aa r X i v : . [ m a t h . F A ] F e b REGULAR ORBITAL MEASURES ON LIE ALGEBRAS
ALEX WRIGHT
Abstract.
Let H be a regular element of an irreducible Lie Al-gebra g , and let µ H be the orbital measure supported on O H . Weshow that ˆ µ kH ∈ L ( g ) if and only if k > dim g / (dim g − rank g ). Introduction
Let G be a compact, connected, simple Lie group and g its Lie al-gebra. It is well known that the non-trivial adjoint orbits in g arecompact submanifolds of proper dimension, but geometric propertiesensure that they generate g . Consequently, if H = 0 is in the torus t of g , and µ H is the orbital measure supported on the orbit O H con-taining H , ie, µ H is the unique (up to normalization) G -invariantmeasure on O H , then some convolution power of µ H is absolutelycontinuous to Lebesgue measure on g and even belongs to L ε forsome ε > g con-volution powers sufficed, and this was improved in a series of papersculminating in [2] with the minimal number of convolution powers be-ing k G = rank G for the classical simple Lie algebras of type B n , C n and D n and k G = rank G + 1 for type A n . There it was also shownthat if µ h was the orbital measure supported on the conjugacy class in G containing the non-central element h , then µ k G h ∈ L ( G ).In the simplest case G = SU (2), g = R and the adjoint orbitsare (the two dimensional) spheres centred at the origin. The sum of Mathematics Subject Classification.
Primary 58C35; Secondary 22E60,43A70.
Key words and phrases.
Orbital measures, Lie algebras.This research was supported in part by NSERC.The author would like to thank Kathryn Hare for her generous help in preparingthis note. two such spheres contains an open set and consequently the convo-lution of any two orbital measures is absolutely continuous [7]. Ingeneral, the generic orbits (the so-called regular orbits defined below)have codimension rank G and two convolution powers of such an or-bital measure is absolutely continuous (in either the group or algebracase). Furthermore, for the generic orbital measure µ h on the group,one can use the Weyl character formula to see that µ k h ∈ L ( G ) for k = 1 + rank G/ (dim G − rank G ) (see [4]) and this fact can be trans-ferred to the Lie algebra setting as well [3].In this note we give a direct proof that if µ H is any generic orbitalmeasure on g , then ˆ µ kH ∈ L ( g ) if and only if k > g / (dim g − rank g ). The novelty of our approach is our geometric method, involv-ing the root systems, of handling the singularities which arise in theintegral of the Fourier transform of the measure.Products of generic orbital measures are also studied in [1] and [9];our approach recovers some of what was proven in [9].2. Definitions and Lemmas
Let T be a maximal torus of G and t be the corresponding subalgebraof g , also called the torus. Let Φ be the root system of g with Weylgroup W and positive roots Φ + . Choose a base ∆ = { β , . . . , β n } forΦ and let t + be the associated fundamental Weyl chamber. t + = { H ∈ t : ( H, β j ) > j = 1 , . . . n } Given H ∈ t , the adjoint orbit of H is given by O H = { Ad ( g ) H : g ∈ G } ⊆ g . If H ∈ t + , then H is called regular and O H is a called a regular orbit.The regular orbital measure, µ H , is the G -invariant measure sup-ported on the regular orbit O H , normalized so the Harish-Chandraformula gives b µ H ( H ) = A H ( H ) Q α ∈ Φ + ( α, H ) for H ∈ t + EGULAR ORBITAL MEASURES ON LIE ALGEBRAS 3 where A H ( H ) = X σ ∈W sgn( σ ) e i ( σ ( H ) ,H ) . As µ H is G -invariant, the Weyl integration formula implies thatˆ µ kH ∈ L ( g ) if and only if Z t + | A H ( H ) | k (cid:12)(cid:12)Q α ∈ Φ + ( α, H ) (cid:12)(cid:12) k − dH < ∞ .In this integral some of the inner products ( α, H ) represent removablesingularities on some walls of the Weyl chamber. This is the primaryobstacle in studying this integral and we are able to deal with thesesingularities using geometry and an induction argument.Specifically, we will relate the integrand near a collection of wallsto the integrand for a subroot system. The power of our inductionis hidden in the fact that the the integrand is continuous, and so isbounded on any neighborhood of the origin. Several technical problemsarise; in fact they are necessary adaptations to the proof of a weakerresult (Cor. 1), where the technical results are not necessary. Thecase of a Lie algebra of type A is surprisingly representative, and thegeometric motivation for the results presented here come exclusivelyfrom this case.The notation will get slightly tedious, so we list it all here in advance.Note that from now on we assume Φ is irreducible, but we will considerreducible subroot systems of Φ that are “simple”; these are simply thosesubroot systems for which a subset of ∆ can be chosen as a base. ALEX WRIGHT g An irreducible Lie algebra.Φ The root system of g .∆ = { β , . . . , β n } The simple roots of Φ.Φ + The positive roots of Φ. n The rank of g . W The Weyl group of Φ. t + = { H ∈ g : ( H, β i ) > i } The fund. Weyl chamber of g .Ψ A simple subroot system of Φ. V The Weyl group of Ψ. { γ , γ , . . . , γ m } ⊂ ∆ A base for Ψ.Ψ + The positive roots of Ψ. m Number of simple roots in Ψ. s + = { H ∈ span Ψ : ( H, γ i ) > i > } The fund. Weyl chamber of Ψ.Recall that every H ∈ s + can be written as a non-negative linearcombination of the simple roots γ i . This follows from the fact that, inthe irreducible case, all entries of the inverse of the Cartan matrix arepositive numbers. (See [5], section 13.4, exercise 8.)We will need to break s + up into the regions R i = { H ∈ s + : k H k ≥ , ( γ i , H ) ≥ ( γ j , H ) for all j } . So s + \ B = ∪ mi =1 R i . Now ifΨ = span Z { γ , . . . , γ m } ∩ Ψthen the roots of Ψ will correspond to removable singularities on thewalls of cl( R ) when we calculate the above integral with root systemΨ. Now let V be the Weyl group of Ψ , and c + = { H ∈ span Ψ : ( H, γ i ) > i = 2 . . . m } be the fundamental Weyl chamber of Ψ . Finally, we define P : span Ψ → span Ψ : H |V | X σ ∈V σ ( H ) . Lemma (1) . Let P be as above. Then (i) σ ( P ( H )) = P ( H ) for all σ ∈ V . (ii) P is the projection from span Ψ onto (span Ψ ) ⊥ . So I − P isthe projection from span Ψ onto span Ψ . (iii) I − P in fact maps s + to c + . EGULAR ORBITAL MEASURES ON LIE ALGEBRAS 5 (iv)
There are constants a, b > so that k P ( H ) k ≥ a k H k and k ( I − P ) H k ≤ b k P ( H ) k if H ∈ R . Before reading the proof of this result, the reader is encouraged tographically verify part (ii) for the case Φ = A . Proof. (i) If σ ∈ V then σ ( P ( H )) = 1 |V | X σ ∈V σ ( σ ( H )) = 1 |V | X σ ∈ σ V σ ( H ) = P ( H ) . (ii) Write H = s + r , where s ∈ span Ψ and r ∈ (span Ψ ) ⊥ . If α ∈ Ψ then σ α ( r ) = r − r, α )( α, α ) α = r. Since V is generated by reflections of the form σ α , α ∈ Ψ , it followsthat σ ( r ) = r for all σ ∈ V . Hence P ( H ) = P ( r ) + P ( s ) = r + P ( s ) . If α ∈ Ψ then by (i) σ α ( P ( s )) = P ( s ). Since we also have σ α ( P ( s )) = P ( s ) − P ( s ) , α )( α, α ) α we get that P ( s ) ∈ (span Ψ ) ⊥ . But P ( s ) ∈ span Ψ so P ( s ) = 0.Putting all of this together, we get that P ( H ) = r is the projection of H onto (span Ψ ) ⊥ .Of course it follows that H − P ( H ) is the projection of H ontospan Ψ .(iii) If k > H ∈ s + then( γ k , H − P ( H )) = ( γ k , H ) > P ( H ) ∈ span { γ , . . . , γ m } ⊥ .(iv) Suppose, in order to obtain a contradiction, that H ∈ cl( R )and P ( H ) = 0. Then H ∈ span Ψ and H ∈ cl( s + ) so we can write H = c γ + . . . + c m γ m with all c i ≥
0. Thus(
H, γ ) = c ( γ , γ ) + . . . + c m ( γ m , γ ) . ALEX WRIGHT
We also have ( γ i , γ j ) ≤ i = j , so in fact ( H, γ ) ≤
0. Since H ∈ s + ( H, γ ) ≥
0. Combining these we get (
H, γ ) = 0. From the definitionof R we get, for each i = 1 , . . . , m , that0 ≤ ( H, γ i ) ≤ ( H, γ ) = 0which contradicts the fact that k H k ≥
1. Thus P ( H ) = 0 on cl( R ). Inparticular, P ( H ) is nonzero on the compact set cl( R ) ∩{ H : k H k = 1 } ,so there is an a > a ≤ k P ( H ) k if k H k = 1, H ∈ R . Thuswe see that a k H k ≤ k P ( H ) k on R . Finally, we can take b = a + 1. (cid:3) We commented earlier that the roots of Ψ will cause problems in R when integrating. As it turns out, all the other roots of Ψ are verywell behaved on R . (It is quite helpful to think of the roots of Ψ asthe “good” roots on R , and the roots of Ψ \ Ψ as the “bad” roots.) Lemma (2) . There exists
C > such that for all α ∈ Ψ + \ Ψ +1 and forall H ∈ R ( H, α ) ≥ C k H k . Proof.
Take α ∈ Ψ + \ Ψ +1 . Write α = P a i γ i with all a i ≥
0. Since α / ∈ Ψ , a >
0. Now if H ∈ cl( R )( H, α ) = X a i ( H, γ i ) ≥ a ( H, γ ) > . Thus the function f ( H ) = ( H, α )is non zero on the compact set cl( R ) ∩{ H : k H k = 1 } . Hence it attainsa positive minimum M α . We can take C = min α ∈ Ψ + \ Ψ +1 M α . (cid:3) We will be interested in subroot systems of Φ of the form { a α + a α + . . . + a m α m : a i ∈ Z for all i } ∩ Φwhere { α , α , . . . α m } ⊂ ∆. We will call these simple subroot sys-tems. Note that Ψ is a simple subroot system of Φ. Simple subrootsystems are the only type of subroot systems that will come up in ourinduction. Restricting our attention to simple subroot systems makesthe verification of the following technical lemma easier. EGULAR ORBITAL MEASURES ON LIE ALGEBRAS 7
Lemma (3) . Suppose Φ is an irreducible root system, with simple sub-root system Ψ with m simple roots, where m < n . Then n | Φ | < m | Ψ | Proof.
When we look at this result for a particular m , it is clearlysufficient to prove it for the largest Ψ with m simple roots. We listthese subroot systems in Appendix A, along with the ratios in question.See [5] for basic facts needed about subroot systems. (cid:3) It is worth noting that this lemma is not true if we allow Φ to bereducible. For example, consider a subroot system of Lie type B ( m | Ψ | = ) in a root system of Lie type B × A × A × A ( n | Φ | = ).We now set ǫ > ǫ < m | Ψ | − n | Φ | for all propersimple subroot systems Ψ of Φ. We will need this ǫ later for technicalreasons. 3. The main result
Theorem.
Let Φ be an irreducible root system. Then ˆ µ kH ∈ L ( g ) ifand only if k > n | Φ | = dim g / (dim g − rank g ) . Corollary (1) . If µ is a regular orbital measure, then ˆ µ ∈ L ( g ) . Corollary (2) . If µ is a regular orbital measure then µ ∈ L p ( g ) forall p < dim( g )rank( g ) .Proof. Our arguments show that ˆ µ ∈ L p ′ for p ′ < n | Φ | . By theHausdorff-Young inequality, µ ∈ L p for all p < dim( g )rank( g ) . (cid:3) It is worth noting that Corollary 2 is sharp when g = su (2) by aresult of Ragozin (see [7], Prop A.5). Proof. (Of main theorem.) We prove a related result for all simplesubroot systems Ψ of Φ. All the notation will be as before, includingthe definition of ǫ .Our induction hypothesis: For all simple proper subroot systems Ψof Φ, if k < n | Φ | + ǫ then Z s + ∩ B r | A H ( H ) | k dH | Q α ∈ Ψ + ( α, H ) | k − = O ( r n − ( k − | Ψ | ) . ALEX WRIGHT
By this we mean that this integral is bounded above, as a function of r , by Cr n − ( k − | Ψ | for some C > m = 1, Ψ is of Lie type A and we get Z r | e itH − e − itH | k | t | k − dt. Hence when k < the integrand is O ( r − k ) and 2 − k > − k < the integral is O ( r − k − ). Lemma 3 tells us that1 + = 1 + n | Φ | + ( − n | Φ | ) > n | Φ | + ǫ .Now we assume the result for all simple subroot systems of rank m − m . We will describe the growthof the integral on R . Since we have not specified any particular orderamong the R i , and the integrand is continuous, this is sufficient.Let σ , . . . , σ t be representatives from the left cosets of V ≤ V . Webreak up | A H | by cosets of Ψ . Z R ∩ B r | P tj =1 P σ ∈V sgn( σ j σ ) e i ( σ j σ ( H ) ,H ) | k dH | Q α ∈ Ψ + ( α, H ) | k − ≤ k t X j =1 Z R ∩ B r | P σ ∈V sgn( σ j σ ) e i ( σ j σ ( H ) ,H ) | k dH | Q α ∈ Ψ + ( α, H ) | k − For convenience we forget about the constant, and just write the termof the σ j coset. We start by factoring out (cid:12)(cid:12) sgn( σ j ) e i ( σ j ( P ( H )) ,H ) (cid:12)(cid:12) = 1 toget Z R ∩ B r | P σ ∈V sgn( σ ) e i ( σ j σ ( H ) ,H ) − i ( σ j ( P ( H )) ,H ) | k dH | Q α ∈ Ψ + ( α, H ) | k − . Since P ( H ) = σ ( P ( H )) (for σ ∈ V ) and ( σ ( v ) , w ) = ( v, σ ( w )) thisintegral equals Z R ∩ B r | P σ ∈V sgn( σ ) e i ( σ ( H − P ( H )) ,σ j ( H )) | k dH | Q α ∈ Ψ + \ Ψ +1 ( α, H ) | k − | Q α ∈ Ψ +1 ( α, H ) | k − . Now we apply Lemma 2 to get the upper bound Z R ∩ B r C k H k ( k − | Ψ |−| Ψ | ) | P σ ∈V sgn( σ ) e i ( σ ( H − P ( H )) ,σ j ( H )) | k dH | Q α ∈ Ψ +1 ( α, H ) | k − . EGULAR ORBITAL MEASURES ON LIE ALGEBRAS 9
At this point we can safely replace σ j ( H ) with H ′ = ( I − P ) σ j ( H ).Since P ( H ) is orthogonal to Ψ , we can change the inner products from( α, H ) to ( α, H − P ( H )). If we also recall the bound k P ( H ) k ≥ a k H k for all H ∈ R from Lemma 1, this gives Z R ∩ B r C k H k ( k − | Ψ |−| Ψ | ) | P σ ∈V sgn( σ ) e i ( σ ( H − P ( H )) ,H ′ ) | k dH | Q α ∈ Ψ +1 ( α, H − P ( H )) | k − ≤ Z R ∩ B r C ′ k P ( H ) k ( k − | Ψ |−| Ψ | ) | P σ ∈V sgn( σ ) e i ( σ ( H − P ( H )) ,H ′ ) | k dH | Q α ∈ Ψ +1 ( α, H − P ( H )) | k − .P maps onto a one dimensional subspace, say span v , k v k = 1. Wecan do a change of variables so that we are integrating first with respectto H ′ = H − P ( H ) ∈ c + and then s , where P ( H ) = sv . If a and b areas in Lemma 1 then s ≥ a and k ( I − P ) H k ≤ b k P ( H ) k for H ∈ R .Note that H ( P ( H ) , ( I − P ) H ) is an orthogonal change of variablesso the Jacobian is a constant.If we now use Fubini’s Theorem to rewrite our integral (and forgetthe constant) we get Z ra s ( k − | Ψ |−| Ψ | ) Z c + ∩ B bs | P σ ∈V sgn( σ ) e i ( σ ( H ′ ) ,H ′ ) | k dH ′ | Q α ∈ Ψ +1 ( α, H ′ ) | k − ds. Note that no element of Ψ annihilates σ j ( H ), so it is regular. It followsthat no element of Φ annihilates H ′ . Thus we can apply the inductionhypothesis. Since m < n ,1 + m | Ψ | = 1 + n | Φ | + ( m | Ψ | − n | Φ | ) > n Φ + ǫ . So if k < n | Φ | + ǫ we have that the above integral is at most Z ra s ( k − | Ψ |−| Ψ | ) O ( s m − −| Ψ | ( k − ) dr = O ( r m −| Ψ | ( k − ) . At some point in our induction we get that n = m and Ψ = Φ. At thispoint our full induction hypothesis does not hold, but we have that actual integral we are interested in is at most Z rδ O ( r n − −| Φ | ( k − ) dr if k < n | Φ | + ǫ . This integral converges if1 + n | Φ | + ǫ > k > n | Φ | . Hence ˆ µ H ∈ L ( g ) if k > n | Φ | .Now we show the necessity of the condition k > n | Φ | .We can rewrite | P σ ∈W sgn( σ ) e i ( σ ( H ) ,H ) | k | Q α ∈ Φ + ( α, H ) | k − as 1 k H k | Φ | ( k − | P σ ∈W sgn( σ ) e i k H k ( σ ( H k H k ) ,H ) | k (cid:12)(cid:12)(cid:12)Q α ∈ Φ + (cid:16) α, H k H k (cid:17)(cid:12)(cid:12)(cid:12) k − and consider this as r −| Φ | ( k − f ( r, φ , . . . , φ n − ), where f is a functionin polar coordinates. f ( r, φ , . . . , φ n − ) = | P σ ∈W sgn( σ ) e ir ((1 ,φ ,...,φ n − ) ,H ) | k | Q α ∈ Φ + ( α, (1 , φ , . . . , φ n − )) | k − As before, we will integrate in t + with a ball around the origin removed,so we will always assume r ≥ φ , . . . , φ n − ), we see that f Φ ( r ) := f ( r, φ , . . . , φ n − )is (the absolute value of) the sum of continuous functions that areperiodic in r . Thus f is almost periodic in r .Since k µ kH k 6 = 0 and f is continuous, we can find a point ( r , ψ , . . . , ψ n − ),a δ > ǫ > U = { ( r, φ , . . . , φ n − ) : |k φ i − ψ i k ≤ δ ∀ i, | r − r | ≤ δ } ⊂ t + then f > ǫ > U .We will have to change to polar coordinates to use this observation.The Jacobian of this change of variables is∆ = r n − sin n − φ . . . sin n − φ n − . EGULAR ORBITAL MEASURES ON LIE ALGEBRAS 11
If necessary, we can modify U so that | ∆ | ≥ Cr n − on U , for someconstant C . We then get that our integral greater or equal to Z ψ + δψ − δ . . . Z ψ n − + δψ n − − δ Z ∞ C r | Φ | ( k − f ( r, φ , . . . , φ n − ) r ( n − drdφ n − . . . dφ . Say that f Φ has an ǫ almost period in every interval of size M . Pick N ≥ M + 2 δ . We know that f Φ ( r ) ≥ ǫ on [ r − δ, r + δ ]. Pick τ n , an ǫ almost period of f Φ in the interval [ nN − r + δ, ( n + 1) N − r − δ ],where n >
0. Hence f Φ ≥ ǫ on [ r + τ n − δ, r + τ n + δ ] ⊂ [ nN, ( n + 1) N ].If χ F is the indicator function of F = S n [ r + τ n − δ, r + τ n + δ ], thenthe inner integral is at least Z ∞ Cǫχ F r − ( k − | Φ | + n − dr ≥ Z ∞ Cǫχ E r − ( k − | Φ | + n − dr where E = S n [ nN, nN + 2 δ ]. This integral is at least ∞ X n =1 δCǫ ( nN ) − ( k − | Φ | + n − . If k ≤ n | Φ | , this diverges. Thus the inner integral is infinite for all φ , . . . , φ n − in the appropriate range. So if k ≤ n | Φ | , our integral isinfinite and µ k / ∈ L ( g ). (cid:3) Remark (1) . A similar result holds when
Φ = Φ × . . . × Φ m is reducible.Say that the number of simple roots in Φ i is r i , and the fundamentalWeyl chamber of Φ i is t + i . In this case the integrand splits to give Z t +1 Z t +2 . . . Z t + m | A H ( t + t + . . . + t m ) | k | Q α ∈ Φ + ( α, t + t + . . . + t m ) | k − dt m . . . dt . This factors as Z t +1 A Φ H ( t ) | Q α ∈ Φ +1 ( α, t ) dt Z t +2 A Φ H ( t ) | Q α ∈ Φ +3 ( α, t ) dt . . . Z t + m A Φ m H ( t m ) | Q α ∈ Φ + m ( α, t m ) dt m . Since none of these factors can be zero, this is finite iff all the integralsconverge. Hence µ H ∈ L ( g ) iff k > max { r | Φ | , r | Φ | , . . . , r m | Φ m | } . Remark (2) . A measure µ is called L p -improving if there is some p < such that the operator T µ : f µ ∗ f is bounded from L p ( g ) to L ( g ) .Using sophisticated arguments Ricci and Travaglini [9] prove that fora regular, orbital measure µ , T µ maps L p ( g ) to L ( g ) if and only if p ≥ g ) / (2 dim( g ) − rank( g )) = p ( g ) . The same reasoningas given in [4] Corollary 12 shows that our arguments give the weakerresult: T µ is bounded from L p ( g ) to L ( g ) for any p > p ( g ) . Appendix A Φ n | Φ | Ψ m | Ψ | A n n +1 A m , m < n m +1 B n n B m , m < n m C n n C m , m < n m D n n − D m , m < n m − E D m , m < m − E D m , m < m − E E E D m , m < m − E E E E F B m , m < m G A References [1] A. Dooley, J. Repka and N. Wildberger,
Sums of adjoint orbits , Linear andmultilinear algebra (1993), 79–101.[2] S. Gupta and K. Hare, Singularity of orbits in classical Lie algebras , Geom.Func. Anal. (2003), 815-844.[3] S. Gupta, K. Hare and S. Seyfaddini, L2 - singular dichotomy for orbital mea-sures of classical simple Lie algebras , preprint.[4] K. Hare,
The size of characters of compact Lie groups , Studia Mathematica (1998), 1–18.[5] J. Humphreys,
Introduction to Lie Algebras and Representation Theory .Springer-Verlag, New York, 1972.[6] D. Ragozin,
Central measures on compact simple Lie groups , J. Func. Anal. (1972), 212–229.[7] D. Ragozin, Rotation invariant measure algebras on Euclidean space , IndianaUniv. Math. J. (1973/74), 1139–1154.[8] F. Ricci and E. Stein, Harmonic analysis on nilpotent groups and singularintegrals. II. Singular kernels supported on submanifolds , J. Func. Anal. (1988), 56–84. EGULAR ORBITAL MEASURES ON LIE ALGEBRAS 13 [9] F. Ricci and E. Travaglini, L p - L q estimates for orbital measures and Radontransforms on compact Lie groups and Lie algebras. J. Funct. Anal. (1995),132–147.
Department of Pure Mathematics, University of Waterloo, Water-loo ON Canada, N2L 3G1
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