The subgroup $PSL(2,R)$ is spherical in the group of diffeomorphisms of the circle

0.11 MB·English

by  Yury A. Neretin

The subgroup PSL (R) is spherical in the group of
2
diffeomorphisms of the circle
Yury A. Neretin1
16 We show that the group PSL2(R) is a spherical subgroup in the group of C3-
0 diffeomorphismsofthecircle. Also,thegroupofautomorphismsofaBruhat–Titstree
2 is a spherical subgroup in thegroup of hierarchomorphisms of the tree.

y 1. Sphericity. Let G be a topological group, K be a subgroup. An
a irreducibleunitaryrepresentationρofGinaHilbertspaceH iscalledspherical
M
if there is a unique up to a scalar factor non-zero K-invariant vector v in H.
The matrix element
8
Φ(g):=hρ(g)v,vi
2
is called a spherical function on G. A subgroup K in G is called spherical if
]
T for any irreducible unitary representation of G the dimension of the space of
R K-invariantvectors is 61.
. Forvarioustypesofsphericalpairsinthissense,see[5],[8],[18],[3],[16],[15].
h
For all known examples the group K is compact or is an infinite-dimensional
t
a analog of compact groups as U(∞), O(∞), Sp(∞), S(∞) etc. (’heavy groups’
m
in the sense of [13]).
2 [ dete2r.minSatnatte1m, leetntPsS.LL(Ret)SbLe2i(tRs)quboetiethntewgirtohupresopfe2ct×to2threeaclenmtaert,riScLes(wRi)t∼h
2 2
v betheuniversalcoveringgroup. DenotebyDiff (respectivelybyDiff3)thegroup
0 ofC∞-smooth(resp. C3-smooth)orientationpreservingdiffeomorphismsofthe
2 ∼ ∼
circle. Denote by Diff the universal covering of Diff, we realize Diff as the
8
5 group of smooth diffeomorphisms q of the line R satisfying the condition
0
. q(ϕ+2π)=q(ϕ)+2π.
1
0
∼
5 The Bott cocycle c(q1,q2) on Diff is defined by the formula
1
v: c(q ,q )= 2πlnq′(q (ϕ)dlnq′(ϕ).
1 2 Z 1 2 2
i 0
X
∼
r Consider the central extension Diff of Diff determined by the Bott cocycle
a
3
(see, e.g., [4], §3.4). By Diff wegdenote the similar central extension of Diff3.
g
Theorem 1 The subgroup PSL (R) is spherical in the group Diff3.
2
Theorem 2 The subgroup PSL∼(R) is spherical in the group Diff3.
2
Remark. a) Several series of spherical representations ofgDiff and Diff∼
were constructed in [11], see also [13], §IX.6. All such spherical representations
are continuous in the C3-topology.
1SupportedbythegrantFWF,ProjectP25142
1
b) Sobolev diffeomorphisms of the circle of the class s > 3/2 form a group
(see [7], Theorem 1.2 and Appendix B). Our proof survives for the group of
Sobolev diffeomorphisms of the class H4.5. ⊠.
Next,consideracombinatorialanalogofDiff. Fixanintegern>2. Consider
the Bruhat–Tits tree T , i.e., the infinite tree such that each vertex is incident
n
to n+1 edges. Let Abs(T ) be its boundary (for detailed definitions, see, e.g.,
n
[17], [12]). Denote by Aut(T ) the group of all automorphisms of the graph
n
T . It is a locally compact group, stabilizers of finite subtrees form a base of
n
open-closed neighborhoods of unit.
Denote by Vert(T ) the set of vertices of T . Consider a bijection θ :
n n
Vert(T ) → Vert(T ) such that for all but a finite numbers of pairs of adja-
n n
cent vertices (a,b), vertices θ(a), θ(b) are adjacent. Hierarchomorphism of the
treeT isarehomeomorphismsofAbs(T )inducedbysuchmaps,see[12],[14].
n n
Denote by Hier(T ) the group of all hierarchomorphisms of the tree T .
n n
Remark. a) For a prime n = p the boundary Abs(T ) can be identified
p
with a p-adic projective line. The group Aut(T ) contains the p-adic PSL and
p 2
the representation theory of Aut(T ) is similar to the representation theory of
p
p-adic and real SL (see [2], [17]). The group Hier(T ) contains the group of
2 p
locally analytic diffeomorphisms of the p-adic projective line.
b) Richard Thompson groups (see [1]) are discrete subgroups of Hier(T ).
n
c) Several series of Aut(T )-spherical representations of Hier(T ) were con-
n n
structed in [10], [12]. ⊠
We define a topology on the group Hier(T ) assuming that Aut(T ) is an
n n
open subgroup (the coset space Hier(T )/Aut(T ) is countable).
n n
Theorem 3 The subgroup Aut(T ) is spherical in Hier(T ).
p p
Theorem 4 Let G ⊃ K be a spherical pair. Assume that K does not admit
nontrivial finite-dimensional unitary representations. Let Φ (g), Φ (g) be K-
1 2
spherical functions on G. Then Φ (g)Φ (g) is a spherical function.
1 2
The both K =SL (R)∼ and Aut(T ) satisfy this condition.
2 n
3. Proof of Theorem 1. Fix a point a in the circle. Denote by G ⊂Diff
0
the group of diffeomorphsims q such that q(x)=x in a neighborhood of a. By
∗
G we denote the group of diffeomorphisms that are flat at a, i.e.,
′ ′′ ′′′
q(a)=a, q (a)=1, q (a)=q (a)=···=0
Let ρ be an irreducible unitary representation of Diff in H. Denote by V
the subspace of all PSL (R)-fixed vectors. Let P be the operator of orthogonal
2
projection on V. For h∈PSL (R) we have
2
Pρ(h)=ρ(h)P =P.
Denote
ρ(g):=Pρ(g)P.
b
2
This operator depends only on a double coset of Diff by PSL (R),
2
ρ(h gh ):=ρ(g), h ,h ∈PSL (R).
1 2 1 2 2
Lemma 5 If ρ is cbontinuous inbthe C3-topology, then the operators ρ(g) pair-
wise commute.
b
Proof. The following statement is our key argument:
Let a sequence h ∈ PSL (R) converges to infinity2. Then ρ(h ) weakly
j 2 j
converges to P,seeHowe,Moore[6],Theorem5.1(this is ageneraltheoremfor
semisimple group, for PSL (R) it can be easily verified case-by-case).
2
Let us realize the circle as the real projective line RP1 = R∪∞. Without
loss of generality we can set a=∞. Let U (x)=x+t be a shift on R, we have
t
U ∈PSL (R). Consider diffeomorphisms r, q ∈G . For sufficiently large t the
t 2 0
supports of r and Ut ◦q ◦U−t are disjoint. Therefore, these diffeomorphisms
commute. Hence,
ρ(r)ρ(Ut)ρ(q)ρ(U−t)=ρ(Ut)ρ(q)ρ(U−t)ρ(r).
Therefore,
P ρ(r)ρ(Ut)ρ(q)P =P ρ(q)ρ(U−t)ρ(r)P.
Passing to a weak limit as t→+∞, we get
P ρ(r)P ρ(q)P =P ρ(q)P ρ(r)P.
Thus
ρ(r)ρ(q)=ρ(q)ρ(r), where r, q ∈G .
0
∗
ButG0 is denseinbG .bTherefobrethbe sameidentityholdsforr, q ∈G∗. Indeed,
let r , q ∈ G be sequences convergent to r, q respectively. Passing to the
j j 0
iterated limit
lim lim ρ(r )ρ(q ) = lim lim ρ(q )ρ(r )
j k k j
j→∞(cid:16)k→∞ (cid:17) j→∞(cid:16)k→∞ (cid:17)
b b b b
and keeping in mind the separate weak continuity of the operator product, we
get the desired statement.
Our last argument: the set PSL2(R)· G∗ · PSL2(R) is dense in Diff with
respect to the C3-topology.
Letusprovethis. ChooseacoordinateonRP1 suchthata=0. Letq ∈Diff.
Consider its Schwarzian derivative,
q′q′′′− 3(q′′)2
S(q)= 2 .
(q′)2
ConsiderapointbsuchthatS(q)(b)=0(bythe Ghys theorem,the Schwarzian
derivativeofadiffeomorphismsofthecirclehasatleast4zero,see[19],Theorem
4.2.1). Then for
r :=U−q(b)◦q◦Ub
2I.e.,foranycompactsubsetB,wehavehj ∈/B startingsomenumber.
3
we have r(0)=0, S(r)(0)=0. Consider maps
ux
σ(x)= ,
u−1+vx
such σ ∈PSL (R) fix 0. Choosing parameters u, v, we can achieve
2
′ ′′
(r◦σ)(0)=1, (r◦σ) (0)=0.
Recall the transformation property of the Scwarzian:
S(r◦σ)=(S(r)◦σ)·(σ′)2+S(σ).
′′′
Sinceσislinearfractional,S(σ)=0. ThereforeS(r◦σ)=0,and(r◦σ) (0)=0.
Such rσ can be approximated in C3-topology by elements of G∗. This proves
Lemma 5. (cid:3)
Theorem1 is a corollaryof the lemma. Note, that ρ(g)∗ =ρ(g−1). Thus we
∗
get a family of commuting operators in V, such that an adjoint operator A is
b b
contained in the family together with A. If dimV > 1, then this family has a
properinvariantsubspaceinV,sayW. ConsidertheDiff-cyclicspanofW,i.e.,
the subspace Z spanned by vectors ρ(g)w, where g ∈Diff3 and w ∈W. Then
Pρ(g)w =Pρ(g)P w =ρ(g)w ∈W.
Hence, PZ =W and therefore Z is a proper sbubspace in H.
4. Proof of Theorem 2. Itrepeatsthepreviousproofwithtwoadditional
remarks.
1)Considerthehomomorphismπ :SL (R)∼ →PSL (R)≃SL (R)∼/Z. We
2 2 2
say that a sequence h ∈ SL (R)∼ converges to ∞ if π(h ) → ∞. Then the
j 2 j
Howe–More theorem remains valid.
2)Forapairofdiffeomorphismswithdisjointsupportsp, q theBottcocycle
c(q,p)vanishes,hencethediffeomorphismsp,qcommuteintheextendedgroup.
5. Proof of Theorem 3. First, there is the following analogof the Howe–
Moore theorem: Let a sequence h ∈ Aut(T ) converges to ∞. Then for any
j n
unitary representation ρ of Aut(T ) the sequence ρ(h ) converges to the projec-
n j
tion to the subspace of Aut(T )-fixed vectors, see [9]; this can be easily verified
n
case-by-case starting the classification theorem of [17].
Second, fix a point a ∈ Abs(T ) and denote by G the group of hierar-
n 0
chomorphisms that are trivial in a neighborhood of a. Let q, r ∈ G . Then
0
there is a sequence h ∈Aut(T )∩G such that h tends to ∞ and supports of
j n 0 j
h ph−1 and q are disjoint. We omit a proof, since it is easier to understand its
j j
self-evidence than to read a formal exposition.
Third,
Aut(T )·G ·Aut(T )=Hier(T )
n 0 n p
Now we can repeat the proof of Theorem 1.
6. Proof of Theorem 4. The statement is semitrivial.
4
Lemma 6 Let ν ν be unitary representations of a group Γ. If the tensor
1 2
product ν ⊗ν contains a nonzero Γ-invariant vector, then the both ν and ν
1 2 1 2
have finite-dimensional subrepresentations.
Proof of the lemma. Assume that an invariant vector exists. Denote
the spaces of representations by V , V . We identify V ⊗V with the space
1 2 1 2
′ ′
of Hilbert–Schmidt operators V → V , where V is the dual space to V . An
1 2 1 1
′
invariant vector corresponds to an intertwining operator T : V → V . The
1 2
∗ ∗
operator TT is an intertwining operator in V . Since TT is compact and
2
nonzero,ithasafinite-dimensionaleigenspace,andthissubspaceisG-invariant.
(cid:3)
Proof of the theorem. Let ρ and ρ be K-spherical representations
1 2
of G in H and H . Let v , v be fixed vectors. By the lemma, v ⊗v is a
1 2 1 2 1 2
uniqueK-fixedvectorinH ⊗H . ThecyclicspanW ofv ⊗v isanirreducible
1 2 1 2
subreprepresentation. Indeed, let W = W ⊕W be reducible. Then the both
1 2
projectionsofv ⊗v toW ,W areK-fixed,thereforev ⊗v mustbecontained
1 2 1 2 1 2
inone of summands, sayW , and thus the cyclic spanof v ⊗v is containedin
1 1 2
W , i.e., W =W .
1 1
Now we consider the representation of G in W,
ρ (g)⊗ρ (g) v ⊗v ,v ⊗v =hρ (g)v ,v i ·hρ (g)v ,v i
1 2 1 2 1 2 W 1 1 1 H1 2 2 2 H2
(cid:10)(cid:0) (cid:1) (cid:11)
=Φ (g)Φ (g).
1 2
References
[1] Cannon, J. W., Floyd, W. J.. Parry, W. R. Introductory notes on Richard
Thompson’s groups. Enseign. Math. (2) 42 (1996), no. 3-4, 215–256
[2] Cartier,P.Geo´m´otrie et analyse sur les arbres,LectureNotesinMath.317
(1973) 123–140.
[3] Ceccherini-Silberstein, T.; Scarabotti, F.; Tolli, F. Harmonic analysis on
finite groups. Cambridge University Press, Cambridge, 2008.
[4] Fuks, D. B. Cohomology of infinite-dimensional Lie algebras. Consultants
Bureau, New York, 1986.
[5] Gelfand, I. M. Spherical functions in symmetric Riemann spaces. Doklady
Akad. Nauk SSSR (N.S.) 70, (1950), 5–8.
[6] Howe,R.E.;Moore,C.C.Asymptoticpropertiesofunitaryrepresentations.
J. Funct. Anal. 32 (1979), no. 1, 72–96.
[7] Inci, H., Kappeler, T., Topalov, P. On the regularity of the composition of
diffeomorphisms. Mem. Amer. Math. Soc. 226 (2013), no. 1062.
[8] Kr¨amer, M. Spha¨rische Untergruppen in kompakten zusammenh¨angenden
Liegruppen, Compositio Math. 38 (1979), 129–153.
5
[9] Lubotzky, A., Mozes, Sh. Asymptotic properties of unitary representations
of tree automorphisms. in Harmonic analysis and discrete potential theory
(Frascati, 1991), 289–298,Plenum, New York, 1992.
[10] Neretin, Yu. A. Unitary representations of the groups of diffeomorphisms
of the p-adic projective line. Funct. Anal. Appl. 18 (1984), no. 4, 345–346.
[11] Neretin, Yu. A. Almost invariant structures and constructions of unitary
representations of the group of diffeomorphisms of the circle. Soviet Math.
Dokl. 35 (1987), no. 3, 500–504.
[12] Neretin, Yu. A. Combinatorial analogues of the group of diffeomorphisms
of the circle. Russian Acad. Sci. Izv. Math. 41 (1993), no. 2, 337–349.
[13] Neretin, Yu. A. Categories of symmetries and infinite-dimensional groups.
Oxford University Press, New York, 1996.
[14] Neretin, Yu. A. Groups of hierarchomorphisms of trees and related Hilbert
spaces. J. Funct. Anal. 200 (2003), no. 2, 505–535.
[15] Neretin, Yu. A. Sphericity and multiplication of double cosets for infinite-
dimensional classical groups. Funct. Anal. Appl. 45 (2011),no.3, 225–239.
[16] Nessonov, N. I.Factor-representation of the group GL(∞) and admissible
representations GL(∞), Mat. Fiz. Anal. Geom. (Kharkov Math. J.), 10:4
(2003), 167–187.
[17] Olshanski, G. I. Classification of the irreducible representations of the au-
tomorphism groups of Bruhat-Tits trees. Funct. Anal. Appl. 11 (1977), no.
1, 26–34.
[18] Olshanski,G.I.Unitaryrepresentationsofinfinitedimensionalpairs(G,K)
and the formalism of R. Howe, in: Representation of Lie Groups and Re-
lated Topics, 1990, pp. 269–463
[19] Ovsienko, V., Tabachnikov, S. Projective differential geometry old and
new. From the Schwarzian derivative to the cohomology of diffeomorphism
groups. Cambridge University Press, Cambridge, 2005.
Math.Dept., University of Vienna,
Oskar-Morgenstern-Platz 1, 1090 Wien;
& Institute for Theoretical and Experimental Physics (Moscow);
& Mech.Math.Dept., Moscow State University.
e-mail: neretin(at) mccme.ru
URL:www.mat.univie.ac.at/∼neretin
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