{1} ⊂ ℤ^{n} such that Ψ_{1} ∩ (−Ψ_{1}) = ∅ there exists a basis B in ℝ^{n} with respect to which Ψ_{1} is lexicographically positive.

If n = 1 then this statement is evident.

Without the loss of generality we may assume that Ψ_{1} spans ℝ^{n}.

Consider

Ω = {∑_{i=1}^{m} r_{i}α_{i} | r_{i} ∈ ℝ_{+}, α_{i} ∈ Ψ_{1}, ⟨α_{1}, …, α_{m}⟩ span ℝ^{n}}.

Note that Ω is an open convex cone in ℝ^{n} and Ψ_{1} ⊂ Ω̄. It can be easily seen that Ω ∩ (−Ω) ≠ ∅ implies Ψ_{2} ∩ (−Ψ_{2}) ≠ ∅ which contradicts the previous corollary. Consequently, Ω ∩ (−Ω) = ∅.

Now we use one of the basic theorems of convex analysis which states that if Ω is an open cone in ℝ^{n} such that Ω ∩ (−Ω) = ∅ then Ω and −Ω can be separated by some hyperplane H. Choose as the first vector for basis B the vector normal to H in the direction of Ω. Due to the induction assumption there exists a basis in H such that H ∩ Ψ_{1} consists of lexicographically positive vectors with respect to this basis. This completes the proof of the proposition.

For any set of positive roots Ψ we let Q_{Ψ} denote the subgroup of G generated by all U_{α} such that α ∈ Ψ ∩ Δ^{re}. Then Q_{wΨ} = wQ_{Ψ}w^{−1}, and we get the decomposition for G :

G = BW Q_{Ψ} .

Lemma 2. Let Ψ be a set of positive roots. Then BQ_{Ψ} ∩ W = {e} .

Proof. This follows immediately from representation theory. Consider an irreducible right highest weight 𝔤-module L(Λ) with integral strictly dominant highest weight Λ. This 𝔤-module is integrable. Suppose that w = bq, where w ∈ W, b ∈ B, q ∈ Q_{Ψ} . Then for the highest weight vector v_{Λ} we have v_{Λ}b ∈ 𝔽^{∗}v_{Λ} and consequently v_{Λ}bq has a non-zero Λ-component as according to Lemma 1 no linear combination of elements of Ψ with positive coefficients equals zero. However, v_{Λ}w ∈ (L(Λ))_{w(Λ)}, hence v_{Λ}w has a non-zero Λ-component only if w = e .

Theorem 2. Let Ψ be a set of positive roots, and Q = Q_{Ψ} be the corresponding subgroup of G. Then the map w ↦ BwQ is a bijection of W onto the double cosets B\G/Q .

The proof of this theorem may be easily derived now as a modification of the corresponding proof from [1] for the Bruhat decomposition applying Corollary 1 and the previous lemma.

We now investigate the structure of Q. Let Q_{+} = Q ∩ U_{+} , Q_{−} = Q ∩ U_{−} .

Lemma 3. Let α be a simple root.

i) If U_{α} ⊂ Q, then Q_{+} = U_{α}(U′_{s_{α}} ∩ Q). ii) If U_{α} ⊄ Q, then Q_{+} = (U′_{s_{α}} ∩ Q). Proof. As U_{+} = U_{α}U′_{s_{α}}, then i) is evident. Let us prove ii). Consider a lowest weight module L(Λ) , where Λ is a strictly

antidominant integral weight. Consider the action of the subgroup G_{α} ≅ SL_{2}(𝔽) on L(Λ) and let V be the G_{α}-submodule

V = ⊕_{n∈ℤ_{+}}L(Λ)_{Λ+nα} ,

of L(Λ). There is a natural projection p : L(Λ) → V, which is an identity on the appropriate weight spaces and zero on the others.

Consider the action of the subgroups U_{+} = U_{α}U′_{s_{α}} and Q on the vector v_{Λ}. Let u ∈ U_{+}. Note that u ∈ U′_{s_{α}} if and only if p(uv_{Λ}) = p(v_{Λ}). At the same time if a weight γ belongs to the support of qv_{Λ}, then γ − Λ can be represented as a linear combination of elements of Ψ with non-negative coefficients. As α ∉ Ψ, then π(qv_{Λ}) = π(v_{Λ}) for all q ∈ Q. Consequently, U_{+} ∩ Q = U′_{s_{α}} ∩ Q.

Corollary 4. U_{+} ∩ Q = (U_{w} ∩ Q)(U′_{w} ∩ Q).

Theorem 3. Let Q = Q_{Ψ} be associated to a set of positive roots Ψ . Then

Q = Q_{+}Q_{−} .

Corollary 5. An element g ∈ G may be uniquely represented in the form g = Bwq , where B ∈ B, w ∈ W and q ∈ Q ∩ (w^{−1}U_{−}w) .

Proof of the Corollary. By Theorem 1 an element g ∈ G may be presented in the form g = Bwq = B(wqw^{−1})w. Let us consider the positive root system wΨw^{−1}. By Theorem 3, wQ_{Ψ}w^{−1} is the product of subgroups:

wQ_{Ψ}w^{−1} = (U_{+} ∩ wQ_{Ψ}w^{−1})(U_{−} ∩ wQ_{Ψ}w^{−1}),

so wqw^{−1} = q_{+}q_{−} , where q_{±} ∈ U_{±} ∩ (wQ_{Ψ}w^{−1}) .

Hence, g = (Bq_{+})w(w^{−1}q_{−}w). Remark that w^{−1}q_{−}w ∈ Q ∩ (w^{−1}U_{−}w). This proves the existence of this presentation. Let us prove uniqueness. Suppose g = b_{1}w_{1}q_{1} = b_{2}w_{2}q_{2}. By Theorem 2 we have w_{1} = w_{2} . Consequently, b_{1}(w_{1}q_{1}w_{1}^{−1}) = b_{2}(w_{1}q_{2}w_{1}^{−1}) and b_{1}^{−1}b_{2} = w_{1}q_{1}q_{2}^{−1}w_{1}^{−1}. But b_{1}^{−1}b_{2} ∈ B and w_{1}q_{1}q_{2}^{−1}w_{1}^{−1} ∈ U_{−}. As B ∩ U_{−} = {e} , we get b_{1}^{−1}b_{2} = q_{1}^{−1}q_{2} = e . This completes the proof of the corollary.

Proof of Theorem 3. Let q ∈ Q . Consider the Bruhat decomposition for q:

q = b_{0}w′_{0}u_{0}, where u_{0} ∈ U_{w′_{0}}.

Let us prove that q ∈ Q_{+}Q_{−} by induction on l(w′_{0}) .

If l(w′_{0}) = 0, then w′_{0} = u_{0} = e, hence q ∈ B ∩ Q = U_{+} ∩ Q = Q_{+} . Let l(w′_{0}) = n > 0 , w′_{0} = s_{n}s_{n−1} … s_{1} . Consider the procedure of reduction

of b_{0}w′_{0}u_{0} to the form b_{n}w_{n}q_{n} , described in the proof of the Theorem 1, using at each step the following presentations :

q = b_{i}w′_{i}u_{i}w_{i}q_{i} ,  (8)

by gradual elimination of w′_{i} = s_{n} … s_{i+1} . Theorem 2 implies that we shall eventually get w_{n} = e , as q ∈ Q.

If we use Cases 1 or 2 for the reduction of (8) then q_{i+1}q_{i}^{−1} ∈ U_{w_{i}} and w_{i+1} = s_{i+1}w_{i} , hence w′_{i+1}w_{i+1} = w′_{i}w_{i} . Note that if Case 3 works then w_{i+1} = w_{i} . So, if this process involves Cases 1 or 2 only then w_{n} = w′_{n}w_{n} = = … = w′_{0}w_{0} = w′_{0} , which is impossible as w_{n} = e .

Consequently, Case 3 is involved in this process. Suppose that Case 3 occurs in (8) for the first time. Then in (8) we have w_{i} = s_{i}s_{i−1} … s_{1}, q_{i} ∈ Q ∩ U_{w_{i}} .

Let α be a simple root, corresponding to s_{i+1} . Note that w_{i}^{−1}U_{−α}w_{i} ⊂ Q. We have w′_{i} = w′_{i+1}s_{i+1} and u_{i} = u′_{i}u_{α} , where u_{α} ∈ U_{α} and u′_{i} ∈ s_{i+1}U_{w′_{i+1}}s_{i+1} .

Hence, q = b_{i}w′_{i+1}(s_{i+1}u′_{i}s_{i+1})s_{i+1}u_{α}w_{i}q_{i} .

Following the arguments the proof of Theorem 1 we find u_{−α} ∈ U_{−α} such that s_{i+1}u_{α}u_{−α} = H_{α}(λ)u_{α}^{−1} .

Consequently,

q = b_{i}w′_{i+1}(s_{i+1}u′_{i}s_{i+1})s_{i+1}u_{α}u_{−α}u_{−α}^{−1}w_{i}q_{i} =

= b_{i}w′_{i+1}(s_{i+1}u′_{i}s_{i+1})H_{α}(λ)u_{α}^{−1}w_{i}(w_{i}^{−1}u_{−α}^{−1}w_{i})q_{i} .

Hence, q_{i+1} = (w_{i}^{−1}u_{−α}^{−1}w_{i})q_{i} ∈ (w_{i}^{−1}U_{−α}w_{i})(U_{w_{i}} ∩ Q) ⊂

⊂ w_{i}^{−1}(U_{−} ∩ (w_{i}Qw_{i}^{−1}))w_{i}.

Applying Corollary 4 to U_{−}, w_{i}^{−1} and w_{i}Qw_{i}^{−1} we get

U_{−} ∩ (w_{i}Qw_{i}^{−1}) =

= (U_{−} ∩ (w_{i}U_{+}w_{i}^{−1}) ∩ (w_{i}Qw_{i}^{−1}))(U_{−} ∩ (w_{i}U_{−}w_{i}^{−1}) ∩ (w_{i}Qw_{i}^{−1})).

Consequently,

w_{i}^{−1}(U_{−} ∩ (w_{i}Qw_{i}^{−1}))w_{i} ⊂
⊂ (w_{i}^{−1}U_{−}w_{i} ∩ U_{+} ∩ Q)(w_{i}^{−1}U_{−}w_{i} ∩ U_{−} ∩ Q) ⊂
⊂ U_{w_{i}}(U_{−} ∩ Q) = U_{w_{i}}Q_{−}.

Thus, q ∈ Bw′_{i+1}U_{w′_{i+1}}w_{i}U_{w_{i}}Q_{−}. Using Corollary 1 for the Bruhat decomposition we conclude that q ∈ BwU_{w}Q_{−} for some w ∈ W with l(w) ≤ l(w′_{i+1}) + l(w_{i}) < l(w′_{0}) .

Let q = bwuq_{−} , where b ∈ B , u ∈ U_{w} and q_{−} ∈ Q_{−} . We have bwu = qq_{−}^{−1} ∈ Q . As l(w) < l(w′_{0}) we may apply the induction assumption. Hence qq_{−}^{−1} ∈ Q_{+}Q_{−} , consequently q ∈ Q_{+}Q_{−} , as was to be proven.

Remark. If we replace both Borel subgroups with Q in the Bruhat decomposition (3) we do not in general obtain a decomposition for G. In particular, for the affine case if we take Q as in Example 3 then G ≠ QW Q .

Closure patterns. We fix a system of positive roots Ψ and let ≤_{Ψ} denote the partial order on W generated by the relations s_{γ}w <_{Ψ} w for w ∈ W and γ ∈ Δ_{+}^{re} \ wΨ; this is the order denoted ≤_{A} in [6, 1.5], where A is the “initial section” A = { s_{α} | Δ_{+}^{re} \ Ψ } of the reflections of W. An alternative description of ≤_{Ψ} may sometimes be useful. Define a (non-standard) length function l_{Ψ}: W → ℤ by setting

l_{Ψ}(w) = l(w) − 2♯((Δ_{+}^{re} \ Ψ) ∩ w^{−1}(Δ_{−}^{re}))

for w ∈ W. Then by [6, 1.7], one has v ≤_{Ψ} w in W iff there is a sequence v = v_{0}, v_{1}, …, v_{n} = w of elements of W with v_{i}v_{i−1}^{−1} a reflection in W (i.e. a conjugate of a simple reflection) and l_{Ψ}(v_{i}) = l_{Ψ}(v) + i for i = 1, …, n. If Ψ = Δ_{+} (resp., Ψ = Δ_{−}) then ≤_{Ψ} is the usual Chevalley (Bruhat) order on W (resp., reverse Chevalley order).

We may now restate Corollary 1 more familiarly as

Corollary 1′. Let Q = Q_{Ψ}. If s = s_{α} ∈ S where α ∈ Π and w ∈ W , then

BsBwQ = \{ \begin{matrix}BwQ ∪ BswQ & if sw <_{Ψ} w \\ BswQ & if sw >_{Ψ} w. \end{matrix}

Proof. One has w^{−1}(α) ∈ Ψ iff α ∈ wΨ ∩ Δ_{+}^{re} i.e. iff sw >_{Ψ} w.

Example. For Ψ as in example 3, the order ≤_{Ψ} on W is isomorphic to the order on the alcoves of an affine Weyl group considered by Lusztig in [17] (by [7]).

We now wish to extend to the orders ≤_{Ψ} the usual interpretation of Chevalley (Bruhat) order in terms of closure patterns (of Schubert cells, or (B, B)-double cosets). To this end, we introduce on G the Zariski topology defined by strongly regular functions as in [15, 2E]. We recall here that Zariski topology is G- biinvariant, that ϕ_{α}: SL_{2}(𝔽) → G_{α} is a homeomorphism (where SL_{2} has the usual Zariski topology), and that U_{w} ≅ 𝔽^{l(w)}. We now prove

Theorem 4. Let Q = Q_{Ψ}. Then for any w ∈ W,

\overline{QwB} = ⋃_{v:v^{−1}≤_{Ψ}w^{−1}} QvB.

Note that v^{−1} ≤_{Ψ} w^{−1} is not equivalent to v ≤_{Ψ} w in general. The proof of Theorem 4 is essentially the same as that of [14, 3.4], using integrable highest weight modules. We fix a strictly dominant integral weight Λ, and consider the corresponding integrable highest weight (left) 𝔤-module L(Λ), with highest weight vector v_{Λ} ∈ L(Λ)_{Λ}. We endow L(Λ) with the Zariski topology defined by strongly regular functions [15, 3A].

Introduce the partial order ≤_{Ψ} on the W -orbit W Λ of Λ, generated by μ ≤_{Ψ} λ if μ − λ ∈ ℚ_{≥0}Ψ and μ = s_{α}(λ) for some α ∈ Δ^{re}. The map W → W Λ given by w ↦ wΛ is a bijection, and we claim that

wΛ ≤_{Ψ} vΛ iff w^{−1} ≤_{Ψ} v^{−1}.  (9)

To check (9), one may assume by definition of the orders ≤_{Ψ} that w = vs_{α} for some α ∈ Δ_{+}^{re}. Then wΛ ≤_{Ψ} vΛ iff −⟨Λ, α^{∨}⟩vα ∈ ℚ_{≥0}Ψ i.e. iff α ∈ Δ_{+}^{re} \ v^{−1}Ψ, as needed.

Let V = G(𝔽v_{Λ}); as shown in [14], this is a Zariski closed subset of L(Λ) (defined by quadratic “Plucker polynomials”), and the group 𝔽^{∗} × U_{−} acts simply transitively on V \ {0}. We recall an important fact concerning V from [14]. For v ∈ L(Λ), write v = ∑_{λ} v_{λ} with v_{λ} ∈ L(Λ)_{λ}, put supp(v) = {λ | v_{λ} ≠ 0} and let S(v) denote the convex hull of supp(v). Then

for v ∈ V, the vertices of the polyhedron S(v) lie in the W -orbit of λ, and the edges of S(v) are parallel to real roots. (10)

For λ ∈ W Λ, set

V(λ)_{Ψ} = { v ∈ V | λ ∈ supp(v), supp(v) − λ ⊂ ℚ_{≥0}Ψ }.  (11)

Now we have the following analogue of [14, Theorem 1].

Proposition 2. i) V \ {0} is the disjoint union of the V(λ)_{Ψ} for λ ∈ W Λ

ii) for λ = wΛ ∈ W Λ, the group Q_{Ψ} ∩ wU_{−}w^{−1} acts simply transitively on V(λ)_{Ψ}.

iii) for λ ∈ W Λ, one has \overline{V(λ)_{Ψ}} \ \{0\} = ∪_{μ≤_{Ψ} λ} V(μ)_{Ψ}.

Proof. Write Q for Q_{Ψ}. The assertion i) follows from (10) above; note also the V(λ)_{Ψ} are Q-invariant. For ii), observe one has a decomposition

Q = (Q ∩ wU_{−}w^{−1})(Q ∩ wU_{+}w^{−1})

using Theorem 3, so it suffices to show that

𝔽^{∗} × Q acts transitively on V(λ)_{Ψ}.  (12)

This is proved as in [14] by “killing edges of S(v).” More precisely, for v ∈ V(λ)_{Ψ}, set Φ(λ) = { α ∈ Ψ | s_{α}(λ) <_{Ψ} λ) },

Φ′(v) = { α ∈ Φ(λ) | [λ, s_{α}(λ)] is an edge of S(v) }

and Φ(v) = Φ(λ) ∩ ℚ_{≥0}(S(v) − λ). If α ∈ Φ′(v), the argument of loc. cit. shows that there exists t ∈ 𝔽 so α ∉ Φ(x_{α}(t)v) ⊂ Φ(v). Using this repeatedly, one finds u ∈ Q so Φ′(uv) = ∅, so uv ∈ L(Λ)_{λ} by (10), proving (12). Finally, iii) is proved from i), ii) in exactly the same way as Theorem 1(c) of [14].

Proof of Theorem 4. The map ϕ : G → V with g ↦ gv_{Λ} is Zariski continuous, with ϕ^{−1}(Q(L(Λ)_{w(Λ)})) = QwB by Theorems 1 and 2. Taking (9) into account, part (iii) of Proposition 2 gives

\overline{QwB} ⊂ ⋃_{v:v^{−1}≤_{Ψ}w^{−1}} QvB.

To prove the reverse inclusion, its sufficient to show for w ∈ W and α ∈ Δ_{+}^{re} \ w^{−1}(Ψ), that \overline{QwB} ⊃ Qws_{α}B. Since Q_{w^{−1}Ψ} = w^{−1}Qw and the Zariski topology is G-biinvariant, one may even assume in addition that w = 1. Write α = w(β) for some w ∈ W and β ∈ Π. Then U_{±α} = wU_{±β}w^{−1} and we set H_{α} = wH_{α}w^{−1}. Recall ϕ_{β}: SL_{2}(𝔽) → G_{β} is a Zariski homeomorphism. Now Q ⊃ U_{−α} and B ⊃ H_{α}U_{α}, so one has by biinvariance again that

\overline{QB} ⊃ Q\overline{U_{−α}H_{α}U_{α}}B ⊃ Qs_{α}B.

Intersection Patterns. In this section, Ψ is a fixed set of positive roots. For A ⊂ G, write A · B := { gB | g ∈ A } ⊂ G/B. Here, we study intersection patterns amongst the sets By · B, Q_{Ψ}w · B and Q_{−Ψ}x · B for x, y, w ∈ W. The basic result is Theorem 5, which is similar to [4] (cf. also [5], [11]). The proof here is very similar to that in [5], but we give the details involving the ordering ≤_{Ψ}.

Fix y ∈ W and a reduced expression y = s_{1} … s_{k}. Let D_{y} be the set of sequences σ = (σ_{0}, σ_{1}, …, σ_{k}) ∈ W^{k+1} satisfying the conditions (a)–(c) below;

(a) σ_{0} = e, the identity element of W

(b) σ_{j} ∈ {σ_{j−1}, s_{j}σ_{j−1}} for j = 1, …, k

(c) if s_{j}σ_{j−1} >_{Ψ} σ_{j−1}, then σ_{j} = s_{j}σ_{j−1}.

For σ ∈ D, we set m(σ) = ♯{j | σ_{j} >_{ψ} σ_{j−1}}, n(σ) = ♯{j | σ_{j} = σ_{j−1}} and π(σ) = σ_{k}^{−1} ∈ W.

Now we define a map η : U_{y^{−1}} → D_{y} by η(u_{1}) = (σ_{0}, …, σ_{k}) where the σ_{j} ∈ W are determined by u_{1}s_{1} … s_{j} ∈ Q_{Ψ}σ_{j}^{−1}B (the conditions (a)–(c) are easily verified, using Corollary 1′ for (c)).

Theorem 5. i) For any σ ∈ D_{y}, the set η^{−1}(σ) is a locally closed subset of U_{y^{−1}} homeomorphic to 𝔽^{m(σ)} × (𝔽^{∗})^{n(σ)}.

ii) For any x ∈ W,

By · B ∩ Q_{Ψ}x^{−1} · B = ⋃_{σ∈D:π(σ)=x} η^{−1}(σ) · B.

Corollary 6. For x, y ∈ W, one has

i) By · B ∩ Q_{Ψ}x · B = ∅ unless y^{−1} ≤_{Ψ} x^{−1} and x ≤ y, where ≤ denotes Chevalley (Bruhat) order on W

ii)

Q_{−Ψ}y · B ∩ Q_{Ψ}x · B ⊂ ⋃_{v∈I(y,x)} Bv · B

where I(y, x) = { v ∈ W | y^{−1} ≤_{Ψ} v^{−1} ≤_{Ψ} x^{−1} }.

Proof of the Corollary. Part ii) follows immediately from i) on noting that ≤_{−Ψ} is the reverse of the order ≤_{Ψ}. For one part of i), one shows

ByB ⊆ ⋃_{x≤y} Q_{Ψ}xB

by induction on l(y) using Corollary 1. For the remaining part, suppose that the intersection in i) is non-empty. Then Theorem 5 implies that there exists σ = (σ_{0}, …, σ_{k}) ∈ D_{y} with π(σ) = σ_{k}^{−1} = x. Set y_{j} = s_{j} … s_{k} for j = 1, …, k + 1, with y_{k+1} = e. For j = 0, …, k, let t_{j} = y_{j+1}^{−1}σ_{j}. Then t_{j−1} = t_{j} unless σ_{j−1} = σ_{j}, in which case t_{j−1} = y_{j+1}^{−1}s_{j}σ_{j} <_{Ψ} y_{j+1}^{−1}σ_{j} = t_{j} since s_{j}σ_{j−1} <_{Ψ} σ_{j−1} and s_{j}y_{j+1} > y_{j+1}. This gives, as needed,

y^{−1} = t_{0} ≤_{Ψ} t_{1} ≤_{Ψ} … ≤_{Ψ} t_{k} = x^{−1}.

For the proof of the theorem, we keep the notation y_{j} = s_{j} … s_{k} from above, and also abbreviate U_{y_{j}^{−1}} as U_{j} and U′_{y_{j}} as U′_{j}, for j = 1, …, k + 1, so one has U_{+} = U_{j}y_{j}U′_{j}y_{j}^{−1} with uniqueness of expression. For w ∈ W, s ∈ S with l(sw) > l(w), one has U_{w} ⊂ U_{sw} by (5), and then Corollary 4 with Q = w^{−1}s^{−1}U_{+}sw gives U′_{sw} ⊂ U′_{w}. Hence U′_{1} ⊂ U′_{2} ⊂ . . ..

Lemma 4. Fix σ ∈ D_{y} and an integer j with 1 ≤ j ≤ k. Set Ω(σ, j) equal to 𝔽, {0} ⊂ 𝔽 or 𝔽^{∗} according as whether σ_{j} >_{Ψ} σ_{j−1}, σ_{j} <_{Ψ} σ_{j−1} or σ_{j} = σ_{j−1}. Then there is an injective map f_{j} : Ω(σ, j) × U_{j+1} → U_{j} with the following properties:

(a) the image of f_{j} is either U_{j}, s_{j}U_{j+1}s_{j}^{−1} or U_{j} \ s_{j}U_{j+1}s_{j}^{−1} according as σ_{j} >_{Ψ} σ_{j−1}, σ_{j} <_{Ψ} σ_{j−1} or σ_{j} = σ_{j−1}, and f_{j} is a homeomorphism onto its image

(b) one has σ_{j−1}^{−1}f_{j}(t, u_{j+1})y_{j} = b_{j}σ_{j}^{−1}u_{j+1}y_{j+1}v_{j+1} for some b_{j} ∈ B_{Ψ} := HQ_{Ψ} and v_{j+1} ∈ U′_{j+1}.

Proof of the Lemma. Write s_{j} = s_{α}, α ∈ Π.

First, suppose that σ_{j} >_{Ψ} σ_{j−1}, so σ_{j} = s_{j}σ_{j−1} and α ∉ σ_{j−1}Ψ. Define f_{j}(t, u_{j+1}) = x_{α}(t)s_{j}u_{j+1}s_{j}^{−1}. The required properties of f_{j} follow immediately on noting that x_{α}(t) ∈ Q_{σ_{j−1}Ψ} = σ_{j−1}Q_{Ψ}σ_{j−1}^{−1}.

The second case is σ_{j} <_{Ψ} σ_{j−1}; one again has σ_{j} = s_{j}σ_{j−1}. The map f_{j} defined by f_{j}(0, u_{j+1}) = s_{j}u_{j+1}s_{j}^{−1} has the required properties.

The remaining case is that σ_{j} = σ_{j−1}; here, s_{j}σ_{j−1} <_{Ψ} s_{j−1}, so α ∉ σ_{j−1}Ψ. Write x_{α}(t)u_{j+1} = u′_{j+1}y_{j+1}v_{j+1}^{−1}y_{j+1}^{−1} for some (uniquely determined) u′_{j+1} ∈ U_{j+1} and v_{j+1} ∈ U′_{j+1}, and set f_{j}(t, u_{j+1}) = x_{α}(t^{−1})s_{j}u′_{j+1}s_{j}^{−1} ∈ U_{j}. As in the proof of (3.2) in [5], f_{j} is a homeomorphism onto its image U_{j} \ s_{j}U_{j+1}s_{j}^{−1}. Now we compute

\begin{aligned}σ_{j−1}^{−1}f_{j}(t, u_{j+1})y_{j} & = σ_{j−1}^{−1}x_{α}(t^{−1})s_{j}u′_{j+1}y_{j+1} \\ & = σ_{j−1}^{−1}x_{α}(t^{−1})s_{j}x_{α}(t)u_{j+1}y_{j+1}v_{j+1}. \end{aligned}

We may write x_{α}(t^{−1})s_{j}x_{α}(t) = hs_{j}x_{α}(−t^{−1})s_{j}^{−1} for some h ∈ H. Setting

b_{j} = σ_{j−1}^{−1}hs_{j}x_{α}(−t^{−1})s_{j}^{−1}σ_{j},

we have

σ_{j−1}^{−1}f_{j}(t, u_{j+1})y_{j} = b_{j}σ_{j}^{−1}u_{j+1}y_{j+1}v_{j+1}.

Finally, observe that b_{j} ∈ B_{Ψ} since −α ∈ σ_{j−1}Ψ implies

σ_{j−1}b_{j}σ_{j−1}^{−1} ∈ HU_{−α} ⊂ B_{σ_{j−1}Ψ} = σ_{j−1}B_{Ψ}s_{j−1}^{−1}.

Proof of Theorem 5. The map U_{y^{−1}} = U_{1} → By · B given by u → u · B is bijective, and clearly maps η^{−1}(σ) into Q_{Ψ}π(σ) · B by definition of η and π. Since U_{1} is the disjoint union of the sets η^{−1}(σ) for σ ∈ D_{y}, we need only prove (1).

Fix σ ∈ D and define subsets A_{j} ⊂ U_{j} for k + 1 ≥ j ≥ 1 by setting A_{k+1} = {e} and A_{j} = f_{j}(Ω(σ, j) × A_{j+1}) for k ≥ j ≥ 1. The lemma implies that A_{j} is a locally closed subset of U_{j} homeomorphic to 𝔽^{m(j,σ)} × (𝔽^{∗})^{n(j,σ)} where

m(j, σ) = ♯{ p | j ≤ p ≤ k and Ω(σ, p) = 𝔽 },
n(j, σ) = ♯{ p | j ≤ p ≤ k and Ω(σ, p) = 𝔽^{∗} }.

Since m(σ) = m(1, σ) and n(σ) = n(1, σ), the theorem will be proved if we show that A_{1} = η^{−1}(σ). Fix u_{1} ∈ U_{1}.

Suppose for j = 1, …, p we have u_{j+1} ∈ U_{j+1} and t_{j} ∈ Ω(σ, j) with u_{j} = f_{j}(t_{j}, u_{j+1}). Choosing b_{j} ∈ B_{Ψ} and v_{j} ∈ U′_{j} as in the lemma, it follows immediately by induction on j that for 1 ≤ j ≤ p,

u_{1}s_{1}s_{2} … s_{j} = b_{1}b_{2} … b_{j}σ_{j}^{−1}u_{j+1}y_{j+1}v_{j+1} … v_{3}v_{2}y_{j+1}^{−1}.  (13)

Recalling U′_{1} ⊂ U′_{2} ⊂ …, the right hand side of (13) is an element of Q_{Ψ}σ_{j}^{−1}B. In particular, if u_{1} ∈ A_{1}, one can take p = k in the above and deduce that η(u_{1}) = σ.

Conversely, suppose that u_{1} ∈ η^{−1}(σ). We prove we have u_{j+1} and t_{j} as in the previous paragraph, for j = 1, …, p, by induction on p. Suppose inductively this is true for p − 1, so in particular (13) holds if j = p − 1. Choose δ ∈ W so that u_{p}s_{p} ∈ Q_{σ_{p−1}Ψ}δ^{−1}B = σ_{p−1}Q_{Ψ}σ_{p−1}^{−1}δ^{−1}B. Then, recalling U′_{j} ⊂ U′_{j+1}, one has u_{1}s_{1} … s_{p−1}s_{p} ∈ Q_{Ψ}σ_{p−1}^{−1}δ^{−1}B so δσ_{p−1} = σ_{p}. To complete the proof, we just need to show that u_{p} is in the image of f_{p}, which is given in the Lemma. Write s_{p} = s_{α} for α ∈ Π. We must show that

i) if δ = s_{p} and α ∉ σ_{p−1}Ψ, then u_{p} ∈ s_{p}U_{p+1}s_{p}^{−1} and

ii) if δ = 1 then u_{p} ∈ U_{p} \ s_{p}U_{p+1}s_{p}^{−1}

(the other case being trivial).

Consider the situation i). We have s_{p}^{−1}u_{p}s_{p} ∈ Q_{s_{p}σ_{p−1}Ψ}B = Q_{σ_{p}Ψ}B. Since u_{p} ∈ U_{p}, we may write s_{p}^{−1}u_{p}s_{p} = x_{−α}(t)v for some t ∈ 𝔽 and v ∈ U_{p+1}, and we must show t = 0. But if t ≠ 0, we get

s_{j}B = x_{α}(−t^{−1})(x_{−α}(t)v)v^{−1}x_{α}(−t^{−1})B ⊂ Q_{σ_{p}Ψ}B

(since α ∈ σ_{p}Ψ ∩ Δ_{+}^{re}), contrary to Theorem 2. In the other situation ii), one has u_{p}s_{p} ∈ Q_{σ_{p−1}Ψ}B so s_{p}^{−1}u_{p}s_{p} ∈ Q_{s_{p}σ_{p−1}Ψ}s_{p}B. Write s_{p}^{−1}u_{p}s_{p} = x_{−α}(t)v as in case i). Then x_{−α}(t) ∈ Q_{s_{p}σ_{p−1}Ψ}s_{p}B so Theorem 2 implies t ≠ 0, s_{p}^{−1}u_{p}s_{p} ∉ U_{p+1} as required. This completes the proof.

Additional Remarks. We make some remarks concerning non-standard Schubert-type decompositions of G/B. Fix a strictly dominant integral weight Λ, and endow L(Λ) with the Zariski topology. Recall the notations V, V(λ)_{Ψ} from the proof of Theorem 4, and note that G/B injects naturally into the set ℙ(L(Λ)) of lines of L(Λ) as in [14].

For any λ ∈ W Λ, one has the closed subset \overline{C}_{λ,Ψ} := ℙ(\overline{V(λ)_{Ψ}}) of the (infinitedimensional) projective space ℙL(Λ). For Ψ_{0} = Δ_{+}, \overline{C}_{λ,Ψ_{0}} is the the finite Schubert variety \overline{C}_{λ} and \overline{C}_{λ,−Ψ_{0}} is the the cofinite Schubert variety \overline{C}^{λ} of [14, 15]. In general, \overline{C}_{λ,Ψ} is the directed union of its intersections with the finite Schubert varieties, these intersections being finite-dimensional projective varieties.

For certain Ψ of particular interest for the representation theory of 𝔤, every interval in the order ≤_{Ψ} on W is finite (see [6, 6.4 and 6.7]). We suppose henceforward for simplicity that Ψ has this property. Since the ordinary Chevalley order is directed, Corollary 6(b) implies that each (closed) intersection

\overline{C}_{λ,μ,Ψ} := \overline{C}_{λ,Ψ} ∩ \overline{C}_{μ,−Ψ}  (15)

will be contained in some finite Schubert variety, and hence aquires a natural structure of finite-dimensional projective variety (essentially independent of the choice of Λ, since this is known for the finite Schubert varieties). One has that \overline{C}_{λ,μ,Ψ} is non-empty precisely when μ ≤_{Ψ} λ (but see the example following).

By [6,7] the (non-empty, finite) open intervals (μ, λ) in the order ≤_{Ψ} on W Λ (or ≤_{Ψ} on W) are of two types; spherical (i.e the order complex of the open interval is a combinatorial sphere) or non-spherical (the order complex is a combinatorial ball); many very interesting constructions (e.g. Kazhdan-Lusztig polynomials, see [6]) can be extended to the former, but either fail or give pathological results for non-spherical intervals. It can be shown (cf. [8]) that for a non-empty spherical interval (v, w) in W, one has

♯{ α ∈ Δ_{+}^{re} | v ≤_{ψ} s_{α}u ≤_{Ψ} w } ≥ l_{Ψ}(w) − l_{Ψ}(v)  (16)

for all v ≤_{Ψ} u ≤_{Ψ} w in W, but that this result need not hold for a non-spherical interval. Now (16) has been established for ordinary Chevalley order [3] by studying H-invariant curves in the intersections of finite and cofinite Schubert varieties,, and one might therefore expect similar results to apply to \overline{C}_{λ,μ,Ψ} for spherical intervals (μ, λ) in W Λ in the order ≤ Ψ. It seems likely that (also only for spherical intervals) there should be a decomposition of \overline{C}_{λ,μ,Ψ} into locally closed subsets (parametrized by the data in [6, 3.1] used to construct the R-polynomial) similar to that in Theorem 5.

Example. Here we show that, in contrast to the classical situation (where Ψ = Δ_{+}^{re}), one can have C_{λ,μ,Ψ} := V(λ)_{Ψ} ∩ V(μ)_{−Ψ} = ∅ even though μ ≤_{Ψ} λ (notation as in the above remark). First, note that by (2) in the proof of Theorem 4 and the definition of ≤_{Ψ}, one has that

C_{λ,μ,Ψ} = { v ∈ V | {μ, λ} ⊂ supp(v) ⊂ [μ, λ] }

where I = [μ, λ] = { ν ∈ W Λ | μ ≤_{Ψ} ν ≤_{Ψ} λ }; also, for v in C_{λ,μ,P si}, the edges of supp(v) are parallel to real roots. Suppose now that the poset I is a chain of cardinality three (by [6] any finite non-spherical interval in an order ≤_{Ψ} contains such a chain as a subinterval). Then μ − λ is a linear combination with strictly positive coefficients of two “adjacent” real roots lying on a plane in 𝔥, so it is not a multiple of a real root, and hence C_{λ,μ,Ψ} = ∅. As a specific example, for Ψ as in Example 3 in the case of SL_{2}, every length two subinterval of ≤_{Ψ} is a chain of cardinality three.

To conclude these remarks, we make an observation on a relationship of the decompositions here to certain Hecke algebra modules associated to the orders ≤_{Ψ} on W.

The results of this paper excepting Theorems 4 and Proposition 2 can be proved by essentially identical arguments for the version of G in [11]. (For the topological statements in Theorem 5, one should take the field algebraically closed; then the sets η^{−1}(σ) are locally closed subvarieties of U_{y^{−1}} in a natural structure of unipotent algebraic group. We haven’t pursued rationality questions over other fields.) For the following remark, consider the group G from [11] for 𝔽 a finite field of q elements.

Let F denote the set of complex-valued functions on G/B. For x ∈ W, define a ℂ-linear map T_{x}: F → F by the formula

(T_{x}f)(g · B) = ∑_{z∈g(Bx·B)} f(z)

for g ∈ G.

It is easily checked (cf. [12]) that the T_{x} span over ℂ a copy H of the Iwahori-Hecke algebra of W, with parameter q. In fact, for s ∈ S and x ∈ W one has

T_{s}T_{x} = \{ \begin{matrix}T_{sx} & if sx > x \\ qT_{sx} + (q − 1)T_{x} & if sx < x \end{matrix}

where ≤ denotes Chevalley order.

Setting t_{w} = q^{l_{Ψ}(w)}χ_{A_{w}} for w ∈ W, where χ_{A_{w}} is the characteristic function of the subset A_{w} := Q_{Ψ}w^{−1} · B of G/B, one can show similarly (or deduce from Theorem 5) that for s ∈ S,

T_{s}t_{w} = \{ \begin{matrix}t_{sw} & if sw >_{Ψ} w \\ qt_{sw} + (q − 1)t_{w} & if sw <_{Ψ} w. \end{matrix}

In [6], a module H_{A} for the generic Iwahori-Hecke algebra of W was associated to the initial section A = { s_{α} | α ∈ Δ_{+}^{re} \ Ψ}. Specializing the indeterminate there to q ∈ ℂ, the resulting H-module is a natural “completion” of the H-submodule of F spanned here by the t_{w} for w ∈ W.

References

1. N. Bourbaki, Groupes et algèbres de Lie, Chapitres IV, V, VI. Hermann, Paris, 1968.

2. F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Publ.Math. IHES, 41(1972).

3. J. Carrell and D. Peterson, The Bruhat graph of a Coxeter group, a conjecture of Deodhar and rational smoothness of Schubert varieties, preprint.

4. C. W. Curtis, A finer decomposition of Bruhat cells, preprint.

5. V. Deodhar, On some geometric aspects of Bruhat ordering I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), 499–511.

6. M. Dyer, Hecke algebras and shellings of Bruhat intervals II; twisted Bruhat orders, Contemp. Math. 139 (1992), 141–165.

7. M. Dyer, Quotients of twisted Bruhat orders, to appear, Jour. of Alg.

8. M. Dyer, The nil Hecke ring and Deodhar’s conjecture on Bruhat intervals, Invent. Math. 111 (1993), 571–574.

9. V. Futorny and H. Saifi, Affine Lie algebras and Verma type modules, Preprint, 1992.

10. H.P. Jacobsen and V.G. Kac, A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras, II, J.Funct. Anal., 82(1989), 69-90.

11. Z. Hadad, A Coxeter group approach to Schubert varieties, pp. 157–165 in “Infinite Dimensional groups with Applications,” edited by V. G. Kac, MSRI Publications Volume 4, Springer-Verlag, 1985.

12. Z. Hadad, Infinite-dimensional flag varieties, Ph. D. thesis, M. I. T. (1984).

13. V.G. Kac and D.H. Peterson, Defining relations of certain infinite dimensional groups, Asterisque, Numero hors serie, 1985.

14. D. H. Peterson and V. G. Kac, Infinite dimensional flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. U. S. A.,80 (1983)1778–1782.

15. V. G. Kac and D. H. Peterson, Regular functions on certain infinite-dimensional groups, pp. 141–166 in “Arithmetic and Geometry” Progress in Math. 36, Birkhauser, Boston, 1983.

16. V.G. Kac, Infinite-dimensional Lie algebras, Birkhauser, Boston, 1983.

17. G. Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns, Adv. in Math. 37 (1980), 121–164.

18. R.V. Moody and K. Teo, Tits’ systems with crystallographic Weyl groups, J.Algebra, 21(1972), 178-190.