On the existence of spines for Q-rank 1 groups

0.28 MB·English

by  Dan Yasaki

ON THE EXISTENCE OF SPINES FOR Q-RANK 1 GROUPS
6
DANYASAKI
0
0
2
Abstract. LetX=Γ\G/Kbeanarithmeticquotientofasymmetricspaceof

n non-compacttype. InthecasethatGhasQ-rank1,weconstructΓ-equivariant
a deformation retractions of D = G/K onto a set D0. We prove that D0 is a
J spine, having dimension equal to the virtual cohomological dimension of Γ.
In fact, there is a (k−1)-parameter familyof such deformations retractions,
4
where k isthenumber of Γ-conjugacy classes ofrational parabolicsubgroups

of G. The construction of the spine also gives a way to construct an exact
]
T fundamental domainforΓ.
N
.
h
t 1. Introduction
a
m
Let D =G/K be a symmetric space ofnon-compacttype, where G is the group
[ of real points of an semisimple algebraic group G defined over Q. Let Γ be an

1 arithmetic subgroup of the rational points G(Q). Let (E,ρ) be a Γ-module over
v R. If Γ is torsion-free, the locally symmetric space Γ D is a K(Γ,1) since D is
\
3 contractible, and the group cohomology of Γ is isomorphic to the cohomology of
7 the locally symmetric space, i.e. H∗(Γ,E) = H∗(Γ D;E), where E denotes the
0 ∼ \
local system defined by (E,ρ) on Γ D. When Γ has torsion, the correct treatment
1 \
involvesthe language of orbifolds, but the isomorphismof cohomologyis still valid
0
6 by using a suitable sheaf E as long as the orders of the torsion elements of Γ are
0 invertible in R.
h/ Thevirtualcohomologicaldimension (vcd)ofGisthesmallestintegerpsuchthat
t cohomologyofΓ Dvanishesindegreesabovep,whereΓ G(Q)isanytorsion-free
a \ ⊂
arithmetic subgroup. Borel and Serre [9] show that the discrepancy between the
m
dimension of D and the vcd(G) is given by the Q-rank of G, the dimension of a
:
v maximalQ-splittorusinG. ThusonecanhopetofindaΓ-equivariantdeformation
i retract D D of dimension equal to the virtual cohomological dimension of G.
X 0 ⊂
When such a subset exists, it is called a spine.
r
a Spines have been constructed for many groups [1,8,14,18,20,21,25,27]. In
[3], Ash describes the well-rounded retract, a method for constructing a spine for
all linear symmetric spaces. Ash and McConnell extend [3] to the Borel-Serre
compactification in [7]. The well-rounded retract works for algebraic groups G
where the real points are isomorphic to a product of the following groups [15]:
(i) GL (R).
n
(ii) GL (C).
n
(iii) GL (H).
n
(iv) O(1,n 1) R×.
− ×
2000 Mathematics Subject Classification. Primary11F57;Secondary 53C35.
Key words and phrases. spine,locallysymmetricspace,cohomologyofarithmeticsubgroups.
The original manuscript was prepared with the AMS-LATEX macro system and the XY-pic
package.
1
2 DANYASAKI
Figure 1. Spine for SL (R)
2
(v) The non-compact Lie Group with Lie algebra e R.
6(−26)
⊕
Theretracthasbeenusedinthecomputationofcohomology[2,4–6,18,21,23,25–27].
The well-rounded retract proves the existence and gives a method of explicitly
describing spines in linear symmetric spaces. However, for non-linear symmetric
spaces, no general technique to construct spines is known. In fact, there were no
examples until MacPherson and McConnell [20] constructed a spine in the Siegel
upper half-space for the Q-rank 2 group Sp (R).
4
In this paper, we deal with the case when G has Q-rank 1, and use a family
of exhaustion functions to define a spine. We use the exhaustion functions to
construct a deformation retraction of D onto a spine. Each exhaustion function
can be thought of as a measure of height with respect to a rational parabolic
subgroupof G. The existence of suchfunctions is not new. Siegel defined a notion
ofdistance froma cuspfor SL ( ) [24]. One canshowthat his distance functions
2 k
O
are a power of our exhaustion functions. More directly, the exhaustion functions
come out of Saper’s work on tilings in [22]. In fact, our exhaustion functions are
nothing more than the composition of his normalized parameters (in the Q-rank
1 case) with the rational root. We use the exhaustion functions to construct a
deformation retraction of D onto a spine. More precisely,
Main Result. Let D =G/K be a symmetric space of non-compact type, where G
is the group of real points of a semisimple algebraic group G defined over Q with
Q-rank 1. Let Γ be an arithmetic subgroup of the rational points G(Q). There
exists a Γ-equivariant retract of the symmetric space onto a set D D with the
0
⊂
following properties:
(i) D is a locally finite union of semi-analytic sets.
0
(ii) dim(Γ D )=vcd(Γ).
0
\
(iii) D hasadecompositionD = D′( ),where rangesovercertainsubsets
0 0
I I
(of order at least 2) of parabolic Q-subgroups.
(iv) The decomposition satisfies γ`D′( )=D′(γ ) for every γ Γ.
· I I ∈
(v) The quotient Γ D is compact.
0
\
In fact, there is a (k 1)-parameter family of different retractions, where k is the
−
number of Γ-conjugacy classes of parabolic Q-subgroups.
For G=SL , our technique yields the same deformation retraction as the well-
2
roundedretract,which is an infinite trivalent tree in the Poincar´eupper half-plane
ON THE EXISTENCE OF SPINES FOR Q-RANK 1 GROUPS 3
(Figure 1). In this case, the exhaustion function corresponding to the cusp i is
∞
simply f (z)=Im(z). Furthercomparisonswith otherknownresultsaregivenin
i∞
Section 8.
SinceeachD′( )isasemi-analyticset,itfollowsfrom[19]thatthedecomposition
I
of the spine D = D′( ) may be refined to a regular cell complex. Thus, the
0
I
cohomology can be described by finite combinatorial data. In [29] we develop
`
machinery for the computation of cohomology using D , and use it together with
0
theresultsofthispapertoinvestigatethecohomologyofSU(2,1)overtheGaussian
integers.
Sections2and3setnotationanddefinethemainobjectsofthispaper. Sections4
and 5 give another interpretation of the exhaustion functions and prove some of
their properties. The main results are presented in Section 6. Section 7 introduces
the notion of a strictly separated linear algebraic group, and show that for these
groups, each D′( ) is a smooth, contractible submanifold. Section 8 concludes by
I
looking at a few examples of spines in low dimensional cases.
I wouldliketothank mythesis advisor,LesSaper,forhis insightintothis work.
I would also like to thank Paul Gunnells for helpful conversations.
2. Notation and background
In order to set notation, we briefly recall without proof some standard results
regarding algebraic groups over Q, the geodesic action, and the Borel-Serre com-
pactification [9]. We follow the exposition of [22]. Throughout this paper, G is
the identity componentofthe realpoints ofa semisimple Q-rank1 algebraicgroup
defined over Q. In order to lighten the notation and exposition, we will notate the
algebraic groupand its group of real points by the same Roman type (G=G(R)),
and we may refer to the algebraic group when properly we should referring to the
group of real points. For example, we will speak of parabolic Q-subgroups of G,
when we properly should be referring to the group of real points of a parabolic
Q-subgroupof the algebraic group G.
2.1. AlgebraicgroupsdefinedoverQ. ForareductivealgebraicgroupH which
is defined over Q, let S denote the maximal Q-split torus in the center of H, and
H
set A =S (R)0. Set
H H
0H = ker(χ2),
χ∈X\(H)Q
where X(H) denotes the rational characters of H defined over Q. Then H splits
Q
as a direct product
H =A 0H.
H
×
The group 0H H contains all compact and arithmetic subgroups of H.
⊂
For a parabolic subgroup P G, let N denote its unipotent radical of and let
P
⊂
ν :P L =P/N
P P P
→
denote the projection to the Levi quotient. Let M denote the group 0L . If
P P
S L denotes the maximal Q-split torus in the center of L , then L splits
P P P P
⊂
as a commuting direct product L = A M , where A = S (R)0. Note that L
P P P P P P
is the centralizer of A , and the connected component of the center of M is a
P P
Q-anisotropic torus.
4 DANYASAKI
Let ∆ = α denote the simple root of the adjoint action of A on n , the
Q P P P P
{ }
Lie algebra of N . The roots will be viewed as characters of A and as elements
P P
of a∗. Since G has Q-rank 1, α gives an isomorphism A R . Let
P P P → >0
(1) ψ :R A
P >0 P
→
be the isomorphism of groups given by ψ (t)=α−1(t).
P P
FortworationalparabolicsubgroupsP andQ,thereis acanonicalisomorphism
∼
A A
P Q
→
induced by an element of G(Q). In particular, the characters α can be
P P∈P
{ }
identified and will be denoted α.
Anylifti:L P determinesaLanglandsdecomposition,whichisasemi-direct
P
→
product
P =N i(A M ).
P P P
Note that L is the centralizer of A and the connected component of the center
P P
of M is a Q-anisotropic torus.
P
Let K G be a maximal compact subgroup and define D = G/K. There is a
⊂
uniquebasepointx D suchthatK =Stab (x). ThischoiceofK alsodetermines
G
∈
the following data [17]:
(i) A maximal compact subgroup K = K P G and a diffeomorphism
P
∩ ⊂
P/K D.
P
→
(ii) A Cartan involution θ :G G such that K is the subgroup fixed by θ .
x x
→
(iii) Auniqueθ -equivariantlifting i :L P. Fora subsetT L , denote
x x P P
→ ⊂
its lift by T(x).
Definition 2.1. [16] Let P G be a rational parabolic subgroup and x a point
⊂
in D. If the lift L (x) is an algebraicsubgroupof P, then x is a rational basepoint
P
for P.
If the basepoint x can be chosen so that the associated maximal compact sub-
group K is defined over Q, then x is rational for all rational parabolic subgroups
of G.
Definition2.2. LetS beamaximalQ-splittorusofG. TheWeyl group over Qor
Q-Weyl group is the quotient of the normalizer (S) of S by the centralizer (S)
N Z
of S, and is denoted
W = (S)/ (S).
Q
N Z
The following is the rational version of the standard Bruhat Decomposition for
Lie groups.
Theorem 2.3. [11] Let P G be a minimal rational parabolic subgroup. Then
⊂
G(Q)is thedisjoint unionoftheclassesP(Q)wP(Q)with[w] W. Inparticular,
Q
∈
given g G(Q), there exists u N , [w] W, and p P such that g =u wp .
g P Q g g g
∈ ∈ ∈ ∈
2.2. Borel-Serre compactification. [9] Let x denote the point of D fixed by K.
ThenP actstransitivelyonD,soeverypointz D canbewrittenasz =p x,
∈P ∈ ·
for some p P. The geodesic action of A on D is given by
P
∈
a z =(pa˜) x,
◦ ·
where a˜ is the image of a in A (x). The geodesic action commutes with the usual
P
action of P on D and is independent of the choice of basepoint. Let A 0P act
P
×
ON THE EXISTENCE OF SPINES FOR Q-RANK 1 GROUPS 5
on D by (a,p) z =p (a z)= a (p z). Then there is an analytic isomorphism
· · ◦ ◦ ·
of A 0P-homogeneous spaces
P
×
∼
(2) (a ,q ):D A e(P),
P,x P P
→ ×
where e(P)=A D is the quotient of D by the geodesic action of A . Normalize
P P
\
a sothata (x)=e. Via(2),D isatrivialprincipalA -bundlewithcanonical
P,x P,x P
cross-sections given by the orbits of 0P.
TheBorel-Serrecompactificationisthenconstructedasfollows: Thesimpleroot
α inducesanisomorphismA ∼ R definedbya aα. Thepartialbordification
P P >0
→ 7→
Dc(P) associated to P is defined to be A D. Equivalently, extend (2) to
P ×AP
(a ,q ):Dc(P) ∼ A e(P).
P,x P P
→ ×
TheBorel-SerrecompactificationD Dc(P)isthengiventhe unique struc-
≡ P∈P
tureofananalyticmanifoldwithboundarysothateachDc(P)isanopensubman-
S
ifold with boundary.
The action of G(Q) on D extends to an action on D, and for an arithmetic
subgroup Γ G(Q), the quotient Γ D is compact.
⊂ \
2.3. Γ-conjugacy classes of parabolic Q-subgroups. Fix a proper parabolic
Q-subgroup P of G. The parabolic Q-subgroups of G are all conjugate to P via
elementsofG(Q). Thus, the setofQ-parabolicsubgroups isinone-to-onecorre-
P
spondence with points of G(Q)/P(Q), where [g] G(Q)/P(Q) corresponds to the
parabolicQ-subgroupgP =Pg−1 gPg−1. Fora∈narithmeticsubgroupΓ G(Q),
≡ ⊂
the double coset space Γ G(Q)/P(Q) is finite [12] and the number of elements in
\
the space is known as the class number. In particular,there are only finitely many
Γ-conjugacyclassesofparabolicQ-subgroupsofG. ForP ,letΓ Γ P 0P.
P
∈P ≡ ∩ ⊂
Then there is a bijection between Γ/Γ and the parabolic subgroups that are Γ-
P
conjugate to P via γΓ γP.
P
7→
3. Main definitions
In this section, we define a family f of exhaustion functions depending
P P∈P
{ }
onaΓ-invariant parameter. We alsodefine subsetsofD associatedtothefamily of
exhaustionfunctions. InSection6,these functions areusedtogivea Γ-equivariant
retraction of D onto a codimension 1 set D .
0
Definition 3.1. A parameter is a family O of closed submanifolds of D
P P∈P
{ }
such that each O has the form
P
O =0P x for some x D.
P P P
· ∈
A parameter is Γ-invariant if
γ O =O for all γ Γ and P .
P γP
· ∈ ∈P
Recall that by conjugation by G(Q), one can canonically identify the A and
P
corresponding simple root α for different P . Denote the simple root by α
P
∈ P
and view it as a character on each A .
P
For P G a rational parabolic subgroup, let
⊂
(3) a :D D A
P P
× →
be the map a (z,x) = a (z). Thus a (z,x) can be viewed as the amount of
P P,x P
geodesic action required to push 0P x to 0P z. The following is immediate from
· ·
the definitions and the 0P-invariance of a .
P,x
6 DANYASAKI
Proposition 3.2. [9] Let P G be a rational parabolic subgroup. Then
⊂
(i) a (z,x)=a (x,z)−1 for all z,x D.
P P
∈
(ii) a (g z,g x)=a (z,x) for all z, x D and g G(Q).
gP P
· · ∈ ∈
(iii) a (z,x)=a (p z,x)=a (z,p x) for all p 0P.
P P P
· · ∈
(iv) a (z,s)a (s,x)=a (z,x) for all z, x, and s in D.
P P P
Given a point x D and rational parabolic subgroup P , consider the
∈ ∈ P
function f : D R given by f(z) = a (z,x)α. Proposition 3.2 implies that f
>0 P
→
only depends on the orbit 0P x. This leads to the following definitions.
·
Definition 3.3. The exhaustion functions associated a parameter 0P x is
P P∈P
{ · }
the family f of functions f :D R given by
P P∈P P >0
{ } →
f (z)=a (z,x )α.
P P P
A family of exhaustion functions associated to a Γ-invariant parameter is called a
family of Γ-invariant exhaustion functions.
Notice that Proposition 3.2 implies that
(4) f (g z)=f (z)f (g x ) for all g G(Q),
gP P gP P
· · ∈
sothatinparticular,forexhaustionfunctionsassociatedtoaΓ-invariantparameter,
(5) f (γ z)=f (z) for all γ Γ.
γP P
· ∈
For a parabolic P, define D(P) D to be the set of z D such that f (z)
P
⊂ ∈ ≥
f (z) for every Q P . More generally, for a subset ,
Q
∈P \{ } I ⊆P
(6) E( )= z D f (z)=f (z) for every pair P,Q
P Q
I { ∈ | ∈I}
(7) D( )= D(P)
I
P∈I
\
(8) D′( )=D( ) D( ′).
I I \ I
I′)I
[
It follows that D′( ) D( ) E( ) and D( )= D′(˜).
I ⊆ I ⊂ I I I˜⊇I I
Definition 3.4. The set D′( ) will be called a de`generate tile. Let fI denote the
I
restriction to E( ) of f for P .
P
I ∈I
Definition 3.5. Let , P , and z E( ). Then z is called a first contact
I ⊆P ∈I ∈ I
for if f (z) is a global maximum of f on E( ).
I I
I I
Definition 3.6. A subset is called admissible if D( ) is non-empty and
I ⊂ P I
strongly admissible if D′( ) is non-empty.
I
Proposition 3.7. Let denote the collection of strongly admissible subsets of .
S P
Then the symmetric space has a Γ-invariant degenerate tiling
D = D′( ),
I
I∈S
a
such that γ D′( )=D′(γ ) for all γ Γ and .
· I I ∈ I ∈S
Definition3.8. GivenafamilyofΓ-invariantexhaustionfunctions,defineasubset
D D by
0
⊂
D = D′( ).
0
I
I∈S
a
|I|>1
ON THE EXISTENCE OF SPINES FOR Q-RANK 1 GROUPS 7
Let f denote the function on D given by
D0 0
(9) f (z)=f (z) for z D( ).
D0 I ∈ I
4. Exhaustion functions via representation theory
In this section we describe a systematic way to construct exhaustion functions.
Fix a parabolic Q-subgroup P G. Choose a rational basepoint x∗ D for P
⊂ ∈
sothattheθx∗-stableliftAP(x∗)ofAP toP istheconnectedcomponentofthereal
points of a maximally Q-split torus of G. Let MP(x∗) denote the θx∗-stable lift of
M =0L toP,andleta denotetheLiealgebraofA (x∗). ThesimpleQ-rootα
P P P P
canbeviewedasalinearfunctionalona . Letg denotethecomplexificationofg.
P C
FixaCartansubalgebracontainingthecomplexificationofa ,andlet α˜ ,...α˜
P 1 m
{ }
be the simple roots. Order the simple roots so that α˜j|aP = α for 1 ≤ j ≤ l and
α˜j|aP = 0 for l +1 ≤ j ≤ m. Let {ω˜1,...,ω˜m} be the associated fundamental
weights. Consider the weight ω˜ = s l ω˜ . Then for s a sufficiently large
j=1 j
positive integer, ω˜ is the highest weight of a finite-dimensional (strongly rational)
P
representationof G [13].
Let (V,π) denote this representation, and ω the restriction of ω˜ to a . The
P
restriction to a of weights are called restricted Q-weights. Every restricted Q-
P
weightof V is of the form ω jα, where j is a non-negative integer and the lowest
−
restrictedQ-weightis ω. ThusV hasadecompositionasadirectsumofrestricted
−
Q-weight spaces,
N
(10) V = V = V , where ω Nα= ω.
λ ω−kα
− −
λ k=0
M M
Fix an admissible inner product , on V [10], that is, one for which
h· ·i
π(g)∗ =π(θx∗g)−1 for all g G,
∈
so that in particular,
π(k)∗ =π(k)−1 for all k K.
∈
Since the Q-Weyl group W is finite, by averaging over W one can arrange
Q Q
thattheinnerproductisalso W-invariant. Thedata(V,π, , ,x∗)willbecalled
Q
h· ·i
a P-adapted representation.
Proposition4.1. Let P G be a parabolic Q-subgroupwith rational basepoint x∗,
⊂
and let (V,π, , ,x∗) be a P-adapted representation. Then M (x∗) preserves the
P
h· ·i
restricted Q-weight spaces of V. The highest and lowest restricted weight spaces are
one-dimensional, and M (x∗) preserves length on both.
P
Proof. Since M (x∗) centralizes A (x∗), it follows that M (x∗) preserves the re-
P P P
stricted weight space decomposition of V. Since ω˜ is orthogonal to α˜ for l+1
j
≤
j m, it follows that the highest weight space V is 1-dimensional. Furthermore,
ω
≤
thisimpliesthatthe connectedcomponentofM (x∗)actstriviallyonV . M (x∗)
P ω P
only has finitely many connected components because M A = L , the real
P × P ∼ P
points of the Levi quotient, which has only finitely many connected components
since it is Zariski connected [28]. It follows that M (x∗) preserves length when
P
restricted to V . An analogous argument shows that V is 1-dimensional, and
ω −ω
M (x∗) preserves length on V . (cid:3)
P −ω
8 DANYASAKI
Write z D as z =p x with p P. Using Langlandsdecomposition, writep as
∈ · ∈
ua˜ (z,x∗)m, where u N , a˜ (z,x∗) A (x∗), andm M (x∗). Fix g G(Q).
P P P P P
∈ ∈ ∈ ∈
From (10) there exists unit vectors v V and constants c (z) C such that
k ω−kα k
∈ ∈
(11) π(p−1g)v =π(a˜ (z,x∗))−1 c (z)v .
P k k
k
X
Notethatc (z)isonlydefineduptomultiplicationbynorm1scalars. SinceM (x∗)
k P
centralizes A (x∗), the c (z) only depend on m and u. Thus for each fixed g, the
P k
norm c can be viewed as a function from D R that is invariant under the
k ≥0
| | →
geodesic action of A .
P
Proposition 4.2. Fix g G(Q) and two distinct rational parabolic subgroups P
∈
and Q = gP. Fix a rational basepoint x∗ for P and let (V,π, , ,x∗) be a P-
h· ·i
adapted representation. Then the functions c :D R (defined above) satisfy
k
| | →
f (g x )f (z)
f (z)= Q · P P for all z D.
Q 1/N ∈
N c (z)2f (x∗)−2kf (z)2k
k=0| k | P P
(cid:16) (cid:17)
Proof. From the defiPnition of the P-adapted representation, α= 2ω. Write z D
N ∈
asz =p x∗,where p P. Letv V denote anorm1highestweightvector. Since
· ∈ ∈
N fixes v, Proposition4.1 implies
P
(12) π(p−1)v −2/N = π(m−1a˜ (z,x∗)−1u−1)v −2/N =a (z,x∗)α.
P P
k k k k
Then from the definition of f and (12),
P
(13) f (z)=a (z,x∗)αf (x∗)= π(p−1)v −2/Nf (x∗).
P P P P
k k
Then (4), (13), and the orthogonality of the restricted weight spaces imply,
f (z)=f (g−1 z)f (g x )
Q P Q P
· ·
N 2 −1/N
=f (g x )f (x∗) π(a˜ (z,x∗))−1 c (z)v
Q P P P k k
·
(cid:18)(cid:13) k=0 (cid:13) (cid:19)
(cid:13) X (cid:13)
(cid:13) N (cid:13)2 −1/N
=f (g x )f (x∗) (cid:13) c (z)a (z,x∗)−ω+kαv (cid:13)
Q P P k P k
·
(cid:18)(cid:13)k=0 (cid:13) (cid:19)
(cid:13)X (cid:13)
(cid:13)N (cid:13) −1/N
=f (g x )f (x∗) (cid:13) c (z)2a (z,x∗)−2ω+2kα(cid:13)
Q P P k P
· | |
(cid:18)k=0 (cid:19)
X
N −1/N
=f (g x )f (x∗)a (z,x∗)2ω/N c (z)2a (z,x∗)2kα
Q P P P k P
· | |
(cid:18)k=0 (cid:19)
X
N −1/N
=f (g x )f (z) c (z)2f (x∗)−2kf (z)2k . (cid:3)
Q P P k P P
· | |
(cid:18)k=0 (cid:19)
X
Proposition 4.3. Let P and Q = gP be two distinct Q-parabolic subgroups with
g G(Q). Letg =u wp betheQ-Bruhatdecomposition ofg asinProposition 2.3.
g g
∈
Use the Langlands decomposition to express z D as z = ua˜ (z,x∗)m x∗ for a
P
∈ ·
rational basepoint x∗ for P, u N , m M (x∗), and a˜ (z,x∗) A (x∗). Then
P P P P
∈ ∈ ∈
c (z) =a (p x∗,x∗)ω.
N P g
| | ·
If c (z) =0 for k =0,...,N 1, then
k
| | −
ON THE EXISTENCE OF SPINES FOR Q-RANK 1 GROUPS 9
(i) u=u .
g
(ii) z is a first contact for P,Q .
{ }
(iii) f (z)=f (g x )1/2a (p x∗,x∗)−α/2f (x∗).
P Q P P g P
· ·
Proof. From the definition of c (z),
k
| |
c (z) = pr (π(m−1u−1g)v)
| k | k Vω−kα k
(14) = pr (π(m−1u−1u wp )v)
k Vω−kα g g k
=a (p x∗,x∗)ω pr (π(m−1u−1u w)v) .
P g· k Vω−kα g k
Notice that π(w)v V and for n N ,
−ω P
∈ ∈
π(nw)v =π(w)v+higher weight vectors.
Since M (x∗) preserves weight spaces and preserves length on V by Proposi-
P −ω
tion 4.1,
(15) c (z) =a (p x∗,x∗)ω pr (π(m−1u−1u w)v) =a (p x∗,x∗)ω.
| N | P g· k V−ω g k P g·
Now suppose c (z) =0 for k =0,...,N 1. Then from (14), π(m−1u−1u w)v
k g
| | − ∈
V . Since π(m−1)preservesweightspacesbyProposition4.1,π(u−1u )preserves
−ω g
V . It follows that u=u .
−ω g
Note that since z E( P,Q ),
∈ { }
N c (z)2 c (z)2
f (g x )N = | k | f (z)2k | N | f (z)2N.
Q · P f (x∗)2k P ≤ f (x∗)2N P
P P
k=0
X
Since c (z) =0 and is independent of z from (15),
N
| |6
f (g x )1/2f (x∗)
Q P P
(16) f (z) · on E( P,Q ).
P ≤ c (z)1/N { }
N
| |
The bound is attained when u(z)=u and
g
f (g x )1/2f (x∗) f (g x )1/2f (x∗)
f (z)= Q · P P = Q · P P (cid:3)
P c (z)1/N a (p x∗,x∗)α/2
N P g
| | ·
5. Properties of Exhaustion Functions
Notice that for a family of Γ-invariant exhaustion functions,
(17) f (γ x )=1 for every γ Γ.
γP P
· ∈
Proposition 3.2 implies the following.
Proposition 5.1. Let P G be a rational parabolic subgroup and f a
P P∈P
⊂ { }
family of exhaustion functions associated to a Γ-invariant parameter 0P x .
P P∈P
{ · }
Then
(i) f (x )=1.
P P
(ii) f (a z)=aαf (z) for all z D and a A .
P P P
◦ ∈ ∈
(iii) f (z)=a (z,x)αf (x) for all z, x D.
P P P
∈
(iv) f (p z)=f (z) for all z D and p 0P.
P P
· ∈ ∈
(v) f (z)=f (g−1z)f (g x ) for all z D and g G(Q).
gP P gP P
· ∈ ∈
The following proposition follows from reduction theory [12].
10 DANYASAKI
Proposition 5.2. Let f be a Γ-invariant family of exhaustion functions.
P P∈P
{ }
Then there exists a constant C >0, depending on the Γ-invariant parameter, such
that
supf (z) C for all z D.
P
≥ ∈
P∈P
Lemma 5.3. Let M be a Riemannian manifold and ψ an isometry of M. Let f
and h be smooth functions on M such that h(x)=f(ψ(x)). Then
h(x)=(dψ−1) f(ψ(x)) for all x M.
ψ(x)
∇ ∇ ∈
Proposition 5.4. Let x, x′ D. If f (x)=f (x′) then f (x) = f (x′) .
P P P P
∈ k∇ k k∇ k
Proof. If f (x) = f (x′), then there exists p 0P such that p x = x′. Since
P P
∈ ·
f is 0P-invariant, f (p z) = f (z) for all z D. The result then follows from
P P P
Lemma 5.3 by noting tha·t 0P G acts by isom∈etries on D. (cid:3)
⊂
Lemma 5.5. Let P G be a rational parabolic subgroup. The geodesic action of
⊂
A acts as an isometry on the 1-dimensional tangent space T (A z).
P z P
◦
Proof. Since the geodesic action of A is equal to the regular action of its θ -
P z
invariant lift on the orbit A z, the geodesic action acts as an isometry on the
P
◦
1-dimensional tangent space T (A z). Furthermore, the geodesic action of A
z P P
commutes with the regular action of◦P on D. The result follows. (cid:3)
Proposition 5.6. Fix two distinct rational parabolic subgroups P and Q. Then
f (z) f (z) =f (z) f (z) for all z D.
P Q Q P
k∇ k k∇ k ∈
In particular, if z E( P,Q ), then f (z) = f (z) .
P Q
∈ { } k∇ k k∇ k
Proof. There exists a g G(Q) such that Q=gP. By Proposition 5.1,
∈
(18) f (z)=f (g−1 z)f (g x ).
Q P Q P
· ·
Then Lemma 5.3 implies that
(19) ∇fQ(z)=fQ(g·xP)(dLg)g−1·z∇fP(g−1·z),
where L : D D is the isometry given by left translation by g G. Since f is
g P
→ ∈
0P-invariant,
(20) fp(g−1 z)=(dLp−1)pg−1·z fP(pg−1 z) for any p 0P.
∇ · ∇ · ∈
There exists a p 0P such that pg−1z =t z, where t=ψ fP(g−1z) . Note that
∈ ◦ P fP(z)
(cid:16) (cid:17)
f (g−1z)
P
(21) f (t z)= f (z).
P P
◦ f (z)
(cid:18) P (cid:19)
Then by Lemma 5.3 and (21)
f (g−1z)
(22) f (pg−1 z)= P d(t ) f (z).
P z P
∇ · f (z) ◦· ∇
(cid:18) P (cid:19)
Combining (19), (20), and (22),
f (z)
Q
(23) ∇fQ(z)= f (z)(dLgp−1)pg−1·zd(t◦·)z∇fP(z).
P
Thus by Lemma 5.5, f (z) f (z) =f (z) f (z) . (cid:3)
P Q Q P
k∇ k k∇ k