September 12, 1999 Preliminary Version

DISPLACEMENT FUNCTIONS ON THE BRUHAT–TITS BUILDING

Allen Moy

Abstract. Suppose G is a semisimple Lie group and K a maximal compact subgroup. Let ℬ be the symmetric Riemannian space ℬ = G/K. If g an isometry of ℬ, the displacement function d_{g} is the function on ℬ which measures the geodesic distance from a point x in ℬ to the translated point gx. A very striking property of the displacement function d_{g} is that it is geodesically convex. We show displacement functions for the Bruhat-Tits building also exhibit this convexity property and remark on its use in representation theory.

1. Introduction

1.1. Let G be a semisimple Lie group and K a maximal compact subgroup. For g ∈ G, the displacement function d_{g} is the function on the symmetric Riemannian space ℬ = G/K which measure the geodesic distance from a point x ∈ ℬ to the translated point gx, that is d_{g}(x) := dist(x, gx). Recall that any two distinct points x_{a}, x_{b} ∈ ℬ, can be connected by a unique geodesic path x(t) parametrized by arclength so that x(0) = x_{a} and x(c) = x_{b} (c = dist(x_{a}, x_{b})). A function f on ℬ is geodesically convex if (1 − α)f(x(0)) + αf(x(c)) ≥ f(x(αc)) for all 0 ≤ α ≤ 1. A very striking property of the displacement function d_{g} is that it is geodesically convex [Ms:§5].

Let k denote a nonarchimedean local field of arbitrary characteristic and let G be a connected reductive algebraic group defined over k. Denote by 𝒢 the group G(k) of krational points of G. We equip 𝒢 with the natural locally compact topology induced from that on k. Let ℬ = ℬ(𝒟G, k) be the Bruhat–Tits building of the derived group of G/k. In the nonarchimedean setting, ℬ is an analogue of the symmetric Riemannian space. Some of the properties shared by both symmetric Riemannian spaces and buildings are

i) They are topologically retractible to a point, i.e. homotopically equivalent to a point. ii) Given two points in ℬ (either the symmetric Riemannian space or the building),

there is a unique geodesic line segment connecting the two points. iii) The real and p-adic groups act isometrically on the respective spaces. In the p-

adic case, the building ℬ is a polysimplicial complex and the p-adic group acts ploysimplicially.

iv) In the real case the elements which leave invariant a maximal flat [Ms] is a group whose connected component is a toral group. In the nonarchimedean case the same is true for an apartment in the building.

Set Aut(ℬ) to be the full group of polysimplicial isometries of ℬ. Our main goal here is to prove the following result.

1.2. Displacement Function Convexity. Suppose ℬ = ℬ(𝒟G, k). For any g ∈ Aut(ℬ), the displacement function d_{g} is a geodesically convex function on ℬ.

A consequence of the convexity of the displacement function d_{g} is that for any r ≥ 0, the set d_{g}^{−1}([0, r]) = {x ∈ ℬ|d_{g}(x) ≤ r} is convex. Set

d(g) := inf_{x∈ℬ} d_{g}(x).

For semisimple Lie groups, Mostow showed [Ms:5.3] that d_{g} obtains its infinimum if and only if g is a semisimple element. In the nonarchimedean setting we show the infinimum is obtained for any g. The difference in the two situations can be heuristically explained by the observation that any unipotent element of 𝒢 is contained in a compact subgroup, while in a semisimple Lie group the identity element is the only unipotent element contained in a compact subgroup.

More generally, if Ξ is any nonempty bounded set of 𝒢, motivated by analogies with depth functions and building sets for the Lie algebra introduced in [MP3], we define the function

d_{Ξ}(x) := sup_{h∈Ξ} d_{h}(x).

In particular, d_{{g}} = d_{g}. The function d_{Ξ} is convex and we show it attains its infinimum. The set ℬ(Ξ) where d_{Ξ} achieves its minimum we call the building set of Ξ. For any x ∈ ℬ, the functions d_{g} allow us to define a filtration of the group 𝒢 by compact modulo center sets 𝒢_{x,r} (r ≥ 0), which are analogues of the filtrations defined in [MP1,MP2]. Building sets permit us to define a notion of nondegenerate and degenerate cosets in 𝒢 and to then prove a descent condition which is analogous to the descent condition established for the Lie algebra and its dual in [MP1,MP2].

We remark that it was the author’s desire to formulate a group analogue of the Lie algebra descent condition which motivated investigation into the displacement function. In their proofs of the Howe Conjecture for the Lie algebra, Howe [Ho] and Harish-Chandra [HC] use notions of depth and descent as key tools, but unfortunately neither was able to generalize their approach to prove the group Howe Conjecture. Later, using a very different method, Clozel proved the group version of the Howe Conjecture in [C] for arbitrary reductive groups. It is rather interesting that for SL(2), the displacement function and the descent condition can be used as tools to give a new proof of the group Howe Conjecture along lines very close to Howe’s and Harish-Chandra’s proof for the Lie algebra. This is done in the final section. The basic idea of the proof can in fact be extended to other groups besides SL(2). We hope to report on the extension to other groups, which is joint work with Barbasch, soon.

The author thanks Dan Barbasch, Alexandre Barvinok, Stephen De Backer, Samuel Evens, Thomas Hales, Ju-Lee Kim, Dan Mostow, Frank Raymond and John Stembridge for discussions during the period this paper was written. The author also thanks Marie-France Vignéras and the Université of Paris VII for the invitation to visit and the hospitality shown May and June 1998 where part of this paper was written. Finally the author was supported in part by the National Science Foundation grant DMS 9801264.

2. Preliminaries on convex functions

2.1. We recall some definitions and facts about continuous convex and concave functions. A subset Z ⊂ ℝ^{n} is convex if for any x, y ∈ Z, the line segment x(t) = tx + (1 − t)y for 0 ≤ t ≤ 1 is contained in Z. A continuous function f : Z ⟶ ℝ on a convex subset Z is convex if

f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) for all 0 ≤ t ≤ 1.

We shall always assume convex functions are continuous. A function is concave if and only if its negative is convex.

2.2. Proposition. Let V = ℝ^{n} with its usual Euclidean metric, ∥z∥ = \sqrt{z · z}. Suppose L ∈ GL(V) and v ∈ V. Let g be the affine transformation gx = g(x) := L(x) + v. Then the displacement function

d_{g}(x) := dist(x, gx) = ∥x − gx∥

is a convex function on V.

Proof. For x, y ∈ V, and α ≥ 0, β ≥ 0, and α + β = 1, we have (α gx + β gy) = α L(x) + α v + β L(y) + β v = L(α x + β y) + v = g(α x + β y). Therefore,

\begin{aligned}α d_{g}(x) + β d_{g}(y) & = α∥x − gx∥ + β∥y − gy∥ ≥ ∥α x − α gx + β y − β gy∥ \\ & = ∥(α x + β y) − (α gx + β gy)∥ = ∥(α x + β y) − g(α x + β y)∥ \end{aligned}

whence d_{g} is a convex function. □

2.3. The next series of results tell us some special but useful situations when the distance function d(t) := dist(p_{1}(t), p_{2}(t)) between two curves p_{1}(t) and p_{2}(t) is convex. Roughly, the basic result is that if p_{1}(t) is a parametrized geodesic line and p_{2}(t) is a ‘bent’ line then d is convex.

2.4 Lemma. Suppose a < b < c are real numbers.

i) Let f : [a, c] ⟶ ℝ be a continuous function. If f is convex on each of the subintervals [a, b] and [b, c], and f is also convex in a neighborhood of b, then f is convex on the entire interval [a, c]. ii) Let f_{1} : [a, b] ⟶ ℝ and f_{2} : [b, c] ⟶ ℝ be convex differentiable functions so that

f_{1}(b) = f_{2}(b), and f′_{1}(b) = f′_{2}(b). Then the function f obtained by splicing f_{1} and f_{2} together, i.e.

f(t) = \{ \begin{matrix}f_{1}(t) for a ≤ t ≤ b \\ f_{2}(t) for b ≤ t ≤ c. \end{matrix}

is convex on the interval [a, c].

2.5 Proposition. Suppose v_{1}, v_{2} ∈ V = ℝ^{n}. For any x_{1}, x_{2} ∈ V, set

(2.5.1)  p_{1}(t) := x_{1} + tv_{1} and p_{2}(t) := x_{2} + tv_{2}

i) The function

f(t) := dist(p_{1}(t), p_{2}(t))

is a convex function on ℝ. ii) Let H be any affine hyperplane in V which contains x_{2} and denote by s the reflection

across the hyperplane H. Set

(2.5.2)  p_{2′}(t) := \{ \begin{matrix}p_{2}(t)  t ≤ 0 \\ s(p_{2}(t)) t ≥ 0 \end{matrix}

Suppose either

(1) x_{1} ∈ H, or

(2) For small positive values of t, the points p_{1}(t) and p_{2}(t) are in the same half space of H in V.

Then the function

h(t) := dist(p_{1}(t), p_{2′}(t))

is a convex function on ℝ.

Proof. The proof of assertion i) is an easy corollary to Proposition 2.2. Indeed, we can find an affine transformation g : V ⟶ V so that g(p_{1}(t)) = p_{2}(t) and thus f(t) = dist(p_{1}(t), p_{2}(t)) = d_{g}(p_{1}(t)) is convex.

To prove ii), we first remark that by i), the function h is convex on each of the intervals (−∞, 0] and [0, +∞). To prove part (1), we consider two cases depending the on whether x_{1} and x_{2} are equal.

Case1: x_{1} = x_{2}. Here, the function h has a (global) minimum value of 0 at t = 0. This and convexity of h on each of the intervals (−∞, 0] and [0, +∞) implies h is convex.

Case 2: x_{1} ≠ x_{2}. The function k(t) = dist(s(p_{1}(t)), s(p_{2}(t))) is obviously a convex function and k(0) = dist(x_{1}, x_{2}) = f(0). Since x_{1} ≠ x_{2}, the functions f and k are smooth in a neighborhood of t = 0. We can resolve the differences p_{2}(t) − p_{1}(t) and s(p_{2}(t)) − p_{1}(t) into vectors parallel and perpendicular to H to obtain

\begin{aligned}p_{2}(t) − p_{1}(t) & = \{(x_{2} − x_{1}) + tv_{h}\} + tv_{a} \\ s(p_{2}(t)) − p_{1}(t) & = \{(x_{2} − x_{1}) + tv_{h}\} + tv_{b} \end{aligned}

The component u(t) := {(x_{2} − x_{1}) + tv_{h}} is parallel to H, is affine and u(0) ≠ 0. The vectors v_{a} and v_{b} are perpendicular to H. An elementary calculation shows that f and k are differentiable at the origin and f′(0) = k′(0). By Proposition 2.3, the ‘spliced’ function which is equal to f(t) for t ≤ 0 and k(t) for t ≥ 0 is convex. This spliced function is precisely dist(p_{1}(t), p_{2′}(t)) = h(t). This completes the proof of (1).

To prove part (2), we calculate that for t ≥ 0 dist(p_{1}(t), sp_{2}(t)) ≥ dist(p_{1}(t), p_{2}(t)). We conclude from this that h is convex in a neighborhood of t = 0. Since we also know h is convex in each of the intervals t ≤ 0 and t ≥ 0, it must in fact be convex. □

3. The retraction map and the Bruhat decomposition

3.1. We recall some notation and facts from [MP2] and [BT2]. Let k denote a nonarchimedean local field of arbitrary characteristic and let G be a connected reductive algebraic group defined over k. Let K be a fixed maximal unramified extension of k and Γ = Gal(K/k). Let 𝔬 and 𝔒 denote the ring of integers of k and K respectively, let 𝔭 be the maximal ideal of 𝔬 and let 𝔣 (resp. 𝔉) be the residue field of k (resp. K). Let 𝒟G be the derived group of G. The Bruhat–Tits buildings ℬ(𝒟G, K) and ℬ(𝒟G, k) of 𝒟G/K and 𝒟G/k respectively are canonically defined. The (enlarged) Bruhat–Tits building ℬ(G, K) (resp. ℬ(G, k)) of G/K (resp. of G/k or of 𝒢) is the product of ℬ(𝒟G, K) (resp. ℬ(𝒟G, k)) with X_{∗}(S) ⊗_{ℤ} ℝ; where S is the maximal K-split (resp. k-split) torus contained in the center of G. There is a natural action of the Galois group Γ on ℬ(G, K), and ℬ(G, k) can (and will) be identified with the subset of points of ℬ(G, K) fixed under Γ.

To each point x of the building ℬ(G, k) of G/k, Bruhat and Tits [BT2] have associated a subgroup of G(K), the parahoric subgroup determined by x, to be denoted by G_{x}, which is a certain subgroup of finite index in the isotropy group at x. (The parahoric subgroup is defined to be the inverse image in the isotropy group at x of the identity component of reduction mod 𝔭 of the 𝔬-group scheme associated with the isotropy subgroup. If G is a semi-simple simply connected group, then the parahoric subgroup coincides with the isotropy subgroup.) The parahoric subgroup is “defined” over k i.e. it is stable under Γ; the subgroup 𝒢 ∩ G_{x} will be denoted by 𝒢_{x} and it is by definition the parahoric subgroup of 𝒢 determined by (or associated with) the point x; it is a compact-open subgroup of 𝒢. A minimal parahoric subgroup is called an Iwahori subgroup. It is known that the Iwahori subgroups are conjugate to each other under 𝒢.

The buildings ℬ(𝒟G, k) ⊂ ℬ(𝒟G, K) are polysimplicial complexes. If F is a facet in ℬ(𝒟G, K), denote by G_{F} the parahoric subgroup of G(K) fixing the facet F. Recall that a point x ∈ F is an interior point if x does not lie in any facet E ⊊ F. It is the case that G_{F} = G_{x} for any interior point x in F. If the facet F lies in ℬ(𝒟G, k), 𝒢_{F} = G_{F} ∩ ℬ is the maximal parahoric subgroup of 𝒢 which fixes F.

3.2. Let S be a maximal k-split torus of G and let A = A(S) be the apartment in ℬ associated to the torus S. Let Z and N denote respectively the centralizer S and normalizer of S in G. Since S is defined over k, so are Z and N. If F is a facet in A, define

W_{F} (S) := (N(K) ∩ 𝒢_{F})/(Z(K) ∩ 𝒢_{F}).

Then W_{F} (S) is naturally isomorphic to the Coxeter group generated by the reflections in A of affine root hyperplanes containing the facet F.

3.3. Denote by G_{F,+} the maximal normal pro-p-subgroup of G_{F}. The quotient group G_{F} /G_{F,+} can be identified with the group of 𝔉-rational points of the maximal reductive quotient M_{F} of the reduction mod 𝔭 of the 𝔬-group scheme associated to the parahoric subgroup G_{F}. The Galois group Γ acts on the groups G_{F}, G_{F,+} and M_{F} with fixed points 𝒢_{F}, 𝒢_{F,+} and 𝒢_{F} /𝒢_{F,+} ≅ M_{F} (𝔣). Let S(K)_{c} be the maximal bounded subgroup of S(K). The image of S(K)_{c} in M_{F} (𝔉) is the 𝔉-rational points of a maximal split 𝔣-torus S_{F} ⊂ M_{F}. The 𝔣-rational points of the 𝔣-torus S is canonically the image of S(K)_{c} ∩ 𝒢_{F}.

3.4. Given two facets there is an apartment containing them both. Let A = A(S) be an apartment and C ⊂ A a chamber. For every apartment A′ containing C there is a unique isometry ϕ_{A′} : A′ ⟶ A which fixes C. All these isometries patch together in a coherent manner to yield a mapping ρ_{A,C} : ℬ ⟶ A called the retraction of ℬ onto A with center C. Clearly, geodesic segments are mapped by ρ_{A,C} onto broken lines, i.e. connected union of finitely many geodesic segments.

3.5. Proposition. (BT1:Proposition 7.4.20) Suppose C is a chamber in the apartment A. Set ρ = ρ_{A,C}. Then for all y, z ∈ ℬ

dist (ρ(y), ρ(z)) ≤ dist (y, z).

Suppose A = A(S) is an apartment in the building ℬ. If ψ is an affine root with respect to S, set

(3.5.1)  H_{ψ} := {x|ψ(x) = 0}.

If 𝒴 is a nonempty subset of A, set

(3.5.2) 𝒞(𝒴) := {x ∈ A | ψ(x) ≥ 0 whenever ψ is an affine root such that ψ|_{𝒴} ≥ 0}.

The set 𝒞(𝒴) is the smallest convex set which contains 𝒴 and is a union of closed affine Weyl facets of A. We call 𝒞(𝒴) the convex chamber closure of 𝒴. In particular, if 𝒴 contains an open subset of A, then 𝒞(𝒴) is a union of chambers.

3.6. Lemma. Let C be a chamber and F a facet in an apartment A = A(S). Let 𝒞(C ∪ F) be the convex chamber closure of C and F. The facet F lies in a unique chamber of 𝒞(C ∪ F).

Proof. Suppose to the contrary that F is a facet for two different chambers C_{a} and C_{b} in 𝒞(C ∪ F). Let H_{ψ} be a root hyperplane in A which separates the two chambers. Relabel the chambers so that C_{a} and C lie in the same half space B of H_{ψ}. Then 𝒞(C ∪ F) ⊂ B, a contradiction to C_{b} ⊂ 𝒞(C ∪ F). □

3.7. Let

(3.7.1)  C(C, F) := the unique chamber in 𝒞(C ∪ F) containing F.

Also, let star(F) be set of all facets containing the facet F. Each facet E ≠ F in star(F) corresponds to a parahoric subgroup 𝒢_{E} ⊊ 𝒢_{F} and hence determines a parabolic subgroup 𝒢_{E}/𝒢_{F,+} in 𝒢_{F} /𝒢_{F,+} ≅ M_{F} (𝔣). Equivalently, each facet E ≠ F determines a facet in the spherical Tits building 𝒯 = 𝒯(M_{F} (𝔣)) of M_{F} (𝔣). Denote by P (E) the parabolic subgroup 𝒢_{E}/𝒢_{F,+} of 𝒢_{F} /𝒢_{F,+}. The topological subspace Star(F) of ℬ which is the union of the facets of star(F) is polysimplicially equivalent to the suspension of the spherical Tits building 𝒯 by the facet F, i.e. the topological join of 𝒯 and F. The space Star(F)\F is homotopically equivalent to 𝒯. The maximal split 𝔣-torus S associated to S determines an apartment 𝒜 in 𝒯. The topological suspension of 𝒜 by F is naturally simpilicially equivalent to A(S) ∩ Star(F). We can and do use the chamber C_{1} = C(C, F) in (3.7.1) as a base chamber to obtain a bijection of the chambers in the apartment 𝒜 with elements of the Weyl group W = W_{F} (S) of S in 𝒢_{F} /𝒢_{F,+}. If w ∈ W, let w̃ ∈ N(S) ∩ 𝒢_{F} be an element which maps to w ∈ W, then the bijection is

(3.7.2)  w ⟷ w̃C_{1}.

When the context is clear, for notational convenience, we shall sometimes identify the chamber with its corresponding Weyl element. Denote the length of an element w ∈ W by ℓ(w). Let ⪰ (and ≻) denote the Bruhat ordering [Hp:5.9] on the elements of W_{F} (S) and let w_{0} be the maximal element. We have ℓ(w_{0}) ≥ ℓ(w). The chamber w̃_{2}w̃_{0}C_{1} is the chamber opposite the chamber w̃_{2}C_{1}.

We recall the following result on the Bruhat ordering.

3.8. Proposition. ([Hp:5.8,5.10,5.11]) Suppose V is an Euclidean space and W the finite Weyl group generated by a root system in V.

i) Let e ∈ W be the identity element and suppose s ∈ W is a reflection across a root hyperplane H in V. For any w ∈ W, a necessary and sufficient condition for the Weyl chambers C_{e} and C_{w} to lie in the same half space of H in V is that w ≺ sw.

ii) Suppose w_{a} ≺ w_{b}. Then there is a sequence of root hyperplanes H_{1}, H_{2}, …, H_{n} with corresponding reflections s_{1}, s_{2}, …, s_{n} so that w_{b} = s_{n} ⋯ s_{1}w_{a} and for i = 1, …, n

(3.8.1)  s_{i−1} ⋯ s_{1}w_{a} ≺ s_{i} ⋯ s_{1}w_{a}.

We remark that the proposition can also be formulated with the reflections on the right instead of the left side.

3.9. The chambers of 𝒯 = 𝒯(M_{F} (𝔣)) are in bijection with the points of the flag variety M_{F} (𝔣)/P (C_{1}) of 𝔣-rational Borel subgroups of M_{F}. The Bruhat decomposition says the P (C_{1}) orbits of the flag variety M_{F} (𝔣)/P (C_{1}) are parametrized by the Weyl group W_{F} (S). A 𝔣-rational Borel subgroup B ∈ M_{F} (𝔣)/P (C_{1}) is in relative position w ∈ W_{F} (S) to P (C_{1}) if B lies in the P (C_{1})-orbit P (C_{1})w̃B, where w̃ is an element of the normalizer N(S) which reduces to w. A Borel subgroup B is opposite to P (C_{1}), if it is relative position w_{0} to P (C_{1}). Equivalently, B is opposite to P (C_{1}) if their intersection B ∩P (C_{1}) is the 𝔣-rational points of a maximal 𝔣-torus.

3.10. Proposition. Let A = A(S) be an apartment in ℬ(G) and C a chamber in A. Suppose F is a facet in A and C_{1} = C(C, F) is the chamber defined in (3.7.1). Let ρ = ρ_{A,C} be the retraction of ℬ to A centered at C. Then for D a chamber in star(F),

position of D relative to C_{1} = position of ρ_{A,C}(D) relative to C_{1}.

Proof. The retraction ρ fixes A pointwise. There exists an element g ∈ 𝒢 so that ρ(x) = gx for all x ∈ D ∪ C_{1}; in particular, g must fix C_{1} pointwise. Hence g lies in the Iwahori subgroup 𝒢_{C_{1}}. In 𝒢_{F} /𝒢_{F,+}, consider the three Borel subgroups P (C_{1}), P (D) and P (ρ_{A,C}(D)). The image \overline{g} of g in 𝒢_{F} /𝒢_{F,+} lies in the Borel subgroup P (C_{1}) and conjugates P (D) to P (ρ_{A,C}(D)). Since conjugation of a Borel subgroup by an element of P (C_{1}) does not change its relative position with respect to P (C_{1}), the assertion is proved. □

3.11. Corollary. Given a chamber C ⊂ A = A(S) and a facet F ⊂ C, suppose D ∈ star(F) is a chamber opposite to C, then the chamber ρ_{A,C}(D) is opposite C.

3.12. We remark that if F is a facet in A, then Proposition 3.10 completely determines the maping ρ_{A,C} on the subset Star(F). Indeed, define the chamber C_{1} as in (3.7.1). If D is a chamber in star(F), express the Borel subgroup P (D) as P (D) = Ad(gn)P (C_{1}), where g ∈ P (C_{1}) and n ∈ N(S). Choose representatives g̃ ∈ 𝒢_{C_{1}} and ñ ∈ N(S) ∩ 𝒢_{F} for g and n respectively. Then

(3.12.1)  ρ_{A,C}(x) = g̃^{−1}x for every x ∈ D.

3.13. Suppose D_{1}, D_{2} ∈ star(F) are chambers so that the Borel subgroups P (D_{1}) and P (D_{2}) are in opposite positions. Let w_{1}, w_{2} ∈ W_{F} (S) be the relative positions of P (D_{1}) and P (D_{2}) with respect to a Borel subgroup P (C). The next result tells us the relationship between w_{1} and w_{2}.

3.14. Proposition. Let D_{1} ∈ star(F) be a chamber so that the Borel P (D_{1}) is in relative position w_{1} to the Borel P (C). A necessary and sufficient condition for there to be a Borel P (D_{2}) opposite to P (D_{1}) and in relative position w_{2} to P (C) is that w_{1} ⪰ w_{2}w_{0}.

Proof. Denote P (C) as B. A pair of Borel subgroups (xB, yB) (x, y ∈ M_{F} (𝔣)/B) is in relative position w if and only if x^{−1}y ∈ BwB. To say P (D_{1}) and P (D_{2}) are in relative positions w_{1} and w_{2} to B means that P (D_{1}) = Ad(g_{1})B and P (D_{2}) = Ad(g_{2})B, where g_{1} ∈ Bw_{1}B and g_{2} ∈ Bw_{2}B respectively. To say the two Borels P (D_{1}) and P (D_{2}) are in relative position w_{0} means g_{2}^{−1}g_{1} ∈ Bw_{0}B and therefore Bw_{0}B ⊂ Bw_{2}^{−1}Bw_{1}B. This is equivalent to the stated condition. □

We remark that the roles of D_{1} and D_{2} are symmetrical; w_{1} ⪰ w_{2}w_{0} is equivalent to w_{2} ⪰ w_{1}w_{0}

4. Proof of Displacement Function Convexity

4.1. We give in this section the proof of our main theorem that a displacement function d_{g} is a convex function on the building. The basic idea of the proof is to calculate the displacement function along a geodesic path v(t). Suppose the geodesic path passes through a chamber C. Let I be the geodesic segment of v(t) in C. To calculate the displacement function along I, take A be an apartment which contains both C and gC. The map g : C ⟶ gC extends to an isometry g̃ of A. The displacement function dg̃ of g̃ on A is equal to d_{g} on the chamber C. Since dg̃ is convex by Proposition 2.2, we conclude d_{g} is convex on C and hence the segment I. Therefore, as a consequence of Lemma 2.4, to prove a displacement d_{g} is convex, it is sufficient to establish it is convex along any geodesic segment consisting of the intersection of v(t) with two adjacent chambers.

4.2. Suppose F is a facet in ℬ and C_{r} and C_{s} are two chambers containing F so that P (C_{r}) and P (C_{s}) are opposite Borel subgroups in M_{F} (𝔣). Furthermore, let x_{r} and x_{s} be points in the interior of C_{r} and C_{s} respectively with the property that the geodesic x(t) from x_{r} to x_{s} meets the facet F, say x(0) = x_{r}, x(b) ∈ F and x(c) = x_{s} where c = dist(x_{r}, x_{s}). The value b is unique since x_{r} and x_{s} are interior points. Let I = x((b − ϵ, b + ϵ)) be a small neighborhood of the point x(b) in the geodesic x. Our goal is to show that the function h(t) = dist(x(t), gx(t)) is convex in a neighborhood of t = b. Convexity in a neighborhood of t = b is obvious if there exists an apartment A which contains both I and gI. Unfortunately, I and gI may not lie in a single apartment. We shall show there is an apartment A containing C_{r} so that the retraction map ρ = ρ_{A,C_{r}} retracts x to a geodesic in A, and gx to a bent geodesic in A with the property that f(t) = dist(ρ(x(t)), ρ(gx(t))) is convex; and since h(t) ≥ f(t), we deduce that h is convex.

For any apartment A containing the chamber C_{r}, let ρ = ρ_{A,C_{r}} be the retraction map of the building ℬ to the apartment A centered at the chamber C_{r}. We claim the retraction of x(t), i.e.

(4.2.1)  v(t) := ρ(x(t))

is a geodesic line segment in A. Indeed, if A′ is any apartment containing C_{r} and C_{s}, the retraction map is the unique isometry of A′ to A which fixes C_{r} pointwise. Note also that by Corollary 3.11, P (C_{r}) and P (ρ_{A,C_{r}}(C_{s})) are opposite Borel subgroups in M_{F} (𝔣).

Suppose g ∈ 𝒢. By Lemma 3.6, the convex set 𝒞(C_{r} ∪ gF) contains a unique chamber C = C(C_{r}, gF) incident with the facet gF. Let A be an apartment containing the chambers C_{r} and C′_{r} := gC_{r} and let ρ = ρ_{A,C_{r}} be the retraction map of the building ℬ to the apartment A centered at the chamber C_{r}. Set C′_{s} = gC_{s}. Both C′_{r} and C′_{s} belong to the set star(gF). Proposition 3.10 tells us that the retraction map on chambers in star(gF) (hence on all facets in star(F)) is completely determined by the relative positions of Borel subgroups in M_{gF} (𝔣) to P (C). Define the path z(t) in A as

(4.2.2)  z(t) := ρ(gx(t)), for 0 ≤ t ≤ c.

Set z_{r} := z(0) and z_{s} := z(c). The path z(t) is a ‘bent’ geodesic path. It is a geodesic line segment in the chamber C′_{r} for 0 ≤ t ≤ b, and in the chamber ρ(C′_{s}) for b ≤ t ≤ c.

4.3. Proposition. For v(t) and z(t) as in (4.2.1) and (4.2.2), the function

(4.3.1)  f(t) := dist (v(t), z(t))

is a convex function.

Proof. We have z(0) = ρ(gx_{r}) ∈ ρ(C′_{r}), z(b) = ρ(gx(b)) = gx(b) ∈ gF, and z(c) = ρ(gx_{s}) ∈ ρ(C′_{s}). Let z_{1}(t) be the geodesic line in the apartment A parametrized by arclength which begins at z(0), i.e. when t = 0 and is at z(b), when t = b; and let z_{2}(t) be the geodesic line which is at z(b) when t = b, and at z(c) when t = c. Each of the functions

f_{1}(t) := dist(v(t), z_{1}(t)) and f_{2}(t) := dist(v(t), z_{2}(t))

is convex. The function f is obtained by ‘splicing’ f_{1} on the interval [0, b] to f_{2} on the interval [b, c]. To show f is convex, it suffices to show it is convex in a neighborhood of the point b. Write the retractions ρ(C′_{r}) and ρ(C′_{s}) as ρ(C′_{r}) = w_{1}C and ρ(C′_{s}) = w_{2}C. The chamber w_{2}C contains z_{2}(c), and the chamber D := w_{1}w_{0}C (in A) opposite to w_{1}C contains z_{1}(c). Since C′_{r} and C′_{s} are opposite chambers with respect to the facet gF, Proposition 3.14 says w_{1}w_{0} ⪯ w_{2}. If w_{1}w_{0} = w_{2}, then z(t) = ρ(g(x(t))) is a geodesic line in A; in particular, f(t) is obviously convex. We may therefore assume w_{1}w_{0} ≺ w_{2}. By Proposition 3.8, there is a sequence of root hyperplanes H_{1}, H_{2}, …, H_{n} with corresponding reflections s_{1}, s_{2}, …, s_{n} so that w_{2} = s_{n} ⋯ s_{1}w_{1}w_{0} and s_{i−1} ⋯ s_{1}w_{1}w_{0} ≺ s_{i} ⋯ s_{1}w_{1}w_{0} for i = 1, …, n. Lift the reflections s_{i} to elements n_{i} ∈ N(S) ∩ 𝒢_{gF}. For i = 1, …, n + 1, define

u_{i}(t) := \{ \begin{matrix}z_{1}(t) 0 ≤ t ≤ b \\ n_{i−1} ⋯ n_{1}z_{1}(t) b ≤ t ≤ c. \end{matrix}

Then, u_{1}(t) = z_{1}(t) and u_{n+1}(t) = z(t). We shall prove by induction on i that the functions

d_{i}(t) := dist(v(t), u_{i}(t))

are convex functions. For i = 0 this is obvious since u_{1}(t) = z_{1}(t) is a geodesic line segment in A. Assume d_{i−1} is a convex function. The facet gF lies on all the hyperplanes H_{i}, and so u_{i}(b) = z(b) for all i. The action of n_{i} on A takes the line segment [z(b), u_{i−1}(c)] to the line segment [z(b), u_{i}(c)]. We consider two cases depending whether or not the facet F lies in H_{i} or in the interior of one of the two half spaces determined by H_{i}.

Case 1. F lies in H_{i}. We apply Proposition 2.5 to conclude that if the function d_{i−1} is convex on the interval [0, c], then so is the function d_{i}.

Case 2. F does not lie in H_{i}. Here, the chambers C_{r}, ρ(C_{s}), C′_{r} = gC_{r} and n_{i−1} ⋯ n_{1}D all lie in the same half space of H_{i}. Hence, it is obvious, using the orthogonal coordinate system provided by H_{i} and a perpendicular line to H_{i} that dist(v(t), u_{i−1}(t)) ≤ dist(v(t), u_{i}(t)) for b ≤ t ≤ c, and therefore d_{i} is a convex function. □

4.4. Displacement Function Convexity Theorem. Suppose g ∈ Aut(ℬ(𝒟G, k)). The displacement function d_{g} : ℬ ⟶ ℝ defined as d_{g}(x) := dist(x, gx) is a convex function.

Proof. As already mentioned in section 4.1, the restriction of d_{g} to any chamber C ⊂ ℬ is a convex function and that global convexity will follow if we can establish convexity on the union C_{a} ∪ C_{b} of two adjacent chambers. In particular, it is enough to show that if x(t) is a geodesic segment which begins in the chamber C_{a} and ends in the adjacent chamber C_{b}, then the function

h(t) := dist(x(t), gx(t))

is a convex function. Furthermore, as it is the case that if a sequence of convex functions converges pointwise, then the limit function is also convex, we need only consider the case where the starting and ending points are interior points. So, suppose x(t) (0 ≤ t ≤ c) is a geodesic segment which starts at a point x_{a} = x(0) ∈ C_{a} and ends at a point x_{b} = x(c) ∈ C_{b} and passes through the facet F = C_{a} ∩ C_{b}, say x(β) ∈ F. Let A be an apartment containing the chambers C_{a} and gC_{a}. Let ρ = ρ_{A,C_{a}} be the the retraction of the building ℬ to the apartment A centered at the chamber C_{a}. The retraction ρ takes the union C_{a} ∪ C_{b} isometrically to C_{a} ∪ ρ(C_{b}); whence, as was already mentioned before (4.2.1), the retraction ρ(x(t)) of x(t) is a geodesic segment in A between the two chambers C_{a} and ρ(C_{b}). We have by Proposition 3.5, that h(t) = dist(x(t), gx(t)) ≥ dist(ρ(x(t)), ρ(gx(t))). Set f(t) := dist(ρ(x(t)), ρ(gx(t))). We have h(t) = f(t) for 0 ≤ t ≤ β. Since the function h is convex on each of the intervals 0 ≤ t ≤ β and β ≤ t ≤ c, it follows that a sufficient condition for h to be convex is that function f is convex. Hence it is enough to show f is convex. But this is precisely the situation of Proposition 4.3. □

5. Filtrations sets and building sets

5.1. Let x ∈ ℬ = ℬ(𝒟G, k) and r ≥ 0. Set

(5.1.1)  𝒢_{x,r} := { g ∈ 𝒢 | d_{g}(x) ≤ r }.

The set 𝒢_{x,r} is the elements in 𝒢 which move the point x a distance at most r. It is thus obvious that 𝒢_{x,0} is the subgroup Stab(x) of elements in 𝒢 which fix the point x; in particular, 𝒢_{x,0} contains the parahoric subgroup 𝒢_{x} which fixes x. We make some remarks. The first is that for r > 0, the sets 𝒢_{x,r} are not subgroups. In particular, they should not be confused with the groups denoted by the same symbol in [MP2]. A definition more compatible with the use of the symbol 𝒢_{x,r} in [MP1] is to replace the definition in (5.1.1) with 𝒢_{x,−r} := { g ∈ 𝒢 | d_{g}(x) ≤ r }. Since this would be somewhat cumbersome in the present setting, we use here the definition (5.1.1) as stated. The sets 𝒢_{x,r} are closed under right multiplication by the group 𝒢_{x,0}. If 0 ≤ r_{1} ≤ r_{2}, then 𝒢_{x,r_{1}} ⊂ 𝒢_{x,r_{2}} and the union of the filtration sets exhaust 𝒢, i.e.

𝒢 = ⋃_{r≥0}𝒢_{x,r}.

The second remark is that if the center of G/k is not anisotropic over k, then the group 𝒢_{x,0} is not compact. The noncompactness is precisely because the k-rational points of the center are contained in 𝒢_{x,0}. In this section we shall assume the center of G/k is anisotropic and therefore 𝒢_{x,r} are compact open subsets of 𝒢 for all r ≥ 0. The third remark is that a reader familar with the notation in [MP1,MP2,MP3] may extrapolate the notation there and believe 𝒢_{x,0} is the parahoric subgroup 𝒢_{x}. In our current situation, 𝒢_{x,0} and 𝒢_{x} are different groups; 𝒢_{x,0} is the stability subgroup of x while 𝒢_{x} is the parahoric subgroup for x.

5.2. For any g ∈ 𝒢, it is obvious that have d_{h}(x) = d_{g}(x) for all h ∈ g𝒢_{x,0}. Futhermore for any R ≥ 0, there are only finitely many cosets g𝒢_{x,0} so that d_{g𝒢_{x,0}} (x) ≤ R. Therefore, the set of values g ↦ d_{g}(x) is a discrete set in ℝ with no accumulation points. For r > 0, define

𝒢_{x,r^{−}} := ⋃_{0≤s<r}𝒢_{x,s}.

As an immediate consequence of the remarks before the definition we have

5.3. Proposition. Suppose x ∈ ℬ.

i) If r ≥ 0, there exists an ϵ > 0 depending on x and r such that 𝒢_{x,r} = 𝒢_{x,s} for every r ≤ s < r + ϵ.

ii) If r > 0, there exists a δ > 0 depending on x and r such that 𝒢_{x,r^{−}} = 𝒢_{x,s} for every r − δ < s < r

5.4. If Ξ is a bounded nonempty subset of 𝒢, set

(5.4.1)  d_{Ξ}(x) := sup{ d_{g}(x) | g ∈ Ξ }.

Since the supremum of a set of convex functions is convex, the function d_{Ξ} is a convex function on ℬ. Set

(5.4.2)  d(Ξ) := inf_{x∈ℬ} d_{Ξ}(x).

For g ∈ 𝒢 shorten d_{{g}} and d({g}) to d_{g} and d(g). The next results says that the function d_{g} achieves d(g) as its minimal value.

5.5. Proposition. Given an nonempty bounded subset Ξ of 𝒢, there is a point x ∈ ℬ so that

d_{Ξ}(x) = d(Ξ).

Proof. We first prove the result when Ξ is a single element g. Choose any point z ∈ ℬ. View [BT1] the set of geodesic rays in ℬ emmanating from the point z as the compact spherical building 𝒯 = 𝒯(𝒟G, k) of 𝒟G/k. Define the continuous function m : 𝒯 ⟶ ℝ to be

m_{g}(r) := infinimum of the function d_{g} on the ray r.

Since 𝒯 is compact, there is a ray r(t), (t ≥ 0), i.e. a point in 𝒯, where the function m achieves its minimum. Clearly m_{g}(r) = d(g). There are two possibilities. Either d_{g} achieves its minimum on the ray r, or lim_{t→∞}d_{g}(r(t)) = d(g).

Case 1: d_{g} achieves its minimum on the ray r. Suppose the minimum is achieved for t = t_{1}. But then d_{g}(r(t_{1})) = d(g), i.e. d_{g} achieves its minimum at the point x_{1} = r(t_{1}).

Case 2: lim_{t→∞}d_{g}(r(t)) = d(g). Here, necessarily d_{g} must decrease along the ray r. The transform gr of the geodesic ray by g is another geodesic ray. By [BT1] for t_{0} ≥ 0 sufficiently large the two geodesic rays r([t_{0}, ∞)) and gr([t_{0}, ∞)) both lie in an apartment A and hence are rays in the apartment. It is the case that

i) The function t ↦ dist(r(t), gr(t)) measures the Euclidean distance between points on the parametrize rays r and gr.

ii) lim_{t→∞}dist(r(t), gr(t)) = d(g).

It is easy to see that the form of the Euclidean distance function forces the existence of an t_{1} so that dist(r(t_{1}), gr(t_{1})) = d(g). Hence, the function d_{g} achieves its minimal value at the point x_{1} = r(t_{1}). This completes the proof of the proposition when Ξ = {g}.

Suppose now Ξ is an arbitrary nonempty bounded set. As in the case of a single element, choose any point z ∈ ℬ. For each geodesic ray r emmanating from z set m_{Ξ}(r) := inf{d_{Ξ}(r)}. Then m_{Ξ} : 𝒯 ⟶ ℝ is a continuous function. Let r be a ray so that m_{Ξ} achieves it minimum. Then m_{Ξ}(r) = d(Ξ). On the ray r, either

(1) d_{Ξ} ∘ r takes on its minimum value at a point t_{1} and therefore for x_{1} = r(t_{1}), we have d_{Ξ}(x_{1}) = d(Ξ), or

(2) the function t ↦ d_{Ξ}(r(t)) decreases with limit d(Ξ). In case (2), for every g ∈ Ξ, we have d_{Ξ} ≥ d_{g} on the ray r. Since d_{g} is a convex function, it must decrease along the ray r. The same argument as in the preceeding paragraph applies and therefore each d_{g} must achieve its minimum on r at some point t_{g} and be constant for t > t_{g}. We can find an open compact subgroup J_{g} so that for all h ∈ gJ_{g}, d_{h} ∘r is constant on some interval [t_{g}, b] of positive length. We conclude that for every g ∈ Ξ, there is an open neighborhood U_{g} := gJ_{g} ∩ Ξ in Ξ about the point g so that for every h ∈ U_{g}, d_{h} ∘ r is constant on [t_{g}, ∞). The collection {U_{g}} is an open cover of the compact set Ξ; hence there is a finite subcover. This implies there is a t_{2} so that for every h ∈ Ξ, the function d_{h} ∘ r is constant on [t_{2}, ∞). We conclude that d_{Ξ}(r(t_{2})) = d(Ξ). □

5.6. Define the building set ℬ(Ξ) of a nonempty bounded subset Ξ ⊂ 𝒢 to be

(5.6.1)  ℬ(Ξ) := { x ∈ 𝒢 | d_{Ξ}(x) = d(Ξ) }.

When Ξ consists of a single element g, we shorten the notation ℬ({g}) to ℬ(g).

5.7. Corollary.

i) ℬ(Ξ) is a convex subset of ℬ.

ii) Suppose g, h ∈ 𝒢 commute. Then hℬ(g) ⊂ ℬ(g). iii) If x ∈ ℬ(g), and n ∈ ℕ is a natural integer, then dist(x, g^{n}x) = n·d(g). In particular,

if d(g) > 0, there is a unique geodesic L(x) which contains all the points g^{i}x, i ∈ ℤ.

Proof. The convexity of ℬ(Ξ) is immediate. For assertion ii), observe that if x ∈ ℬ(Ξ), then dist(hx, g(hx)) = dist(hx, hgx) = dist(x, gx) = d(g). To prove iii), note that a curve in ℬ is a geodesic curve if and only if it is locally a geodesic curve. We claim that if x ∈ ℬ(g), then the curve C obtained by joining the two geodesic segments [x, gx] and [gx, g^{2}x] is a geodesic curve. Indeed, let y be the midpoint of the segment [x, gx]. Obviously, gx is the midpoint of [gx, g^{2}x]. Since dist(y, gx) = dist(gx, gy) = \frac{1}{2}dist(y, gy), we deduce that gx is the midpoint, of the geodesic segment [y, gy] and therefore the curve C is a geodesic curve in a neighborhood of gx; thus C is a geodesic curve. A simple induction shows that for any positive n, the curve formed from the geodesic segments [g^{i−1}x, g^{i}x] , 1 ≤ i ≤ n is a geodesic curve. Assertion iii) is then obvious. □

Remark. Suppose s is a semisimple element which is contained in some split torus S. Then the centralizer L = C_{G}(s) is a Levi subgroup. Corollary 5.7ii) says that the Levi subgroup acts on the building set ℬ(s). An interesting question is whether ℬ(s) is isometric with the extended building of L.

5.8. Proposition. Suppose g ∈ 𝒢 is the product of two commuting elements g = hk, with the property that the smallest closed subgroup K of 𝒢 containing the powers of k is compact. Then d(g) = d(h) and ℬ(g) ∩ ℬ(h) ≠ ∅.

Proof. For each x ∈ ℬ(g), we have

dist(kx, gkx) = dist(kx, kgx) = dist(x, gx)

hence kℬ(g) ⊂ ℬ(g) and therefore ℬ(g) is K invariant. Since ℬ(g) is also a convex set, by the Bruhat-Tits fixed point theorem there is a point y ∈ ℬ(g) so that ky = y. Then

(5.8.1)  d(g) = dist(y, gy) = dist(ky, hky) = dist(ky, khy) = dist(y, hy) ≤ d(h).

Since, h = gk is obviously a product of commuting elements, interchanging the roles of g and h, we see d(g) = d(h). It then follows from (5.8.1) that y ∈ ℬ(g) ∩ ℬ(h). □

5.9. Corollary. Suppose g = su is the Jordan decomposition of an element g ∈ 𝒢, then

i) d(g) = d(s).

ii) The building sets ℬ(g) and ℬ(s) are u invariant, then the u-fixed point sets ℬ(g)^{u} and ℬ(s)^{u} are nonempty, and

ℬ(s)^{u} ⊂ ℬ(g) and ℬ(g)^{u} ⊂ ℬ(s).

Proof. Part i) is obvious from the Proposition. The assertions in part ii) that ℬ(g) and ℬ(s) are u invariant is Corollary 5.7ii). That ℬ(g)^{u} and ℬ(s)^{u} is the nonempty follows from the Bruhat–Tits fixed point theorem. To prove the last assertion of part ii), suppose x ∈ ℬ(s)^{u}. Then ux = x and so sx = sux = gx and thus, d(g) = d(s) = dist(x, sx) = dist(x, gx), and so x ∈ ℬ(g). Similarly, ℬ(g)^{u} ⊂ ℬ(s). □

5.10. An interesting question arising from the Corollary is whether or not ℬ(g)^{u} = ℬ(g).

The function g ↦ d(g) is clearly a class function on 𝒢. Corollary 5.9 says d(g) depends only on the semisimple part of g. In particular, d is completely determined by its restriction to the maximal tori of 𝒢. If T is a maximal k-torus, then Proposition 5.8 says, the function d is constant on cosets of the maximal compact subgroup T (k)_{c} in T (k).

6. A descent property

6.1. In this section we prove a descent property in 𝒢 which is an analogue of a fundamental descent property in the Lie algebra and its dual. Recall that in [MP1,MP2] it was shown that if Ξ = X + 𝔤_{x,−r_{i−1}}^{∗} ∈ 𝔤_{−r_{i}}^{∗}/𝔤_{−r_{i−1}}^{∗} is a coset which contains a nilpotent element, then Ξ ⊂ 𝔤_{y,−s}^{∗} for some s < r_{i}. In [MP3], this result was reinterpreted in terms of the concave depth function d^{Ξ} (notation [MP3]). In the language of depth functions, a nonzero coset Ξ contains a nilpotent element if and only if d^{Ξ} does not attain it maximum at the point x. We show here an analogue of this statement with the role of a concave depth function replaced by a convex displacement function.

Given x ∈ ℬ, let F be the smallest dimensional facet containing the point x.

6.2. Lemma. For any x ∈ ℬ, there exists a neighborhood of the point x fixed by the subgroup 𝒢_{F,+}.

Proof. The set Star(F) is the union of all chambers which contain x and so clearly contains a neighorhood of x. As mentioned in section 3, the subspace Star(F) is the topological suspension of the spherical building 𝒯 of 𝒢_{F} /𝒢_{F,+} by the facet F. The action of 𝒢_{F} on Star(F) is the suspension of the action of 𝒢_{F} on 𝒯 and so Star(F) is fixed by the subgroup 𝒢_{F,+}. □

6.3. Proposition. For any x ∈ ℬ and any h ∈ g𝒢_{x,+}, there exists a neighborhood U of x so that the two functions d_{h} and d_{h𝒢_{x,+}} are equal on U.

Proof. Let U be a neighborhood of x so that for every k ∈ 𝒢_{F,+} and y ∈ U, we have y = ky. Then d_{hk}(y) = dist(y, hky) = dist(y, hy) = d_{h}(y). Hence d_{h𝒢_{F,+}} (y) := sup_{k∈𝒢_{F,+}} d_{hk}(y) =

d_{h}(y). □

For s > 0, define a coset Ξ = g𝒢_{x,+} of 𝒢_{x,+} in 𝒢_{x,s} to be nondegenerate if the depth function d_{Ξ} takes on its minimal value at the point x and degenerate if not.

6.4. Theorem. Suppose s > 0 and Ξ = gG_{x,+} ⊂ G_{x,s}.

i) If Ξ is a degenerate coset, then there exists a point y and s′ < s so that Ξ ⊂ G_{y,s′}. ii) If Ξ is a nondegenerate coset, then for any nonempty subset Y ⊂ Ξ, the function d_{Y}

achieves its minimum at the point x.

Proof. The proof of i) is obvious. If d_{Ξ} does not take on its minimum at the point x, there is a point y so that s′ = d_{Ξ}(y) < d_{Ξ}(x). But, d_{Ξ}(y) = s′ means d_{h}(y) ≤ s′ for all h ∈ Ξ, i.e. Ξ ⊂ 𝒢_{y,s}. To prove ii), select an element h ∈ Y ⊂ Ξ. We have d_{h} ≤ d_{Y} ≤ d_{Ξ}. Proposition 6.3 says that d_{h} = d_{Ξ} in a neighborhood of the point x; hence d_{Y} = d_{Ξ} in a neighborhood of x. If d(Y) < d(Ξ), let y be a point where d_{Y} (y) < d(Ξ). Consider the geodesic segment [y, x]. Since d_{Y} is convex and has values d_{Y} (y) < d(Ξ) = d_{Y} (x), it must be the case that d_{Y} takes on values less than d_{Y} (x) in any neighborhood of the point x. But, this clearly contradicts the two facts that d_{Y} = d_{Ξ} in a neighborhood of x and Ξ is nondegenerate. Thus, d(Y) = d(Ξ), and so Y is nondegenerate. □

7. An example

7.1. We illustrate the displacement function as a tool in a proof of the Howe Conjecture for the group G = SL(2, k), when char(k) = 0. We use the notation of [MP1] to define the filtration subgroups G_{x,r} (r ≥ 0) attached to a point x in the building ℬ(G). Denote by D the diagonal torus in G, and by A the apartment of D. Let ϖ be a prime element in 𝔬_{k} and take δ ∈ GL(2, k) to be the element

δ = ( \begin{matrix}ϖ^{−1} & 0 \\ 0 & 1 \end{matrix} ) .

Let x_{0} and x_{1} be points in A so that G_{x_{0},0} = SL(2, 𝔬_{k}) and G_{x_{1},0} = Ad(δ)G_{x_{0},0}. If we denote by x_{h} the midpoint of the segment [x_{0}, x_{1}], then G_{x_{h},0} is the Iwahori subgroup G_{x_{0},0} ∩ G_{x_{1},0}. Given a compact subset ω ⊂ G, set ω^{G} := {Ad(g)x | x ∈ ω , g ∈ G} and let J(ω) denote the space of G-invariant distributions supported on the closure of ω^{G}. Fix a positive integer r; then G_{x_{0},r} is the r-th congruence subgroup of SL(2, 𝔬_{k}), G_{x_{1},r} = Ad(δ)G_{x_{0},r} and G_{x_{h},r} = G_{x_{0},r} ∩ G_{x_{1},r}. Let 𝒮(r) be the vector space of functions on G which are compactly supported and right G_{x_{h},r}-invariant, i.e. 𝒮(r) = C_{o}^{∞}(G/G_{x_{h},r}).

7.2. Group Howe Conjecture for SL(2). Suppose char(k) = 0. Given a compact subset ω ⊂ SL(2, k) and a positive integer r, the restriction of the G-invariant distributions J(ω) to 𝒮(r) induces a finite dimensional space of linear functionals of 𝒮(r).

Proof. As in [Ho], we view evaluation of an element T ∈ J(ω) on an element f ∈ 𝒮(r) as a pairing ⟨T, f⟩ := T (f) on J(ω) × 𝒮(r). In the dual spaces of 𝒮(r) and J(ω), denote by J(ω)|_{𝒮(r)} and 𝒮(r)|_{J(ω)}, the subspaces induced by J(ω) and 𝒮(r) respectively. Our plan, following both [Ho] and [HC], is to show the characteristic functions of cosets gG_{x_{h},r}, g ∈ G span a finite dimensional space of linear functionals on J(ω).

Set Y = {x_{0}, x_{1}, x_{h}}. The group G_{x_{h},r} is a subgroup of both G_{x_{0},r} and G_{x_{1},r}; and so C_{o}^{∞}(G/G_{x_{0},r}) and C_{o}^{∞}(G/G_{x_{1},r}) are subspaces of 𝒮(r). For g ∈ G and y ∈ Y, set 1_{g,y} to be the characteristic function of the coset gG_{y,r}. These functions obviously span 𝒮(r). We shall use the displacement function to define finite dimensional subspaces W_{y} in each C_{o}^{∞}(G/G_{y,r}), y ∈ Y, and then give a reduction argument modeled on the one in [Ho] for Lie algebras to say that the linear functional of J(ω) induced by any 1_{g,y} is contained in the span of the W_{y}′s.

Recall the notation from section 5 that if h ∈ G, then d(h) is the minimum value of the displacement function d_{h}. Compactness of ω means there is a M = M(ω) > 0 such that d(h) ≤ M for all h ∈ ω. Note also for any N > 0, there are only finitely many cosets gG_{y,r}, y ∈ Y such that d_{gG_{y,r}}(y) ≤ N.

We use the notion of degenerate and nondegenerate cosets described in section 6 to define two subspaces of 𝒮(r). Let 𝒟(r) and 𝒩(r) be the subspaces spanned by elements 1_{g,y} so that gG_{y,0^{+}} is respectively degenerate and nondegenerate. Then, 𝒮(r) = 𝒟(r) + 𝒩(r). For convenience, call a coset gG_{y,r} degenerate or nondegenerate depending on the corresponding behavior of gG_{y,0^{+}}. The crucial property of nondegenerate cosets which we use is that if gG_{y,r} is nondegenerate then d(h) = d(gG_{y,r}) = d_{gG_{y,r}}(y) for every h ∈ gG_{y,r}.

We claim 𝒩(r)|_{J(ω)} is finite dimensional. This assertion follows from our observations that there exists an M > 0 so that d(h) ≤ M for all h ∈ ω and that there are only finitely many cosets gG_{y,r}, whence only finitely many nondegenerate cosets, with d(gG_{y,r}) ≤ M, and the fact that if gG_{y,r} is a nondegenerate with d(gG_{y,r}) > M, we have gG_{y,r} ∩ ω^{G} = ∅, and so 1_{g,y}|_{J(ω)} ≡ 0.

For N > 0, let J_{N}^{∗} be the subspace of the dual space J(ω)^{∗} spanned by 𝒩(r)|_{J(ω)} and the functions 1_{g,y}|_{J(ω)}, where y ∈ Y and gG_{y,r} satisfies d_{gG_{y,r}}(y) ≤ N. The results of the previous paragraph implies J_{N}^{∗} is finite dimensional. We shall show that if gG_{y,r} is a degenerate coset, and R = d_{gG_{y,r}}(y) is sufficiently large, then the linear functional of J(ω) induced by 1_{g,y} in fact lies in the subspace J_{R−1}^{∗}. A proof of this assertion establishes 7.2. To accomplish our goal we first prove a result on degenerate cosets. For convenience, normalize the distance function on ℬ so that dist(x_{0}, x_{1}) = 1. If S is any split torus in G, denote by S_{c} the maximal compact subgroup of S.

7.3. Lemma. Suppose y ∈ Y and gG_{y,0^{+}} is a degenerate coset, then there exists an apartment A(S) containing y, a unipotent element u so that u−1 ∈ 𝔰𝔩(2, k) is a eigenvector for Ad(S), and an element a ∈ S_{c} so that gG_{y,0^{+}} = auG_{y,0^{+}}.

Proof. Call a point z ∈ ℬ a vertex point if G_{z,0^{+}} is a maximal compact subgroup of G. It is elementary to see, e.g. via a graph, that if gG_{y,0^{+}} is a degenerate coset, then gy ≠ y and furthermore

i) d_{gG_{y,0^{+}}} decreases in a neighborhood of the point y in the geodesic [y, gy].

ii) If y ∈ {x_{0}, x_{1}}, then dist(y, gy) is an even integer; while if y = x_{h}, then dist(y, gy) is an odd integer. In any case, the midpoint z of [y, gy] is a vertex.

Let A(S) be any apartment which passes through the points y and z and does not meet the segment (z, gy]. Denote by z_{a}, the vertex in (y, z] closest to y and z_{b} the vertex in A(S) on the opposite side of y at distance dist(y, z_{a}). It is again elementary that there exists a unipotent element, so that u − 1 ∈ 𝔰𝔩(2, k) is a eigenvector for Ad(S), so that the half line of A(S) containing z but not y is fixed, and such that uz_{a} = gz_{a}, uy = gy, and uz_{b} = gz_{b}. It follows that g^{−1}u ∈ G_{z_{a},0} ∩ G_{y,0} ∩ G_{z_{b},0} = S_{c}G_{y,0^{+}}, which immediately implies the lemma. □

We are now ready to complete the proof of 7.2. Suppose y ∈ {x_{0}, x_{1}} and gG_{y,r} is degenerate. Set d_{gG_{y,r}}(y) = R. As our argument for x_{0} and x_{1} are similar, we’ll assume y = x_{0}. Apply Lemma 7.3 to find an apartment A(S) containing y and gG_{y,r} = aukG_{y,r}, where k ∈ G_{y,0^{+}}. Since we are only interested in the linear functional of J(ω) induced by 1_{g,y}, we can replace gG_{y,r} by a conjugate g′G_{y,r} = Ad(k′)gG_{y,r} (k′ ∈ G_{y,0}). By replacing gG_{y,r} by a suitable conjugate g′G_{y,r}, we may assume the torus we obtain from Lemma 7.3 is the diagonal torus D and u is an upper triangular unipotent element. The coset g′G_{y,r} is a disjoint union ∐ g′m_{i}G_{x_{h},r}, where m_{i} runs over G_{y,r}/G_{x_{h},r} and d_{g′m_{i}G_{x_{h},r}}(x_{h}) = R − 1. Hence, 1_{g′,y} = ∑ 1_{g′m_{i},x_{h}} ∈ J_{R−1}^{∗}.

The last case we need to consider is a degenerate coset of the form gG_{x_{h},r}. Set R = d_{gG_{x_{h},0^{+}}} (x_{h}). Arguing as in the previous paragraph, after a possible conjugation, we can assume the torus provided by Lemma 7.3 is the diagonal torus D, and gG_{x_{h},r} = aukG_{x_{h},r}, with k ∈ G_{x_{h},0^{+}}. Our argument is essentially the same for u an upper or lower triangular unipotent element, so we’ll assume u is upper triangular. In this situation, d_{gG_{x_{h},r}}(x_{1}) = R − 1. An extra complication, because G_{x_{h},r} is a proper subgroup of G_{x_{1},r}, is that the coset gG_{x_{h},r} is not a disjoint union of G_{x_{1},r} cosets. In fact, we can choose representatives of the cosets in G_{x_{1},r}/G_{x_{h},r} to be upper triangular unipotent elements. However, provided R (which equals d_{u}(x_{1})) is large enough, we can conjugate gG_{x_{h},r} = aukG_{x_{h},r} by a small open compact subgroup H of D_{c} (which means we effectively only conjugate the unipotent element u by H) so that Ad(H)(gG_{x_{h},r}) = a(Ad(H)u)kG_{x_{h},r} = aukG_{x_{1},r}. It is this step that requires char(k) = 0, so that (k^{×})^{2} contains an open neighborhood of 1. We conclude there exists a L > 0 so that if R > L, then 1_{g,y} ∈ J_{R−1}^{∗}. Finally, if M = M(ω) is the bound we defined during our discussion of nondegenerate cosets, then the space of linear functionals of J(ω) induced by 𝒮(r) equals the finite dimensional space J_{max(L,M)}^{∗}. □

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