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c aiainadtpgahcdetail. typographic 2224–2257 and 6, the pagination by No. published 38, Vol. article 2010, original Statistics the Mathematical of of reprint Institute electronic an is This λ,σ ≥ nttt fMteaia Statistics Mathematical of Institute ADMLEGOPATOSO OPC MANIFOLDS: COMPACT ON ACTIONS GROUP LIE RANDOM ups ie i group Lie a given Suppose applications. and discussion results, Main 1. e od n phrases. and words Key 1 M 00sbetclassifications. subject 2000 AMS 2009. August revised 2008; February Received upre yteDFG. the by Supported 10.1214/10-AOP544 n parameter a and 0 G ∈ · : yCrsinSdladHranSchulz-Baldes Hermann and Sadel Christian By in oteter fpout frno arcsadamodel a t and on matrices presented. A random are measure constant. of wire coupling smooth products quantum the of unique disordered of theory a order the the to w.r.t. of tions integral errors turns to its then up to function manifold smooth equal given be any of to sum replac Birkhoff and in The elements hypothesis lems. algebra irreducibility effectiv Furstenberg’s Lie or This random transience of bu randomness. weak, terms the of i in of regime expressed paper a coupling this in effective of sums ficiently object Birkhoff main associated The of evaluation process. Markov time discrete M → M × G eafml fgopeeet eedn naculn constan coupling a on depending elements group of family a be admLegopato nacmatmnfl eeae a generates manifold compact a on action group Lie random A ETRAIEANALYSIS PERTURBATIVE A Thus, . Universit¨at Erlangen–N¨urnberg M ru cin nain esr,Brhffsum. Birkhoff measure, invariant action, Group σ T λ,σ ayn nsm rbblt pc (Σ space probability some in varying ihu onayadasot,tastv group transitive smooth, a and boundary without 2010 , = hsrpitdffr rmteoiia in original the from differs reprint This . ⊂ G M R 00,3H5 37H15. 37H05, 60J05, exp sahmgnossae utemr,let Furthermore, space. homogeneous a is in GL(
h naso Probability of Annals The 1 n X L, ∞ =1 C λ ,acmat once,smooth connected, compact, a ), n P n,σ ! hswr rvdsaper- a provides work This , 1 eae prob- related ro hison chains arkov , ns is eness pplica- sthe es suf- t the s , fa of p out aue are easures he i section his ,wihis which ), ti,and etail, t 2 C. SADEL AND H. SCHULZ-BALDES where and n,σ are measurable maps on Σ with compact image in the LieR algebra ∈ G g ofP such that G 1/n (2) lim sup sup( n,σ ) < n σ Σ kP k ∞ →∞ ∈ for some norm on g. This implies that λ,σ is well defined and analytic in λ for λ sufficiently small. The expectationT value of the first-order term P1,σ will be denoted by = p(dσ) 1,σ . Let us consider theP product probabilityP space (Ω, P)=(ΣN, pN). Associ- R ated to ω = (σn)n N Ω, there is a sequence ( λ,σn )n N of group elements. ∈ ∈ T ∈ An -valued Markov process xn(λ, ω) with starting point x0 is defined iterativelyM by ∈ M
(3) xn(λ, ω)= λ,σn xn 1(λ, ω). T · − The averaged Birkhoff sum of a complex function f on is M N 1 N 1 1 − 1 − (4) I (f)= E f(x (λ, ω)) = (T nf)(x ), λ,N ω N n N λ 0 n=0 n=0 X X where in the second expression we used the Markov transition operator (T f)(x)= E (f( x)). Here and below, expectation values w.r.t. P (or λ σ Tλ,σ · p) will be denoted by E (or Eω and Eσ). Next, recall that an invariant mea- sure νλ on is defined by the property νλ(dx)f(x)= νλ(dx)(Tλf)(x). The operatorM ergodic theorem [16], Theorem 19.2, then states that I (f) R R λ,N converges almost surely (in x0) w.r.t. any invariant measure νλ and for any integrable function f. In the case that is a projective space and the action is matrix multiplication, one is in the worldM of products of random matrices. If then the group generated by λ,σ, with σ varying in the support of p, is noncompact and strongly irreducible,T Furstenberg, Guivarch and Raugi have proved [2, 9, 11] that there is a unique invariant measure νλ which is, moreover, H¨older continuous [2]. To our best knowledge, little seems to be known in more general situations and also concerning the absolute conti- nuity of νλ (except if p is absolutely continuous [18], for some and under supplementary hypothesis [4, 28]). Let p1 be the distribution of the random variable 1,σ on the Lie algebra g, that is, for any measurable b g one has p (b)=Pp( b ). We are ⊂ 1 {P1,σ ∈ } interested in a perturbative calculation of Iλ,N (f) in λ for smooth functions f with rigorous control on the error terms. This can be achieved if the support of p1 is large enough in the following sense. First, let us focus on the special case = 1 and = 0. R P
Theorem 1. Let λ,σ be of the form (1) and assume = 1, = E( 1,σ)= 0. Let x be the associatedT Markov process on as givenR byP (3) andP let n M RANDOM LIE GROUP ACTIONS 3 v = Lie(supp(p1)) be the smallest Lie subalgebra of g that contains the sup- port of p1. Recall that µ(dx) denotes the Riemannian volume measure on . MCoupling hypothesis: Suppose that the smallest subgroup of containing exp(λ ), v, λ [0, 1] acts transitively on . (This isV a LieG subgroup with{ LieP algebraP ∈ v,∈ but it} may not be a submanifold.)M Then there is a sequence of smooth functions ρm with dµρm = δm,0 M and ρ0 > 0 µ-almost surely, such that for any M N and any function f C ( ), one obtains ∈ R ∈ ∞ M M 1 (5) I (f)= λm µ(dx)ρ (x)f(x)+ , λM+1 . λ,N m O Nλ2 m=0 X ZM 1 M+1 Here, the expression ( Nλ2 , λ ) means that there are two error terms, O 1 M+1 one of which is bounded by C1 Nλ2 and the other by C2λ with C1,C2 depending on f and M. Especially, C2 may grow in M so that we cannot deduce uniqueness of the invariant measure for small λ this way (cf. Remark 1 below). When = 1 or = 0 further assumptions are needed in order to control the BirkhoffR 6 sums.P We 6 assume that and generate commuting compact 1 R P groups, that is, − = Ad ( )= and the closed Abelian Lie groups RPR R P P = k : k Z and = exp(λ ) : λ R) are compact. While is alwayshRi {R connected,∈ } canhPi possibly{ beP disconnected.∈ } However, therehPi exists K N such that hRiK is connected. By considering the suspended Markov ∈ n hR i process (y )n N with yn = xKn corresponding to the family ∈ = Tλ,σ1,...,σK Tλ,σK ···Tλ,σ1 K K for (σ1,...,σK) (Σ , p ), one can always assume that is connected and we shall do∈ so from now on. Note that the product hRi is also a compact, connected, Abelian subgroup of which will be denotedhRihPi by , . G hR Pi All these groups are tori in and their dimensions are L , L and L , . TLR TLGP TLR,P RTL PRL RZPL Hence, ∼= , ∼= and , ∼= , where = /(2π ) is the LhRi-dimensionalhPi torus. The (chosen)hR Pi isomorphisms shall be denoted by R ,R and R , , respectively, for example, R (θ) GL(L, C) for R P R P TLR R ∈ hRi ⊂ θ = (θ1,...,θLR ) . The isomorphism∈ R directly leads to the Fourier decomposition of the function θ TLR f(R (θ) x), notably ∈ 7→ R · ıj θ (6) f(R (θ) x)= fj(x)e · , R · j ZLR ∈X where LR dθ ıj θ fj(x)= e− · f(R (θ) x), j θ = jlθl. LR TLR (2π) R · · Z Xl=1 4 C. SADEL AND H. SCHULZ-BALDES
LP LR,P Similarly, the maps θ T f(R (θ) x) and θ T f(R , (θ) x) ∈ 7→ P · ∈ 7→ R P · lead to Fourier series.
Definition 1. A function f C ( ) is said to consist of only low ∈ ∞ M frequencies w.r.t. if the Fourier coefficients f C ( ) vanish for j hRi j ∈ ∞ M with norm j = LR j larger than some fixed integer J> 0. Similarly, f k k l=1 | l| is defined to consist of only low frequencies w.r.t. or , . P hPi hR Pi The following definitions are standard (see [17] for references).
Definition 2. Let us define θˆ TLR by R (θˆ )= and θˆ RLP R ∈ R R R P ∈ by R (λθˆ ) = exp(λ ). Then is said to be a Diophantine rotation or simplyP DiophantineP ifP there is someR s> 1 and some constant C such that for any nonzero multi-index j ZLR 0 one has ∈ \ { }
ıj θˆR s e · 1 C j − . | − |≥ k k Similar, is said to be Diophantine, or a Diophantine generator of a rota- P tion, if there is some s> 1 and some constant C, such that for any nonzero multi-index j ZLP 0 one has ∈ \ { } s j θˆ C j − . | · P |≥ k k As final preparation before stating the result, let us introduce the measure p on the Lie algebra g obtained from averaging the distribution p1 of the lowest-order terms 1,σ w.r.t. the Haar measure dR on the compact group , , namely for anyP measurable set b g, hR Pi ⊂ 1 p(b)= dR p( σ Σ : R 1,σR− b ). , { ∈ P ∈ } ZhR Pi
Theorem 2. Let be of the form (1) and x the associated Markov Tλ,σ n process on as given in (3). Denote the Lie algebra of , by r and let M hR Pi v = Lie(supp(p), r) be the Lie subalgebra of g generated by the support of p and r. Suppose that the smallest subgroup of containing exp(λ ) : V G { P P∈ v, λ [0, 1] acts transitively on . Further, suppose that f C ( ) and ∈ } M ∈ ∞ M one of the following conditions hold: (i) and are Diophantine and = K/H where K and H K are R P M ⊂ compact Lie groups. (ii) f consist of only low frequencies w.r.t. , . hR Pi RANDOM LIE GROUP ACTIONS 5
Then there is a µ-almost surely positive function ρ0 C∞( ) normalized w.r.t. the Riemannian volume measure µ on , such∈ that M M 1 (7) I (f)= µ(dx)ρ (x)f(x)+ , λ , λ,N 0 O Nλ2 ZM where µ is the Riemannian volume measure on . Moreover, the probability measure ρ µ is invariant under the action of M, . 0 hR Pi M m The probability measures m=0 λ ρmµ in Theorem 1 and ρ0µ in Theo- rem 2 can be interpreted as perturbative approximations of invariant mea- P sures νλ. In fact, integrating (5) over the initial condition x0 w.r.t. any invariant measure νλ and then taking the limit N , shows that for any smooth function →∞ M (8) ν (dx)f(x)= λm µ(dx)ρ (x)f(x)+ (λM+1). λ m O m=0 ZM X ZM This means that the invariant measure is unique in a perturbative sense and, moreover, its unique approximations are absolutely continuous with smooth density. In fact, one obtains the following.
Corollary 1. Let the assumptions of Theorems 1 or 2 be fulfilled and (νλ)λ>0 be a family of invariant probability measures for the Markov pro- cesses xn(λ). Then
w∗- lim νλ = ρ0µ, λ 0 → where w∗- lim denotes convergence in the weak- topology on the set of Borel measures. ∗
Proof. Approximating a continuous function by its Fourier series shows that the set of smooth functions consisting of only low frequencies w.r.t. , is dense in the set of continuous functions w.r.t. the -norm. hRThePi set of probability measures is norm bounded by 1 w.r.t. thke ·dual k∞ norm. Now, let g C( ). For any ε> 0, there is a smooth function g consisting of only low∈ frequenciesM such that f g < ε. Then one has k − k∞ ν (f) ρ µ(f) ν (f g) + ν (g) ρ µ(g) + ρ µ(g f) | λ − 0 | ≤ | λ − | | λ − 0 | | 0 − | 2ε + ν (g) ρ µ(g) . ≤ | λ − 0 | One obtains lim supλ 0 νλ(f) ρ0µ(f) 2ε for any ε> 0, so that by (8) → | − |≤ lim sup νλ(f) ρ0µ(f) = 0, λ 0 | − | → for any continuous function f C( ), which gives the desired result. ∈ M 6 C. SADEL AND H. SCHULZ-BALDES
Remark 1. According to the unique weak- -limit for a family of invari- ∗ ant measures νλ, one might expect uniqueness for the invariant measure at least in a small interval around 0. However, we will briefly describe a simple example satisfying all conditions of Theorem 1 such that for any rational λ the invariant measure is not unique. Let = = S1 = z C : z = 1 and let the Lie group action be the ordinaryG multiplication.M { Furthermore,∈ | | } 1 let =1 and 1,σ be Bernoulli distributed with probability 2 at ıπ and R P 1 ıπ and let n,σ =0 for n 2. Any measure on S which is invariant under −a rotation byPλπ is an invariant≥ measure and for rational λ there are many of them. Therefore, we expect the following to hold: given the conditions of Theorem 2 one finds λ0 > 0 such that for Lebesgue a.e. λ [0, λ0] there is a unique invariant measure. ∈
Remark 2. The main hypothesis of Theorems 1 and 2 is that the Lie group associated to the Lie algebra v acts transitively on . This can roughly be thought of as a Lie algebra equivalent of FurstenbM erg’s irre- ducibility condition or the Goldsheid–Margulis criterion [10]. Let us note that nontrivial , lead to a larger support for p and hence weaken this hypothesis. A secondR P hypothesis is that the group is compact. This ex- cludes many situations appearing in physical modelshRi where hyperbolic or parabolic channels appear. In some particular situations, this could be dealt with [24, 25].
Remark 3. As by the main hypothesis the action of on is transi- tive, is always a homogeneous space and given as a quotientG M of w.r.t. someM isotropy group, but hypothesis (i) requires that is, moreover,G a quo- tient of a compact group (which in the examples of SectionM 5 is a subgroup of ). The assumption that GL(L, C) (or, equivalently has a faithful representation)G is only neededG⊂ for the proof of Theorem 2 underG hypothesis (ii).
Remark 4. Suppose K is a compact subgroup of acting transitively on [which is a special case of the condition in TheoremG 2(i)]. Then the HaarM measure dk on K induces a unique natural K-invariant measure on which one may choose to be µ (which is also the volume measure of theM metric dK K g). It is interesting to examine whether ρ0 = 1 , that is, the lowest-order approximation∗ of the invariant measure is givenM by the natural measure.R The proof below provides a technique to check this. More precisely, in the notation developed below, ˆ∗1 = 0 implies that ρ0 is constant. An example, where this can indeed beL checkedM is developed in Section 5. Note 1 that, if K is as above, then any conjugation K − with an element has another natural measure, given by J µ NwhereN J is the JacobianN of ∈ the G N N map x x. Unless µ is invariant under all of , the equality ρ0 = 1 7→ N · G M RANDOM LIE GROUP ACTIONS 7 is hence linked to a good choice of K. If µ is invariant under , then it is also an invariant measure for the Markov process and under thGe hypothesis of Theorem 2 one therefore has ρ0 = 1 . M
Remark 5. If , acts transitively on , then the measure ρ0µ is uniquely determinedhR byPi the fact that it is invariantM under the action of , and normalized. Moreover, is isomorphic to the quotient of , hRandPi M hR Pi the stabilizer x of any point x (which is a compact Abelian subgroup of , ). Hence,S in this case,∈ Mis a torus and the action is simply the hR Pi M translation on the torus. Consequently, the measure ρ0µ is the Haar measure. Note that, if = 0, this holds independently of the perturbation and is imposed by theP deterministic process for λ = 0.
Remark 6. If the action of on is not transitive, there are many hRi M invariant measures ν0 for the deterministic dynamics (in particular, if = 1 any measure is invariant under ). Under the hypothesis of TheoremsR 1 and 2, the random perturbationshRi and single out a unique pertur- P1,σ P2,σ bative invariant measure ρ0µ.
Remark 7. We believe that condition that and commute is unnec- essary. In fact, we expect that conditions on canR be replacedP by conditions 1 P on ˆ = dR R R− . P hRi P RemarkR 8. Let us cite prior work on the rigorous perturbative evalu- ation of the averaged Birkhoff sums (4). In the case of =SL(2, R), = RP (1) and a rotation matrix in (1), Pastur and FigotinG [20] showedM (7) for the lowest two harmonics wheneverR , 2 = 1. The above result com- bined with the calculations in Section 5RshowsR 6 that± (7) holds also for other 2 functions with ρ0 = 1 . Without the conditions , = 1, Theorem 2 was proved in [24, 26].M Moreover, when K = 1 (atR so-calledR 6 ± anomalies) and for an absolutely continuous distributionR on , Theorem 1 was proved by Campanino and Klein [4]. Quasi-one-dimensionalG generalizations of [20] in the case where is a symplectic group were obtained in [25, 27]. The work [7] is an attempt toG treat higher-dimensional anomalies. To further generalize, the above results to quasi-one-dimensional systems was our main motivation for this work.
Remark 9. Our main application presented in Section 5 is the pertur- bative calculation of Lyapunov exponents associated to products of random matrices of the form (1). Moreover, we show how to choose (cf. Remark N 5) such that ρ0 = 1 . This property is called the random phase property in [22] which is relatedM to the maximal entropy Ansatz in the physics literature. Section 5 can be read directly at this point if Theorem 2 is accepted without proof. 8 C. SADEL AND H. SCHULZ-BALDES
Remark 10. The recent work by Dolgopyat and Krikorian [6] on ran- dom diffeomorphisms on Sd contains results on the associated invariant mea- sure and Lyapunov spectrum which are related to the results of the present paper. The main difference is that [6] assume the random diffeomorphisms to be close to a set of rotations which generate SO(d + 1) while in the present work the diffeomorphisms are to lowest order given by the identity (Theorem 1) or close to one fixed rotation (Theorem 2). As a result, the invariant measure in Proposition 2 of [6] is close to the Haar measure while it is determined by the random perturbations in the present paper. In the particular situation of the example studied in Section 5, the randomness is such that the invariant measure is the Haar measure and as a consequence the Lyapunov spectrum is equidistant, just as in [6].
In order to clearly exhibit the strategy of the proof of the theorems, we first focus on the case = 1 and = 0 in Sections 2 and 3, which corresponds to a higher-dimensionalR anomalyP in the terminology of our prior work [24, 26]. The main idea is then to expand Tλf into a Taylor expansion in λ. This directly leads to a second-order differential operator on of L M the Fokker–Planck type, for which the Birkhoff sums Iλ,N ( f) vanish up to order λ. Under the hypothesis of Theorem 1, it can beL shown to be a sub-elliptic H¨ormander operator on the smooth functions on with a one- dimensional cokernel. Then one can deduce that C + (C ( M)) = C ( ) L ∞ M ∞ M and that the kernel of ∗ is spanned by a smooth positive function ρ0. These are the main elementsL of the proof of Theorem 1 for M =1. Then using the properties of the operators and and a further Taylor expansion of L L∗ Tλf one can prove Theorem 1 by induction. The additional difficulties for other , in Theorem 2 are dealt with in the more technical Section 4. The applicationsR P to Lyapunov exponents are presented in Section 5.
2. Fokker–Planck operator and its properties. In this section, we sup- pose = 1 and = E( ) = 0 in (1) and introduce in this case the back- R P P1,σ ward Kolmogorov operator and its adjoint ∗, called forward Kolmogorov or also Fokker–Planck operatorL [21]. Their useL for the calculation of the av- eraged Birkhoff sum is exhibited and several properties of these operators are studied. One way to define the operator : C ( ) C ( ) is L ∞ M → ∞ M d2 (9) ( f)(x)= (T f)(x). L dλ2 λ λ=0
Let us rewrite this using the smooth vector fields ∂ associated to any P element P g by ∈ d (10) ∂ f(x)= f(eλP x). P dλ · λ=0
RANDOM LIE GROUP ACTIONS 9
Then is given by L E 2 (11) = σ(∂ 1,σ + 2∂ 2,σ ). L P P
Proposition 1. For F C ( ), one has ∈ ∞ M 1 I ( F )= , λ . λ,N L O Nλ2 Proof. For P g, a Taylor expansion with Lagrange remainder gives ∈ F (eP x)= F (x) + (∂ F )(x)+ 1 (∂2 F )(x)+ 1 (∂3 F )(eχP x), · P 2 P 6 P · 2 3 for some χ [0, 1]. Choose P = λ 1,σ + λ 2,σ + λ σ(λ), where σ(λ)= n 3 ∈ P P S S ∞ λ and use that is centered to obtain n=3 − Pn,σ P1,σ E E 2 1 2 3 P σF ( λ,σ x)= F (x)+ σ(λ ( 2 ∂ 1,σ F (x)+ ∂ 2,σ F (x))) + (λ ) T · P P O = F (x)+ 1 λ2 F (x)+ (λ3). 2 L O The error terms depend on derivatives of F up to order 3 and are uniform in x because is compact and 1,σ, 2,σ and σ(λ) are compactly supported by (2). DueM to definition (3), thisP impliesP S N N 1 1 1 − λ2 E F (x (λ, ω)) = E F (x (λ, ω)) + I ( F )+ (λ3). ω N n ω N n 2 λ,N L O n=1 n=0 X X As the appearing sums only differ by a boundary term, resolving for Iλ,N ( F ) finishes the proof. L
Next, let us bring the operator into a normal form. According to Ap- pendix A, one can decompose Linto a finite linear combination of fixed P1,σ Lie algebra vectors i g, i I, with uncorrelated real random coefficients, namely P ∈ ∈ I = v , v R, E (v ) = 0, E (v v ′ )= δ ′ . P1,σ i,σPi i,σ ∈ σ i,σ σ i,σ i ,σ i,i Xi=1 Then (11) implies that is in the so-called H¨ormander form L I 2 = ∂ i + 2∂ , L P Q Xi=1 where = Eσ( 2,σ). Using the main assumption of Theorem 1 (i.e., v u), one canQ show thatP satisfies the strong H¨ormander property of rank r⊂ N [14, 15, 23]. L ∈ 10 C. SADEL AND H. SCHULZ-BALDES
Proposition 2. Under the assumptions of Theorem 1, there exists r N such that satisfies a strong H¨ormander property of rank r, that is, the∈ L vector fields ∂ i and their r-fold commutators span the whole tangent space at every pointP of . M In order to check this, one needs to calculate the commutators of vector fields ∂ , ∂ for P,Q g. Let X , X denote the left-invariant vector fields P Q ∈ P Q on and furthermore introduce for each x a function on by fx( )= f( G x), . Then one obtains ∈ M G T T · T ∈ G d d2 ∂ ∂ f(x)= (∂ f)(eλP x)= f(eµQeλP x) P Q dλ Q · dλ dµ · λ=0 λ,µ=0
= X X f (1), Q P x which implies
(∂P ∂Q ∂Q∂P )f(x) = (XQXP XP XQ)fx(1)= X[Q,P ]fx(1) (12) − − = ∂[Q,P ]f(x), where [Q, P ] denotes the Lie bracket (this is well known, see Theorem II.3.4 in [12]). We also need the following lemma for the proof of Proposition 2.
Lemma 1. Let be a Lie subgroup of that acts transitively on U ⊂ G G M and denote the Lie algebra of by u. Then the vector fields ∂P , P u, span the whole tangent space at eachU point of . ∈ M Proof. First, let us show that there is a dense set of points in for which the vector fields ∂ , P u, span the whole tangent space. Indeed,M for a P ∈ fixed x consider the surjective, smooth map ϕx : , ϕx(U)= U x. A point∈x M is called regular for ϕ if and onlyU if → for M any point in the· ′ ∈ M x preimage of x′ the differential Dϕx is surjective. For each point x′, the hypothesis implies that there is a U such that x′ = ϕx(U)= U x and the regularity of x then shows that∈U the paths λ ϕ (eλP U)= e·λP x , ′ 7→ x · ′ P u, span the whole tangent space at x′. By Sard’s theorem [13], the set of∈ regular points is dense in . Actually, the existence ofM only 1 regular point x implies that all points are regular. In fact, again any other point is of the form x = U x. As ′ · the map x x′ = U x is a diffeomorphism, the push-forward of the paths 7→ · λUPU −1 λ exp(λP ) x, P u, given by the paths λ U exp(λP ) x = e x′, P 7→u, span the· tangent∈ space also at x . 7→ · · ∈ ′ Proof of Proposition 2. Define iteratively the subspaces v g by r ⊂ (13) v1 = span i : 1 i I , vr = span(vr 1 [vr 1, v1]). {P ≤ ≤ } − ∪ − RANDOM LIE GROUP ACTIONS 11
By definition, one has v = span(supp( )). The space v g defined in The- 1 Pσ ⊂ orem 1 is equal to v = Lie(v1). Due to (12), the strong H¨ormander property of rank r is equivalent to the property that ∂P , P vr, spans the whole tangent space at every point x . ∈ ∈ M By the Lemma 1 and the assumption of Theorem 1, this is fulfilled if vr = v for some r. As the vector spaces vr are nested and g is finite dimensional, the sequence has to become stationary. This means, there is some r such that vr = vr+1. Using the Jacobi identity, one then checks that vr is closed under the Lie bracket and therefore vr = v.
Next, we want to recollect the consequences of the strong H¨ormander property of rank r as proved in [14, 15, 23]. The first basic fact is the subelliptic estimate within any chart (14) f C( f + f ), k k(1/r) ≤ kL k(0) k k(0) where denotes the Sobolev norms. Using a finite atlas of , one can k · k(s) M define a global Sobolev space Hs( ) with norm also denoted by (s). Then the estimate (14) holds alsoM w.r.t. these global norms. Moreover,k · k the 2 norm (0) can be seen to be equivalent to the norm in L ( ,µ) where µ is thek Riemannian · k volume measure. As usual, the embeddingM of H ( ) s+ε M in Hs( ) is compact for any ε> 0. TheM second basic fact is the hypoellipticity of . In order to state this L property, let us first extend in the usual dual way to an operator dis on the space = (C ( )) Lof distributions on . Then hypoellipticityL D′ ∞ M ′ M states that, for any smooth function g, the solution f of disf = g is itself smooth. L The Fokker–Planck operator is the adjoint of in L2( ,µ). Because L∗ L M is compact and has no boundary, the domain ( ∗) of ∗ contains the Msmooth functions C ( ). Furthermore, is againD aL second-orderL differen- ∞ M L∗ tial operator with the same principal symbol as . Therefore, ∗ also satisfies the strong H¨ormander condition of rank r. Thus,L the subellipticL estimate as well as the hypoellipticity property also holds for . We, moreover, deduce Ldis∗ that is closable with closure = ∗∗ dis. TheL following proposition recollectsL L properties⊂L of as a densely defined operator on the Hilbert space L2( ,µ). L M
Proposition 3. There exists c0 > 0 such that for c > c0 the following holds: (i) c is dissipative. L− 2 (ii) ( c)(C∞( )) is dense in L ( ,µ). (iii) L−c is maximallyM dissipative. M (iv) L− c is the generator of a contraction semigroup on L2( ,µ). L− M 12 C. SADEL AND H. SCHULZ-BALDES
(v) The resolvent ( c) 1 exists and is a compact operator on L2( ,µ). L− − M Proof. (i) Let us rewrite : L I
f = [div(∂ i (f)∂ i ) div(∂ i )∂ i (f)] + 2∂Q(f). L P P − P P Xi=1 Defining X to be the smooth vector field 2∂ i div(∂ i )∂ i , one has Q − P P I P f = div(∂ i (f) ∂ i )+ X(f). L P P Xi=1 For a real, smooth function f, the divergence theorem and estimate on the negative quadratic term gives I
f f = dµ ∂ i (f) ∂ i (f)+ fX(f) dµ fX(f). h |L i "− P P # ≤ ZM Xi=1 ZM Using 2fX(f)= X(f 2) = div(f 2X) f 2 div(X) and again the divergence theorem, it follows that −
1 2 1 2 (15) f f dµ div(X)f div(X) f 2. h |L i≤−2 ≤ 2k k∞k k ZM As is real, it follows that e f ( c)f 0 for f C∞( ) and c > c0 L 1 ℜ h | L− i≤ ∈ M where c0 = 2 div(X) . By definition, this means precisely that c is dissipative. k k∞ L− (ii) Let h L2( ,µ) such that h f cf = 0 for all f C ( )= ∈ M h |L − i ∈ ∞ M ( ). Then h is in the kernel of dis∗ . By hypoellipticity, it follows that D L L 2 h C∞( ). Therefore, h h = c h 2 contradicting (15) unless h = 0. ∈The statementM (iii) meansh |L thati k therek is no dissipative extension, which follows directly from (i) and (ii) by [5], Theorems 2.24, 2.25 and 6.4. Item (iv) follows from the same reference. Concerning (v), the existence of the resolvent follows directly upon inte- gration of the contraction semigroup. Its compactness follows from the subel- 2 liptic estimate (14) and the compact embedding of Hs( ) into L ( ,µ). M M
The next proposition is based on Bony’s maximum principle for strong H¨ormander operators [1], as well as standard Fredholm theory.
Proposition 4. (i) The kernel of consists of the constant functions on . L M(ii) The kernel of is one dimensional and spanned by a smooth func- L∗ tion ρ0. RANDOM LIE GROUP ACTIONS 13
(iii) Ran = (ker ) and Ran = (ker ) = (ker ) . L L∗ ⊥ L∗ L ⊥ L ⊥ (iv) ρ0 is µ-almost surely positive.
Proof. (i) By Corollaire 3.1 of [1], a smooth function f which has a local maximum and for which f = 0 has to be constant on (the pathwise L connected compact set) . If f lies in the kernel of the closure = , M L L∗∗ then disf =0. As is hypoelliptic, f C∞( ) and therefore f is again constant.L L ∈ M 1 (ii) Choose c > c0 as in Proposition 3 and let K = ( + c)− . Then one has L f = g ( + c)f = cf + g f = cKf + Kg L ⇔ L ⇔ (1 cK)f = Kg, ⇔ − and similarly f = g (1 cK )f = K g. For g = 0, this implies ker = L∗ ⇔ − ∗ ∗ L ker(1 cK) and ker ∗ = ker(1 cK¯ ∗). By the Fredholm alternative (the index− of 1 + cK is 0),L the dimension− of these two kernels are equal and by (i) hence, both one dimensional. The smoothness of the function in the kernel follows from the hypoellipticity of ∗. (iii) For v ker = ker(1 cKL ) and g v =0, one has 0= g v = ∈ L∗ − ∗ h | i h | i g cK∗v = c Kg v , therefore g (ker ∗)⊥ implies Kg ker(1 cK)⊥ and theh | Fredholmi h alternative| i states∈ thatL (1 cK)f = Kg∈is solvable.− Hence − by the above, f = g is solvable. Therefore, Ran = (ker ∗)⊥. The other equality is provedL analogously. L L (iv) Let f 0 be smooth and suppose that dµ ρ0f = 0. According to (ii), (iii) and≥ hypoellipticity this implies that f = F 0 for some smooth R F . Again by Bony’s maximum principle F is constantL ≥ and therefore f = 0. Hence, for any nonvanishing positive function f one has dµ ρ0f > 0. R Even though not relevant for the sequel, let us also prove the following.
Proposition 5. generates a contraction semigroup in (C( ), ), also called a Feller semigroup.L M k·k∞
Proof. This will follow directly from the Hille–Yosida theorem [16], Theorem 19.11, once we verified that ( c)C∞( ) is dense in C( ) for some c> 0 and that satisfies the positive-maximumL− M principle. TheM first property follows fromL the existence of the resolvent (Proposition 3) and the hypoellipticity. For the second, let a smooth f have a positive local maximum at some x . Then one only has to check ( f)(x) 0, which follows because the first∈ M derivatives of f vanish, its secondL derivative≤ is negative and the principal symbol is positive definite. 14 C. SADEL AND H. SCHULZ-BALDES
1 One can rewrite (9) as limλ 0 2 (Tλ 1)f = f in and for f → 2λ − L k · k∞ ∈ C∞( ). Hence, the above statement and [16], 19.28, implies directly the followingM approximation result of the Feller process by the discrete time Markov processes.
t Corollary 2. Let e L denote the Feller semigroup of Proposition 5. Then with convergence in (C( ), ), M k · k∞ [t/(2λ2)] t lim T f = e Lf. λ 0 λ → Finally, let us note yet another representation of the generator following 1 β N L from the two above, namely = limN 2 N ((TN −α ) 1) where β = 2α 1 > 0 and with strong convergence.L →∞ − − 3. Control of Birkhoff sum in the case R = 1, P = 0. The aim of this section is the proof of Theorem 1.
Proposition 6. Let = 1 and = 0. The kernel of is spanned by R P Ldis∗ a nonnegative smooth function ρ0 that is normalized by dµ ρ0 = 1. For f C ( ), M ∈ ∞ M R 1 I (f)= dµ ρ f + , λ . λ,N 0 O Nλ2 ZM Proof. By hypoellipticity, the kernel of coincides with the kernel Ldis∗ of ∗. First, we show C∞( )= C1 + C∞( ). Indeed, let f C∞( ). L M M L M ∈ M Set C = dµ fρ0 and fˆ= f C. Then one has dµ fρˆ 0 = 0 and therefore ˆ M − M ˆ f (ker R ∗)⊥ = Ran by Proposition 4. By hypoellipticity,R f (C∞( )). ∈Now usingL PropositionL 1 and the above decomposition ∈L M 1 2 I (f)= I (fˆ+ C)= C + I ( F )= C + (N − λ− , λ), λ,N λ,N λ,N L O one completes the proof.
In order to prove Theorem 1, let us define the operators M (M) d f(x)= (T f)(x), f C∞( ). L dλM λ ∈ M λ=0
(1) (2) Then =0 as 1,σ is centered and = . Using (1), these operators can beL written as P L L M (M) M! f = Eσ ∂ a ,σ ∂ a ,σf . L m! P 1 · · · P m m=0 a1+ +am=M ! X ···X RANDOM LIE GROUP ACTIONS 15
Hence, (M) is a differential operator of order M. As 1 ker (m) and hence kerL ker (m) for all positive m, one obtains usingM Proposition∈ L 4(iii) L⊂ L (m) (m) Ran ∗ (ker )⊥ (ker )⊥ = Ran ∗. L ⊂ L ⊂ L L Therefore, and as ker ∗ is one dimensional, the functions ρM for M N are iteratively and uniquelyL defined by ∈ M 2 (m+2) (16) ∗ρM = ∗ρM m, dµ ρM = 0, L (m + 2)!L − m=1 X ZM with ρ given by Proposition 6. By induction and hypoellipticity of , it 0 L∗ follows that ρM is a smooth function for all M, therefore the right-hand side of (16) always exists. Now we can complete the following proof.
Proof of Theorem 1. The proof will be done by induction. The case M = 0 is contained in Proposition 6. For the step from M 1 to M, we first − need a Taylor expansion of higher order than done so far. As 1,σ is centered m n P and due to the compact support of n,σ and m n λ − n,σ [uniform for small λ by (2)], one obtains with uniformP error bound≥ P P M+2 1 λm T F (x)= F (x)+ λ2 F (x)+ (m)F (x)+ (λM+3), λ 2 L m! L O m=3 X which using the induction hypothesis implies for Birkhoff sums M 2λm 1 I ( F )= I ( (m+2)F )+ λM+1, λ,N L (m + 2)! λ,N L O λ2N m=1 X M M m − 2λl+m 1 = dµ ρ (m+2)F + λM+1, (m + 2)! lL O λ2N mX=1 Xl=0 Z M m m 2λ (l+2) M+1 1 = dµ ( ∗ρm l)F + λ , (l + 2)! L − O λ2N m=1 X Xl=1 Z M 1 = λm dµ ρ ( F )+ λM+1, . m L O λ2N mX=1 Z The last step follows from the definition (16) of ρm. Now given any smooth function f, we can write it as f = dµ ρ f + F and obtain 0 L M R 1 I (f)= dµ ρ f + λm dµ ρ F + λM+1, λ,N 0 mL O λ2N m=1 Z X Z M 1 = λm dµ ρ f + λM+1, , m O λ2N m=0 X Z 16 C. SADEL AND H. SCHULZ-BALDES where the last step follows from dµ ρ =0 for m 1. m ≥ R 4. Extension to lowest-order rotations. In this section, the lowest-order matrix is an arbitrary rotation and E( 1,σ)= commutes with and generatesR a rotation. For any R , let usP considerP the associatedR diffeo- morphism x R x and its∈ differential G DR. Then the push-forward of ∈ M 7→ · functions f : C and vector fields X = (Xx)x are defined by M→ ∈M 1 (R f)(x)= f(R− x), (R X)R x = DRx(Xx). ∗ · ∗ · 1 The pull-back is then R∗ = (R )− . With this notation, R (Xf) = (R X)(R f) and ∗ ∗ ∗ ∗
R (∂P (R∗f)) = (R ∂P )f = ∂RP R−1 f. ∗ ∗ Furthermore, we set R (XY ) = (R X)(R Y ) for the composition of two vector fields X and Y . ∗ ∗ ∗ Now let be defined as in (11) [note that this is not equal to the right- hand side ofL (9)]. As is a zeroth-order term in λ, the Birkhoff sums are to lowest order givenR by averages along the orbits of . Furthermore the expectation of the first-order term, λ , then leads toR averages over the group to order λ. It is hence reasonableP to expect that an averaged KolmogorovhPi operator has to be considered. In order to define it, recall that there are unique, normalized Haar measures on the compact groups , and , . Averages with respect to these measures will be denotedhRi hPi hR Pi by E , E and E , ; the integration variable will be R. As the Haar measurehRi is definedhPi byhR leftPi invariance and the groups and commute hRi ˜ hPi ˜ by hypothesis, one has E , (g(R)) = E (ˆg(R) forg ˆ(R)= E (g(RR)) and any function g on hR, Pi. Then set hPi hRi hR Pi I 2 2 (17) ˆ = E (R )= E ∂ −1 + ∂ + 2∂ −1 , , , R iR R R L hR Pi ∗L hR Pi P P Q ! Xi=1 where are obtained by decomposing the centered random variable Pi P1,σ − into a sum v such that the real coefficients satisfy E(v v ′ )= P i i,σPi i,σ i ,σ δi,i′ (cf. Appendix A). With this definition, we are able to prove a result similar to PropositionP 1.
Proposition 7. Let f C∞( ) and assume one of the following con- ditions to hold: ∈ M (i) and are Diophantine and = K/H for compact Lie groups K and H RK. P M (ii)⊂f consists of only low frequencies w.r.t. , . hR Pi RANDOM LIE GROUP ACTIONS 17
Then one has 1 I ( ˆf)= , λ . λ,N L O Nλ2 For the proof, we first need the following lemma.
Lemma 2. Let f C∞( ), f0 = E (R∗f) and f˜0 = E (R∗f0)= ∈ M hRi hPi E , (R∗f). If either (i) or (ii) as in the Proposition 7 holds, then hR Pi
(18) f f0 = g ∗g, f0 f˜0 = ∂ E (R∗g˜), − − R − P hRi for smooth functions g, g˜ C ( ). ∈ ∞ M Proof. The group is isomorphic to a torus TLR with isomorphism hRi R (θ) . Furthermore we define θˆ by = R (θˆ ). If f consists of only lowR frequencies∈ hRi w.r.t. , , it can beR writtenR asR finiteR sum of its Fourier coefficients hR Pi ıj θ f = fj where fj(R (θ) x)= e · fj(x), R · j In case (i), g andg ˜ will be defined by the same formulas, but with infinite sums. Thus, we have to show that these sums are well defined and that they define smooth functions on . Let p : K be the projection identifying with K/H and define the smoothM class→ function M F (K,θ)= f(R (θ) p(K)) M R · on the compact Lie group K TLR . We want to compare the Fourier series (6) of f w.r.t. with the Fourier× series of F as given by the Peter–Weyl theorem. By TheoremR 5 in Appendix B, this Fourier series of F is given by ıj θ f(R (θ) p(K)) = F (K,θ)= d(a) Tr( F (a, j)πa(K))e · , R · F a + j ZLR X∈W ∈X where + denotes the set of highest weight vectors of K, πa : K U(d(a)) is the d(aW)-dimensional, unitary representation of K parameterized→ by a, and F (a, j) is a d(a) d(a) matrix given by F × 1 ıj θ F (a, j)= dK dθ F (K,θ)πa(K− )e− · . F TLR ZK Z Here, dθ and dK denote the normalized Haar measures. Comparing this equation with (6), one obtains that the Fourier coefficients w.r.t. satisfy hRi f (p(K)) = d(a) Tr( F (a, j)π (K)). j F a a + X∈W ıj θˆR 1 Let gj(x) = (1 e · )− fj for j > 0. The next aim is to verify that the infinite sum g =− g definesk k a smooth function on . j >0 j M As F is smooth, thek k Fourier coefficients F (a, j) are rapidly decreasing by P F h [29] or Theorem 4 in Appendix B, meaning that lim (a,j) (a, j) F (a, j) = 0 for any natural h. Here, one may choosek somek→∞ normk k forkF which (ka, j) j and F (a, j) denotes the Hilbert–Schmidt norm. As is k k ≥ k k ˆ kF k R Diophantine, eıj θR 1 C j s C (a, j) s for some natural s and the | · − |≥ k k− ≥ k k− ıj θˆR 1 coefficients G(a, j) = (1 e · )− F (a, j) defined for j > 0 are still rapidly decreasing.F Therefore,− F k k ıj θ ıj θ G(K,θ)= d(a) Tr( G(a, j)π (K))e · = g (p(K))e · F a j j >0 a + j >0 kXk ∈WX kXk is a smooth function and the series converges absolutely and uniformly by Theorem 4. Setting θ = 0, this implies that j >0 gj converges uniformly to a smooth function g on satisfying g kgk= f = f f . M − RP∗ j >0 j − 0 As before, we write f = E (R f) as sum of Fourierk k coefficients w.r.t. 0 ∗ P ˜ hRi ˆ 1 ˜ , so f0 = j fj, and letg ˜j = (ıj θ )− fj for j > 0. Consider the hPi · P k k LP function F˜(K,θ)= f0(R (θ) p(K)) on K T , just as above define the P P · × Fourier coefficients F˜(a, j) for a , j ZLP and let G˜(a, j) = (ıj F ∈W+ ∈ F · RANDOM LIE GROUP ACTIONS 19 1 s s θˆ )− F˜(a, j). As j θˆ C j − C (a, j) − the coefficients G˜(a, j) areP rapidlyF decreasing,| · theP |≥ seriesk k ≥ k k F ıj θ ıj θ G˜(K,θ)= d(a) Tr( G˜(a, j)π (K))e · = g˜ (p(K))e · F a j a + j ZLP j >0 ∈WX ∈X kXk ˜ converges absolutely and G is smooth. Thus,g ˜ = j >0 g˜j exists, is smooth and k k P d ıλj θˆP ∂ g˜ = g˜je · = f˜j = f0 f˜0. P dλ − λ=0 j >0 j >0 kXk kXk As f0 f˜0 is -invariant one obtains also ∂ E (R∗g˜)= f0 f˜0. − R P hRi − Lemma 3. If either (i) or (ii), as in Proposition 7 holds, one has 1 Iλ,N (f)= Iλ,N (E , (R∗f)) + λ, . hR Pi O λN Proof. Similarly as in the proof of Proposition 1, a Taylor expansion gives 2 λ 3 EσF ( λ,σ x)= ∗F (x)+ λ∂ ∗F (x)+ ∗F (x)+ (λ ), T · R P R 2 LR O where the error term is uniform in x. For Birkhoff sums, this implies 2 λ 3 1 (19) Iλ,N (F ∗F )= λIλ,N (∂ ∗F )+ Iλ,N ( ∗F )+ λ , . − R P R 2 LR O N Using this for F = g, it therefore follows that Iλ,N (f f0)= Iλ,N (g ∗g)= 1 − − R (λ, N − ). The function F = E (R∗g˜) is ∗-invariant, so that the left- Ohand side of (19) vanishes, and ithRi follows thatR 1 Iλ,N (f0 f˜0)= Iλ,N (∂ E (R∗g˜)) = λ, . − P hRi O λN Combining both estimates completes the proof. As an immediate consequence, one obtains the following. LR,P Corollary 3. The derivative dR , of the isomorphism R , : T LR,P R P R P gives an isomorphism from ıR to the Lie algebra r of , . Let 1,..., be the images of the standard orthonormal basis. ThenhR Pi one has Qexp(2π QLR,P × i) = 1 and the i span r. If either (i) or (ii) as in Proposition 7 holds, Qone has Q LR,P 1 2 1 Iλ,N (∂ i (f)) = λ, implying Iλ,N ∂ i (f) = λ, . Q O λN Q ! O λN Xi=1 20 C. SADEL AND H. SCHULZ-BALDES Proof. First, note that ∂ i f consists of only low frequencies w.r.t. Q , whenever f does. By Lemma 3, it is sufficient to prove E , (R∗(∂ i f)) = hR Pi hR Pi Q 1 0. This can be easily checked to be true as dt exp(2πt i)∗(∂ i f)=0. 0 Q Q The following lemma is only needed forR the proof of Theorem 2 under hypothesis (ii). Lemma 4. For any Lie algebra element P g, smooth function f on C ∈ i N and any x , the map , , R ∂RP R−1 f(x), i , is a trigonometricM polynomial∈ M on ,hR Piwith → uniformly7→ bounded coefficients∈ and uniform degree in x (dependinghR Pi on i though). This implies that the ∈ M function (E , (R∗f)) consists of only low frequencies w.r.t. , . L hR Pi hR Pi Proof. As stated above, , GL(L, C) is isomorphic to TLR,P hR Pi⊂G⊂ and the isomorphism is denoted by R , (θ) , . Furthermore, this group lies in some maximal torus of GL(L,R PC). As∈ hRall maximalPi tori are conju- gate to each other, so that by exchanging with some conjugate subgroup in GL(L, C) one may assume , to be diagonal,G that is, it consists of diag- hRıϕ1(Piθ) ıϕL(θ) onal matrices R(θ) = diag(e ,...,e ). Beneath the ϕ1(θ),...,ϕL(θ) there are maximally L , rationally independent, and each is a linear com- R P bination with integer coefficients of θ1,...,θLR,P . Hence, any trigonometric polynomial in ϕ(θ) is a trigonometric polynomial in θ (possibly of higher degree), that is a trigonometric polynomial on , . hR Pi On g gl(L, C), consider the usual real scalar product e Tr(P ∗Q)= e P⊂ Q , where P denotes the entries of the matrixℜP . Let M = ℜ a,b ab ab ab dimR(g) and B1,...,BM g be some orthonormal basis for g w.r.t. this P scalar product. If R = diag(∈eıϕ1 ,...,eıϕL ) , and P g, then one has ∈ hR Pi ∈ M L 1 m 1 m RPR− = e(B (RPR− ) )B ℜ ab ab m=1 X a,bX=1 M L m ı(ϕa ϕb) m = e(B P e − )B , ℜ ab ab mX=1 a,bX=1 and therefore M L i i m ı(ϕa ϕb) ∂ −1 f = e(B P e − )∂ m f RP R ℜ ab ab B m=1 ! X a,bX=1 is a trigonometric polynomial in ϕ. Thus by definition of , the map R L 7→ R ( (R∗f)) = (R )f is a trigonometric polynomial on , , and there- ∗ L ∗L ˆ ˆ hR Pi fore also R R ( f) for f = E , (R∗f). But this means precisely that 7→ ∗ L hR Pi fˆ consists of only low frequencies w.r.t. , . L hR Pi RANDOM LIE GROUP ACTIONS 21 ˆ Proof of Proposition 7. As = E , (R ), it follows for R L hR Pi ∗L ∈ , that (R ˆ)f = ˆf = (R∗ ˆ)f. This implies R∗( ˆf)= ˆ(R∗f) and hR Pi ˆ ∗L ˆ L L L L ˆ E , (R∗( f)) = (E , (R∗f)). Hence, the Fourier coefficients of f are givenhR Pi by L L hR Pi L (20) ( ˆf) = ˆ(f ). L j L j Therefore, ˆf consists of only low frequencies w.r.t. , whenever f does. L ˆ hR Pi Furthermore, one obtains for f = E , (R∗f) the following equalities: hR Pi ˆ ˆ ˆ ˆ ˆ E , (R∗( f)) = f = E , (R ( (R∗f))) = E , (R ( f)). hR Pi L L hR Pi ∗ L hR Pi ∗ L Now fˆ consists of only low frequencies by Lemma 4. Thus,L applying Lemma 3 twice [the hypothesis are given either by hy- pothesis (i) of Proposition 7 or by (ii) and Lemma 4]. One obtains ˆ ˆ 1 ˆ 1 Iλ,N ( f)= Iλ,N (E , (R∗( f))) + λ, = Iλ,N ( f)+ λ, . L hR Pi L O λN L O λN As ∗fˆ= fˆ and ∂ fˆ= 0, equation (19) for F = fˆ implies R P 1 I ( fˆ)= , λ , λ,N L O λ2N which combined with the above finishes the proof. After these preparations, the proof of Theorem 2 is analogous to the case = 1. R Proof of Theorem 2. Consider the Markov process on induced by the random family M = exp(λ + λ2 ), Tλ,σˆ P1,σˆ P2,σˆ whereσ ˆ = (σ, R, α, β, i) Σ=Σˆ , 1, 1 1, 1 1,...,L , and = (R R 1 ∈ )+ α× hR+ βPi×{−, = R}×{−R 1}×. The { areR de-P } P1,σˆ P1,σ − − P P Qi P2,σˆ P2,σ − Qi fined as in Corollary 3. Σˆ is equipped with the probability measure p dR 1 1 1 × × (δ 1 +δ1) (δ 1 +δ1) (δ1 + +δLR,P ) where dR denotes the Haar 2 − × 2 − × LR,P · · · ˜ ˆ LR,P 2 E measure on , . Let us define = + i=1 ∂ i . As σˆ( 1,σˆ)=0, hR Pi L L Q P LR,P P 2 2 2 2 ˜ = ∂ +E E (∂ −1 +∂ +2∂ −1 )= E (∂ +2∂ ) i , σ R 1,σ R R 2,σ R σˆ 1,σˆ 2,σˆ L Q hR Pi P −P P P P P Xi=1 and span(supp( )) = span(supp(p), r), this new process leads to the op- P1,σˆ ˜ ˆ LR,P 2 erator = + i=1 ∂ i instead of and the whole analysis done for L L Q L P 22 C. SADEL AND H. SCHULZ-BALDES in the case = 1, = 0 is applicable to ˜ now due to the hypothe- sisL of TheoremR2. In particular,P ˜ and ˜ areL hypoelliptic operators, the L L∗ kernel of ˜ consists of the constant functions and the kernel of ˜∗ is one- dimensionalL and spanned by a normalized, smooth function ρ 0.L Further- 0 ≥ more, C∞( )= C1 + ˜C∞( ) and hence for any smooth function f and M M L M C = dµ ρ0f, there is a smooth function g such that f = C + ˜g. AssumeM f consists of only low frequencies, that is, f =0 forL j > J. R j k k Then by (20) one obtains for frequencies j > 0 that f = (f C) = ˜g k k j − j L j and hence ˜gj =0 for j J. Therefore gj is constant, which means gj = 0 as j >J>L 0 and g consistsk k≥ of only low frequencies if f does. Hence, Propo- sitionk k 7 implies for both cases (i) and (ii) the first statement of Theorem 2: 1 I (f)= C + I ( ˜g)= C + λ, . λ,N λ,N L O λ2N To see that the measure ρ0µ is , -invariant, let again f be any smooth function. As mentioned above, therehR existsPi g C ( ) and C C such that ∈ ∞ M ∈ ˜g = f C. For all R , , this implies ˜R∗g = R∗ ˜g = R∗f C and henceL f− R f = ˜(g ∈R hRg) Pi(ker ˜ ) whichL gives L − − ∗ L − ∗ ∈ L∗ ⊥ dµ ρ (f R∗f) = 0. 0 − ZM This is precisely the stated invariance property of the measure ρ0µ. 5. An application to random Jacobi matrices. 5.1. Randomly coupled wires. Here, we consider a family Hλ of random Jacobi matrices with matrix entries of the form CL Z (Hλψ)n = ψn+1 ψn 1 + λWσn ψn, ψ = (ψn)n Z ( )× , − − − ∈ ∈ where the (Wσn )n Z are independently drawn from an ensemble of Hermitian ∈ L L matrices, for which all the entries Wi,j C, 1 i < j L, and Wk,k R,× 1 k L, are independent and centered random∈ ≤ variables≤ with variances∈ satisfying≤ ≤ (21) E(W 2 ) = 0, E( W 2) = 1, E(W 2 ) = 1. i,j | i,j| k,k This is equivalent to having E(Wi,jWk,l)= δi,lδj,k. This model is relevant for the quantum mechanical description of a disordered wire, consisting of L identical subwires (all described by a one-dimensional discrete Laplacian) which are pairwise coupled by random hopping elements having random magnetic phases. Moreover, within each wire there is a random potential of the Anderson type. This is similar to a model considered by Wegner [30] RANDOM LIE GROUP ACTIONS 23 and Dorokhov [8]. We are interested in the weak coupling limit of small randomness. Next, we show how this model leads to a question which fits the framework of the main theorems of this work. For a given fixed energy E ( 2, 2), the associated transfer matrices [2, 20] are ∈ − λW E1 1 ˆ E = σ − − . Tλ,σ 1 0 Let us introduce the symplectic form , the Lorentz form and the Cayley transformation by J G C 0 1 1 0 1 1 ı1 = − , = , = − . J 1 0 G 0 1 C √2 1 ı1 − Then the transfer matrix ˆ E is in the Hermitian symplectic group, namely Tλ,σ it satisfies ˆ ˆ = . Hence, its Cayley transform ˆ E is in the gener- T ∗J T J CTλ,σC∗ alized Lorentz group U(L,L) of signature (L,L) consisting by definition of the complex 2L 2L matrices ˆ satisfying ∗ = . As a first step, let bring the transfer× matrix in itsT normal formT (thisGT corresponG ds to a change of conjugation as in the proof of Lemma 4). Setting E = 2 cos(k) and − 1 sin(k)1 0 = , N cos(k)1 1 sin(k) − where E < 2, sin(k) = 0, itp is a matter of computation to verify | | 6 E 1 λ σ = ˆ − ∗ = e P U(L,L), Tλ,σ CN Tλ,σN C Rk ∈ where ık e− 1 0 ı Wσ Wσ (22) k = ık , σ = . R 0 e 1 P 2 sin(k) Wσ Wσ − − Note that the group generated by is a subgroup of the group consisting Rk of all θ for θ T. Furthermore, E(Wσ)=0. TheR group U(∈ L,L) naturally acts on the Grassmanian flag manifold of -isotropic subspaces of C2L [2]. In order to describe the flag manifold,M letG us introduce the set of isotropic frames I = Φ Mat(2L L, C):Φ∗Φ= 1; , Φ∗ Φ = 0 . { ∈ × G } I 1/2 U One readily checks that each Φ is of the from Φ = 2− V with U, V U(L). I ∈ ∈ Hence, ∼= U(L) U(L) and it has a natural measure given by the product of the Haar measures.× The column vectors of Φ then generate a flag. Two isotropic frames Φ1 and Φ2 span the same flag if and only if there is an upper triangular L L matrix S such that Φ1 =Φ2S. Due to the above, S is also unitary so× that it has to be a diagonal unitary. These diagonal 24 C. SADEL AND H. SCHULZ-BALDES unitaries can be identified with the torus TL and thus I is a TL-cover of the flag manifold, namely = I/TL = U(L) U(L)/TL. Consequently is a symmetric space and itM also carries a natural× measure µ. The groupM action of U(L,L) on U(L) U(L) is given × A B U AU + BV (23) = S, C D · V CU + DV where S is an upper triangular matrix such that (AU + BV )S is unitary; then automatically also (CU + DV )S is unitary. This also defines an action on the quotient and one readily checks that µ is invariant under the action of the subgroupM U(L,L) U(2L). Let us recall how the general∩ framework of the Introduction is applied in the present situation: the Lie group is = U(L,L) acting on the compact G flag manifold by (23); equation (22) shows that the rotation is = k and the randomM perturbation = , while =0 for n 2.R ObjectsR P1,σ Pσ Pn,σ ≥ of interest are now the L positive Lyapunov exponents γl,λ(E), l = 1,...,L [2]. It can be shown that p N 1 (24) γl,λ(E) = lim E fp,λ(xn) = lim Iλ,N (fp,λ), N N N →∞ n=1 →∞ Xl=1 X where xn is the Markoff process on the compact manifold and fp,λ will be defined next. Actually, we may also consider the actionM on the cover I and then f is a class function, defined for Φ = (φ ,..., Φ ) I by p,λ 1 L ∈ f (Φ) = E log( φ φ pC2L ) p,λ σ kTλ,σ 1 ∧···∧Tλ,σ pkΛ = Eσ det(1p LΦ∗ λ,σ∗ λ,σΦ1L p), p × T T × for 1 p L, where 1p L = (1, 0) is a p L matrix and 1L p = 1p∗ L. Hence, ≤ ≤ × × × × γl,λ(E) are all given by a Birkhoff sum. Applying Theorem 2, one obtains the following. Proposition 8. As long as E = 2 cos(k) = 0 and E < 2, the lowest- 6 | | order approximation ρ0µ of the invariant measure is the Haar measure on , that is, ρ0 = 1. The pth greatest Lyapunov exponent γp(E) is then given byM 2 1 + 2(L p) 3 (25) γp(E)= λ − + (λ ). 8sin2(k) O For L =1, (25) is proved in [20]. At the band center E =0, the methods below show that the lowest-order invariant measure is not the Haar measure. In the case L = 1, the measure was explicitly calculated in [26]. A formula RANDOM LIE GROUP ACTIONS 25 similar to (25) was obtained in [8]. It shows, in particular, that the Lyapunov spectrum is equidistant. Distinctness of the Lyapunov exponents can also be deduced from the Goldscheid–Margulis criterion. The first step of the proof is to expand fp,λ w.r.t. λ for any p. To deal with the expectation values, the following identities are useful. Lemma 5. Let P,Q Mat(L, C). Then one has ∈ 2 E(Wσ) = 0, E(Wσ )= L1, E(Tr(PWσ) Tr(QWσ)) = Tr(PQ), t E(WσPWσ) = Tr(P )1, E(WσQW σ)= Q . Using this, some calculatory effort leads to 2 λ 3 (26) fp,λ(Φ) = Fp(Φ) + (λ ), 8sin2(k) O 1/2 U where, setting Φ = 2− V , the class function Fp is defined by 1 2 Fp(Φ) = 2Lp + L Tr(1L;p(V ∗U + U ∗V )) + 2 [Tr(1L;p(U ∗V ))] 1 2 2 + [Tr(1 V ∗U)] p , 2 L;p − L where 1L;p = 1L p1p L is the projection on the first p entries in C . There- fore, × × p 2 λ 3 (27) γl,λ = lim Iλ,N (Fp)+ (λ ). 8sin2(k) N O →∞ Xl=1 Note that F is a polynomial of second degree in the entries of (U, V ), and U e−ıθ U hence consists of only low frequencies w.r.t. to k as θ V = eıθ V . Thus, in order to apply Theorem 2 we just needhR toi checkR the coupling hypothesis. 5.2. Verifying the coupling hypothesis for Theorem 2. First, we introduce a connected, transitively acting subgroup such that the space v as defined in Theorem 2 fulfills u v, where Uu ⊂is G the Lie-algebra of . Then is also a subgroup of the group⊂ as defined in Theorem 2 andU acts transitivelyU as required. Set V V = diag(U, V ) : U, V U(L) and UV SU(L) U(L,L). U { ∈ ∈ }⊂ Its Lie algebra is given by u = diag(u, v) : u, v u(L), Tr(u + v) = 0 . { ∈ } Now the action of via (23) on I is not transitive, but it is indeed transitive on the quotient U= I/TL. M 26 C. SADEL AND H. SCHULZ-BALDES Proposition 9. The Lie algebra u is contained in the Lie algebra v 1 generated by the set − : k , supp( σ) , where σ is given in (22). {RPR R ∈ hR i P∈ P } P Proof. We obtain 2ık 1 ı Wσ e− Wσ k σ k− = 2ık . R P R 2 sin(k) e Wσ Wσ − − Hence, 1 1 1 cos(2k) ıWσ 0 2 cos(2k) σ + k σ −k + k− σ k = − . − P R P R R P R sin(k) 0 ıWσ − ıW 0 Therefore, the space v contains all matrices ( 0 ıW ) where W = W ∗. The [ıV,ıW ] 0 − commutator of such two matrices is ( 0 [ıV,ıW ]), hence also obtained in v. As su(L) is a simple Lie-algebra and ıV and ıW are arbitrary elements of u(L), the commutators [ıV, ıW ] contain any element of su(L). Therefore, taking linear combinations of these terms shows that u v. ⊂ Thus, Theorem 2 applies and equation (25) follows readily from (27) once L one has shown that ρ0µ is the Haar measure on = U(L) U(L)/T for E = 0. Furnishing with a left invariant metric,M the Haar× measure is the 6 M volume measure so that we have to show ρ0 = C1 with some normalization ˆ M constant C. This is equivalent to verifying that ∗1 = 0. Using ∂P∗ = ∂P ˆ 2 L M − − div(∂ ) and the special form = E E (∂ − ) of the Fokker–Planck P σ R σ R 1 operator in the present situation,L onehRi gets P 2 ˆ − − ∗1 = E Eσ(([∂R σ R 1 + div(∂R σ R 1 )] )1 ) (28) L M hRi P P M 2 − − − = E Eσ(∂R σ R 1 (div(∂R σ R 1 )) + (div(∂R σ R 1 )) ). hRi P P P In order to calculate this further, one needs a formula for the divergence of a vector field ∂ , which is the object of the next section. P 5.3. Divergence of vector fields. Let A B = u(L,L), A∗ = A, D∗ = D. P B∗ D ∈ − − The aim of this section is to calculate the divergence of the vector field ∂ I P on . It can be lifted to a vector field on ∼= U(L) U(L). At the point (U,M V ), ∂ is given by the path × P t (U(1 + t[U ∗AU + U ∗BV + S]),V (1 + t[V ∗DU + V ∗B∗U + S])) (29) 7→ + (t2). O RANDOM LIE GROUP ACTIONS 27 The upper triangular matrix S is determined by the fact that it has reals on the diagonal such that U ∗AU + U ∗BV + S is in the Lie algebra u(L). This leads to S + S∗ = U ∗BV V ∗B∗U. In order to calculate S S∗, let us define the following R−-linear function− on Mat(L, C), − t t (30) w(A)= [E (A + A∗) E E (A + A∗) E ], j,k j,k − k,j k,j Xj = δ(ui,vi) Tr(ui∗U ∗AU + vi∗V ∗DV ) Xi 1 Tr (u + v )∗w(U ∗BV ) − 2 i i 1 + Tr((u v )∗(U ∗BV V ∗B∗U)) . 2 i − i − Now as u = u , one obtains i∗ − i 2 2 δ Tr(u∗U ∗AU) = Tr(u∗(u∗U ∗AU +U ∗AUu )) = Tr(U ∗AU(u u )) = 0. (ui,vi) i i i i i − i Thus, one has δ Tr(u U AU) = 0 and analogously δ Tr(v V i (ui,vi) i∗ ∗ i (ui,vi) i∗ ∗ × DV ) = 0. Next, consider δ(u ,v ) Tr((ui +vi)w(U ∗BV )). It is easy to check P i i i P P 28 C. SADEL AND H. SCHULZ-BALDES that for j = k one has u E v¯ = u¯ E v =0 and u¯ E u = 6 i i j,k i i i j,k i i i j,k i i uiEj,ku¯i = Ek,j. The same holds with vi and ui exchanged. From these equations, the cyclicity ofP the trace andP the definition of w oneP obtains after someP calculatory effort 1 δ Tr((u +v )w(U ∗BV )) = Tr((E E )(U ∗BV +V ∗B∗U)). 2 (ui,vi) i i k,k − j,j Xi Xj 5.4. Volume measure to lowest order. For E = 0, we now want to show 6 ˆ1 =0 using (28). As the group k is a closed subgroup of the torus L M hR i consisting of all θ for θ T = R/2πZ, the Haar measure of k can be considered as a probabilityR ∈ measure on T. Expectations w.r.t. tohR thisi measure with integration variable θ T will be denoted by E . Then for any function ∈ θ f on k , one has E (f( )) = Eθ(f( θ)). hR i R R R Lemma 6. Away from the band center E = 0, one has 6 2ıθ 4ıθ Eθ(e± ) = 0, Eθ(e± ) = 0. k Proof. If k is an irrational angle, that is, 2π is irrational, then the closed group generated by k is just the set of all θ and the measure Eθ R R2ıθ 4ıθ is the Haar measure of the torus T implying Eθ(e± )= Eθ(e± )=0. If k is a rational angle, then the closed group generated by k is finite and ıθ R consists of all θ such that e is a sth root of 1 for some natural s. The Haar measure isR just the point measure giving each point the same mass. As 2ıθ sin(k) = 0, we get s> 2 which gives Eθ(e± ) = 0. Similarly, as long as s = 4 6 4ıθ 6 one also obtains E(e± )=0. If s = 4 which means k = π/2 and E = 0, then 4ıθ Eθ(e )=1. RANDOM LIE GROUP ACTIONS 29 Define A = B = ıWσ and D = A . Then σ σ 2sin2(k) σ − σ 2ıθ 1 Aσ e− Bσ − = . RθPσRθ e2ıθB D σ∗ σ 2 From now on, we assume E = 0. First, consider the term [div(∂ −1 )] θ σ 6 R P Rθ appearing in (28). By (31), it is equal to 4ıθ 2 4ıθ 2 e− Tr(CU ∗BσV ) + e Tr(CV ∗Bσ∗U) + 2 Tr(CU ∗BσV ) Tr(CV ∗Bσ∗U). By Lemmas 6 and 5, one obtains 2 2 1 Tr(C ) E E −1 σ(div(∂ σ )(U, V )) = 2 Tr(VCU ∗UCV ∗)= 2 . R RP R 2sin k 2sin (k) Next, we need to calculate the average of ∂ −1 div(∂ −1 ) which θ σ θ θ σ θ equals R P R R P R 2ıθ e Tr(e− 2CU ∗(A∗ B + B D )V + C(V ∗B∗B V + U ∗B B∗U) ℜ σ σ σ σ σ σ σ σ 4ıθ 2ıθ e− 2U ∗B VU ∗B V e− U ∗B V (Cw∗ + w C)), − σ σ − σ θ,σ θ,σ 2ıθ where wθ,σ = w(e− U ∗BσV ) and w is defined as in (30). Averaging over − k and σ one gets by Lemma 6 and Lemma 5 that E Eσ(∂ σ 1 hR i R RP R × − div(∂ σ 1 )) is equal to RP R L Tr(C) 2ıθ EθEσ e(e− Tr(U ∗BσV (Cw∗ + wθ,σC))). 2sin2(k) − ℜ θ,σ The last term with w consists of terms of the form e 4ıθ Tr(U B V E (U θ,σ − ∗ σ k,j ∗ × BV )tE C) and Tr(U B V E U tB V E C). The latter one gives 1 k,j ∗ σ j,k σ j,k 4sin2(k) × t 1 Tr(U ∗UEk,jV V Ej,kC)= 4sin2(k) Tr(Ek,kC) after averaging over σ. There- fore and by a similar result for the term with wθ,σ∗ as well as the definition of C, one obtains Tr((E E )C) 2ıθ j Therefore the lowest-order invariant measure ρ0µ on is given by the Haar measure. M 30 C. SADEL AND H. SCHULZ-BALDES APPENDIX A: VECTOR-VALUED RANDOM VARIABLES t n Lemma 7. Let a = (a1, . . ., an) :Σ R be a centered, vector-valued random variable on a probability space (Σ→, p), and each a L2(Σ, p). Then k ∈ there exist a linear decomposition a = i vibi over finitely many fixed vectors n 2 bi R with coefficient vi which are centered random variables vi L (Σ, p) ∈ 2 P ∈ that are uncorrelated E(vivi′ )= E(vi )δi,i′ . Proof. One can assume that the random variables ak as elements on L2(Σ, p) are linearly independent [otherwise one takes a basis for the vector space span(supp(a)) and rewrites the random variable a as vector using this basis]. Let us introduce λk,j for k > j and write the Ansatz vk = ak + k 1 i=1− λk,iai. Inverting the matrix form of these equations gives 1 P 1 0 0 − a ·. · · . v 1 λ 1 .. . 1 . 2,1 . . = . . . . . . .. .. 0 an vn λ λ 1 n,1 n,n 1 · · · − Hence, one can write a as a sum k vkbk where the bk’s are the vectors of the inverted matrix. The vk’s are pairwise uncorrelated, if E(vkai) = 0 P for all i < k, as this implies E(vkvi) = 0 for all i < k. Now E(vkai)=0 for i = 1,...,k 1 is guaranteed if − E(aka1) E(a1a1) E(a1ak 1) λk,1 . . ·. · · . − . . = ...... − . . . . E(akak 1) E(ak 1a1) E(ak 1ak 1) λk,k 1 − − · · · − − − If the appearing matrix is invertible, one can resolve this equation to get λk,i for all i < k. So it remains to show that this matrix is invertible which is equivalent to the property that the columns are linearly independent. Now let ξ R such that i ∈ k 1 k 1 − − ξiE(ajai)= E aj ξiai = 0 ! Xi=1 Xi=1 2 for all j = 1,...,k. The vector 1 i k 1 ξiai is then orthogonal in L (Σ,µ) ≤ ≤ − to any vector in the subspace spanned by a1, . . . , ak 1 and it therefore has P − to be zero. As the random variables ai are linearly independent, one gets ξ = 0 for all i = 1,...,k 1. i − APPENDIX B: FOURIER SERIES ON COMPACT LIE GROUPS First, let us summarize some facts about the representation theory of compact Lie groups. All this is well known and proofs can be found in the RANDOM LIE GROUP ACTIONS 31 literature, for example, [3], but we need to introduce the notation for the proof of Theorem 5. Let K be a compact Lie group equipped with its normalized Haar measure Tr and let T K be some maximal torus T ∼= , where r is called the rank of K. The continuous⊂ irreducible representations of the torus T are given by the characters, that is, the homomorphisms into the group S1 = U(1) C. Let ⊂ us denote them by X∗(T). They form a Z-module isomorphic to the lattice r Z and hence X∗(T) is a lattice in the vector space = R Z X∗(T), the tensor product over the ring Z. This is an abstract descriptionV ⊗ of the fact, r r ıj θ that the characters of the torus T are given by the maps θ T e · for a fixed j Zr. In this case, = Rr. ∈ 7→ ∈ V Define some AdK-invariant scalar product on the Lie algebra k of K, where AdK denotes the adjoint representation, and adopt with an scalar product V , such that the norm of a X∗(T) coincides with the operator norm of theh· ·i derivative da acting on t,∈ the Lie algebra of T. Let p be the orthogonal complement in k of t, the Lie algebra of T. Then the group T acts on the complexification pC = C R p by the adjoint rep- resentation and linearity. This representation of T⊗can be decomposed into irreducible continuous representations, which means pC = a Φ pa where ∈ pa is the set of P pC such that AdT(P )= a(T )P for all T T. One can ∈ L∈ show that the spaces pa are one-dimensional complex vector spaces. The appearing characters a Φ X∗(T) are called roots of K. If a Φ is a root, then also a Φ. Note∈ that⊂ the character a as a map on T∈ is given by − ∈ 1 − ( a)(T ) = (a(T ))− . −One can divide the vector space in an upper half space and a lower half space in such a way that there is no rootV on the boundary. A root in the upper half space is then called a positive root. The set of vectors v that satisfy v, a 0 for all positive roots a is a so-called positive Weyl∈V chamber . h i≥ C+ An element of the lattice X∗(T) lying in the positive Weyl chamber is called a highest weight. The set of highest weights will be denoted by +. There is a one-to-one correspondence between the irreducible representationsW and the highest weight vectors. Theorem 3. Any irreducible (unitary) representation of K induces (by restriction) a representation of T, which when decomposed into irreducible representations of T contains exactly one highest weight a . For any ∈W+ highest weight vector a +, there is exactly one irreducible representation of K containing a. ∈W Let πa : K U(d(a)) for a + be the corresponding irreducible unitary representation→ of dimension ∈Wd(a). By Schur orthogonality and the Peter– Weyl theorem the matrix coefficients πa(K)k,l, where 1 k, l d(a), of these representations, considered as functions on K, form an≤ orthogonal≤ basis for 32 C. SADEL AND H. SCHULZ-BALDES 2 2 1/2 L (K). The L norm of such a matrix coefficient is d(a)− . Therefore, the orthogonal projection of f onto the space spanned by the matrix coefficients of the irreducible representation πa is given by d(a) d(a) 1 dK˜ (f(K˜ )πa(K˜ )k,l)πa(K)k,l = dK˜ (f(K˜ )πa(K˜ − )l,k)πa(K)k,l K K k,lX=1 Z k,lX=1 Z = d(a) Tr( f(a)π (K)), F a where 1 f(a)= dK f(K˜ )π (K− ). F a ZK Hence Schur orthogonality and the Peter–Weyl theorem imply the following. Corollary 4. Let f L2(K), then one obtains with convergence in L2(K) ∈ f(K)= d(a) Tr( f(a)π (K)). F a a + ∈WX As shown in [29], one can characterize the smooth functions on K by their Fourier series. Theorem 4. A function f on K is smooth if and only if its Fourier coefficients are rapidly decreasing, which means that h> 0 : lim a h f(a) = 0. ∀ a k k kF k k k→∞ Here f(a) denotes the Hilbert–Schmidt norm. If this is fulfilled, then the FourierkF seriesk converges absolutely in the supremum norm on K. Note that the definition of f(a) to be rapidly decreasing is independent F of the chosen norm on + . Now let us consider theW compact⊂V group K TL with the maximal torus T TL and its Lie algebra t RL. The characters× of this torus also factorize by× L ×L L X∗(T T )= X∗(T) Z . As 1 T lies in the center, the direct product of the× scalar product× on k and{ the}× canonical scalar product on RL give a scalar product on k RL that is invariant under the adjoint representation of the group K TL×. Therefore, the induced scalar product on the vector space RL spanned× by the characters also factorizes. As theV× adjoint representation of 1 TL is trivial, the roots of K TL consist of elements (a, 0) where a is a{ root}× of K. Therefore, the positive× roots RANDOM LIE GROUP ACTIONS 33 of K TL are simply the positive roots of K and, as the scalar product on ×L L L R factorizes, the positive Weyl chamber for K T is given by + R . V× × L C × Hence, the highest weight vectors are given by + Z . W ıj×θ Now for a + the mapping (K,θ) πa(K)e · is an irreducible rep- resentation of∈WK TL which contains the7→ highest weight vector (a, j) and by Theorem 3, it× is the unique one containing this weight. Thus, we have shown the following. Theorem 5. The highest weight vectors of K TL are given by ZL, × W+ × where + are the highest weight vectors of K. The irreducible representation parameterizedW by (a, j) ZL is given by ∈W+ × ıj θ π(a,j)(K,θ)= πa(K)e · . Hence, the Fourier series of F is given by ıj θ F (K,θ)= d(a) Tr( F (a, j)π (K))e · F a a + j ZL ∈WX X∈ with convergence in L2(K TL), where × 1 ıj θ F (a, j)= dK dθ F (K,θ)πa(K− )e− · . F TL ZK Z APPENDIX C: DIVERGENCE OF VECTOR FIELDS Let H K be some compact subgroups of the unitary group U(L) and let = K/H⊂be the homogeneous quotient and π : K . On the Lie algebra M → M u(L) and hence on the Lie algebra k of K, the Killing form (u, v) = Tr(u∗v) defines a bi-invariant metric. At each point K K, the Lie algebra h of H form the vertical vectors, that is, the kernel of the∈ differential of π. Hence, the tangent space at π(K) can be identified with the horizontal vectors, h⊥, the orthogonal complement of h in k. This identification depends on the choice of K. Two horizontal lifts of some tangent vector on to two different preimages differ by a conjugation and therefore have theM same length due to the invariance of the metric. Thus, there is a unique metric on such that the projection π : K is a Riemannian submersion. This metricM is invariant under the action→ of MK. Let Si be some orthonormal basis for h⊥, then the push forward, π (Si) forms an orthonormal basis at π(K). (This basis vectors may differ for∗ two different preimages.) Let X be some smooth vector field on and denote M the horizontal lift to K by Xˆ which then is also smooth. As π is a Riemannian submersion, the covariant derivative of X with respect to π (Si) is given by ˆ ∗ π ( Si X). Let (Bj) denote some orthonormal basis of k and identify Bj with∗ ∇ the left invariant vector field. Furthermore, we identify any vector field 34 C. SADEL AND H. SCHULZ-BALDES Y with a function Y : K k such that the vector at K is given by the path K exp(tY (K)). With →Xˆ , we denote the covariant derivative of the vector ∇S field Xˆ and with δSXˆ the derivative of the function w.r.t. to the left-invariant vector field S. Then one has 1 Xˆ = Tr(B∗Xˆ)B = Tr(B∗Xˆ) [S,B ]+ δ Xˆ . ∇S ∇S j j j 2 j S Xj Xj If g denotes the metric on , then the divergence of X at π(K) is given by M ˆ div(X) π = g(π (Si), π∗Si X) π = Tr(Si∗ Si X), ◦ ∗ ∇ ◦ ∇ Xi Xi where we used that Si is horizontal so that g(π (Si),π (Y )) = Tr(Si∗Y ) for all Y . Using the identity above and the fact that∗ S =∗ S which implies i∗ − i Tr(Si∗[Si,Bj]) = 0, the expression reduces to (32) div(X) π = δ Tr(S∗Xˆ). ◦ Si i Xi ˆ As Tr(Si∗Y ) = 0 for any vertical vector Y h, the lifted vector field X does not need to be horizontal for the last equation∈ to hold. Acknowledgments. We thank A. Bendikov for many helpful discussions and hints to the literature at an early stage of this work, and the referee for a number of helpful suggestions. REFERENCES [1] Bony, J.-M. (1969). 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