E Ban van den

J Dixmier

NS Poulsen

Z Rudnick

English

Bernhard Krötz

Henrik Schlichtkrull

Crossref

On Function Spaces on Symmetric Spaces

Krötz, Bernhard

Schlichtkrull, Henrik

Copenhagen University Research Information System

On function spaces on symmetric spaces

Krötz, B.

Schlichtkrull, H.

MPG.PuRe

arXiv:0711.1087v1  [math.RT]  7 Nov 2007ON FUNCTION SPACES ON SYMMETRIC SPACESBERNHARD KRO¨TZ AND HENRIK SCHLICHTKRULLAbstract. Let Y = G/H be a semisimple symmetric space. It isshown that the smooth vectors for the regular representation of Gon Lp(Y ) vanish at infinity.Date: April 9, 2013.121. Vanishing at infinityLet G be a connected unimodular Lie group, equipped with a Haarmeasure dg, and let 1 ≤ p <∞. We consider the left regular represen-tation L of G on the function space Ep = Lp(G).Recall that f ∈ Ep is called a smooth vector for L if and only if themapG→ Ep, g 7→ L(g)fis a smooth Ep-valued map.Write g for the Lie algebra of G and U(g) for its enveloping algebra.The following result is well-known, see [3].Theorem 1.1. The space of smooth vectors for L isE∞p = {f ∈ C∞(G) | Luf ∈ Lp(G) for all u ∈ U(g)}.Furthermore, E∞p ⊂ C∞0 (G), the space of smooth functions on G whichvanish at infinity.Our concern is with the corresponding result for a homogeneousspace Y of G. By that we mean a connected manifold Y with a tran-sitive action of G. In other wordsY = G/Hwith H ⊂ G a closed subgroup. We shall request that Y carries aG-invariant positive measure dy. Such a measure is unique up to scaleand commonly referred to as Haar measure. With respect to dy weform the Banach spaces Ep := Lp(Y ). The group G acts continuouslyby isometries on Ep via the left regular representation:[L(g)f ](y) = f(g−1y) (g ∈ G, y ∈ Y, f ∈ Ep) .We are concerned with the space E∞p of smooth vectors for this repre-sentation. The first part of Theorem 1.1 is generalized as follows, see[3], Thm. 5.1.Theorem 1.2. The space of smooth vectors for L isE∞p = {f ∈ C∞(Y ) | Luf ∈ Lp(Y ) for all u ∈ U(g)}.We write C∞0 (Y ) for the space of smooth functions vanishing atinfinity. Our goal is to investigate an assumption under which thesecond part of Theorem 1.1 generalizes, that is,(1.1) E∞p ⊂ C∞0 (Y ).Notice that if H is compact, then we can regard Lp(G/H) as aclosed G-invariant subspace of Lp(G), and (1.1) follows immediatelyfrom Theorem 1.1.FUNCTION SPACES 3Likewise, if Y = G regarded as a homogeneous space for G×G withthe left×right action, then again (1.1) follows from Theorem 1.1, sincea left×right smooth vector is obviously also left smooth.However, (1.1) is false in general as the following class of examplesshows. Assume that Y has finite volume but is not compact, e.g.Y = Sl(2,R)/Sl(2,Z). Then the constant function 1Y is a smoothvector for Ep, but it does not vanish at infinity.2. Proof by convolutionWe give a short proof of (1.1) for the case Y = G, based on thetheorem of Dixmier and Malliavin (see [2]). According to this theorem,every smooth vector in a Fre´chet representation (π,E) belongs to theG˚arding space, that is, it is spanned by vectors of the form π(f)v,where f ∈ C∞c (G) and v ∈ E. Let such a vector L(f)g, where g ∈Ep = Lp(G), be given. Then by unimodularity(2.1) [L(f)g](y) =∫Gf(x)g(x−1y) dx =∫Gf(yx−1)g(x) dx.For simplicity we assume p = 1. The general case is similar. Let Ω ⊂ Gbe compact such that |g| integrates to < ǫ over the complement. Then,for y outside of the compact set suppf · Ω, we haveyx−1 ∈ suppf ⇒ x /∈ Ω,and hence|L(f)g(y)| ≤ sup |f |∫x/∈Ω|g(x)| dx ≤ sup |f | ǫ.It follows that L(f)g ∈ C0(G).Notice that the assumption Y = G is crucial in this proof, since theconvolution identity (2.1) makes no sense in the general case.3. Semisimple symmetric spacesLet Y = G/H be a semisimple symmetric space. By this we mean:• G is a connected semisimple Lie group with finite center.• There exists an involutive automorphism τ of G such that His an open subgroup of the group Gτ = {g ∈ G | τ(g) = g} ofτ -fixed points.We will verify (1.1) for this case. In fact, our proof is valid also underthe more general assumption that G/H is a reductive symmetric spaceof Harish-Chandra’s class, see [1].4 BERNHARD KRO¨TZ AND HENRIK SCHLICHTKRULLTheorem 3.1. Let Y = G/H be a semisimple symmetric space, andlet Ep = Lp(Y ) where 1 ≤ p <∞. ThenE∞p ⊂ C∞0 (Y ) .Proof. A little bit of standard terminology is useful. As customarywe use the same symbol for an automorphism of G and its derivedautomorphism of the Lie algebra g. Let us write g = h + q for thedecomposition in τ -eigenspaces according to eigenvalues +1 and −1.Denote by K a maximal compact subgroup of G. We will and mayassume that K is stable under τ . Write θ for the Cartan-involutionon G with fixed point group K and write g = k + p for the eigenspacedecomposition of θ. We fix a maximal abelian subspace a ⊂ p ∩ q.The simultaneous eigenspace decomposition of g under ad a leads toa (possibly reduced) root system Σ ⊂ a∗\{0}. Write areg for a with theroot hyperplanes removed, i.e.:areg = {X ∈ a | (∀α ∈ Σ) α(X) 6= 0} .Let M = ZH∩K(a) and WH = NH∩K(a)/M .Recall the polar decomposition of Y . With y0 = H ∈ Y the basepoint of Y it asserts that the mappingρ : K/M × a → Y, (kM,X) 7→ k exp(X) · y0is differentiable, onto and proper. Furthermore, the element X in thedecomposition is unique up to conjugation by WH , and the inducedmapK/M ×WH areg → Yis a diffeomorphism onto an open and dense subset of Y .Let us return now to our subject proper, the vanishing at infinity offunctions in E∞p . Let us denote functions on Y by lower case romanletters, and by the corresponding upper case letters their pull backs toK/M × a, for example F = f ◦ ρ. Then f vanishes at infinity on Ytranslates into(3.1) limX→∞X∈asupk∈K|F (kM,X)| = 0 .We recall the formula for the pull back by ρ of the invariant measuredy on Y . For each α ∈ Σ we denote by gα ⊂ g the correspondingroot space. We note that gα is stable under the involution θτ . Definepα, resp. qα, as the dimension of the θτ -eigenspace in gα according toeigenvalues +1,−1. Define a function J on a byJ(X) =∣∣∣∣∣∏α∈Σ+[coshα(X)]qα · [sinhα(X)]pα∣∣∣∣∣ .FUNCTION SPACES 5With d(kM) the Haar-measure on K/M and dX the Lebesgue-measure on a one then gets, up to normalization:ρ∗(dy) = J(X) d(k,X) := J(X) d(kM) dX .We shall use this formula to relate certain Sobolev norms on Y andon K/M × a. Fix a basis X1, . . . , Xn for g. For an n-tupel m =(m1, . . . , mn) ∈ Nn0 we define elements Xm ∈ U(g) byXm := Xm11 · . . . ·Xmnn .These elements form a basis for U(g). We introduce the Lp-Sobolevnorms on Y ,Sm,Ω(f) :=∑|m|≤m[∫Ω|L(Xm)f(y)|p dy]1/pwhere Ω ⊂ Y , and where |m| := m1 + . . . +mn. Then f ∈ E∞p if andonly if Sm,Y (f) <∞ for all m.Likewise, for V ⊂ a we denoteS∗m,V (F ) :=∑|m|≤m[∫K×V|L(Zm)F (kM,X)|p J(X) d(k,X)]1/pHere Z refers to members of some fixed bases for k and a, acting fromthe left on the two variables, and again m is a multiindex.Observe that for Z ∈ a we have for the action on a,[L(Z)F ](kM,X) = [L(Zk)f ](k exp(X) · y0)where Zk := Ad(k)(Z) can be written as a linear combination of thebasis elements in g, with coefficients which are continuous on K. Itfollows that there exists a constant Cm > 0 such that for all F = f ◦ ρ,(3.2) S∗m,V (F ) ≤ CmSm,Ω(f)where Ω = ρ(K/M, V ) = K exp(V ) · y0.Let ǫ > 0 and setaǫ := {X ∈ a | (∀α ∈ Σ) |α(X)| ≥ ǫ} .Observe that there exists a constant Cǫ > 0 such that(3.3) (∀X ∈ aǫ) J(X) ≥ Cǫ .We come to the main part of the proof. Let f ∈ E∞p . We shall firstestablish that(3.4) limX→∞X∈aǫF (eM,X) = 0 .6 BERNHARD KRO¨TZ AND HENRIK SCHLICHTKRULLIt follows from the Sobolev lemma, applied in local coordinates, thatthe following holds for a sufficiently large integer m (depending onlyon p and the dimensions of K/M and a). For each compact symmetricneighborhood V of 0 in a there exists a constant C > 0 such that(3.5)|F (eM, 0)|≤ C∑|m|≤m[∫K/M×V|[L(Zm)F ](kM,X)|p d(k,X)]1/pfor all F ∈ C∞(K/M × a). We choose V such that aǫ + V ⊂ aǫ/2.Let δ > 0. Since f ∈ Ep, it follows from (3.2) and the propernessof ρ that there exists a compact set B ⊂ a with complement Bc ⊂ a,such that(3.6) S∗m,Bc(F ) ≤ CmSm,Ω(f) < δwhere Ω = K exp(Bc) · y0.Let X1 ∈ aǫ ∩ (B + V )c. Then X1 + X ∈ aǫ/2 ∩ Bc for X ∈ V .Applying (3.5) to the functionF1(kM,X) = F (kM,X1 +X),and employing (3.3) for the set aǫ/2, we derive|F (eM,X1)|≤ C∑|m|≤m[∫K/M×V|[L(Zm)F1](kM,X)|p d(k,X)]1/p≤ C ′∑|m|≤m[∫K/M×Bc|[L(Zm)F ](kM,X)|p J(X) d(k,X)]1/p= C ′S∗m,Bc(F ) ≤ C′δ,from which (3.4) follows.In order to conclude the theorem, we need a version of (3.4) whichis uniform for all functions L(q)f , for q ∈ Q ⊂ G a compact subset.Let δ > 0 be given, and as before let B ⊂ a be such that (3.6) holds.By the properness of ρ, there exists a compact set B′ ⊂ a such thatQK exp(B) · y0 ⊂ K exp(B′) · y0.We may assume that B′ is WH-invariant. Then, for each k ∈ K,X /∈ B′ and q ∈ Q we have that(3.7) q−1k exp(X) · y0 /∈ K exp(B) · y0,FUNCTION SPACES 7since otherwise we would havek exp(X) · y0 ∈ qK exp(B) · y0 ⊂ K exp(B′) · y0and hence X ∈ B′.We proceed as before, with B replaced by B′, and with f , F replacedby fq = Lqf , Fq = fq ◦ p. We thus obtain for X1 ∈ aǫ ∩ (B′ + V )c,|Fq(eM,X1)| ≤ CS∗m,(B′)c(Fq) ≤ CCmSm,Ω′(fq)where Ω′ = K exp((B′)c) · y0.Observe that for each X in g the derivative L(X)fq can be writtenas a linear combination of derivatives of f by basis elements from g,with coefficients which are uniformly bounded on Q. We conclude thatSm,Ω′(fq) is bounded by a constant times Sm,Q−1Ω′(f), with a uniformconstant for q ∈ Q. By (3.7) and (3.6) we conclude that the latterSobolev norm is bounded from the above by δ.We derive the desired uniformity of the limit (3.4) for q ∈ Q,(3.8) limX→∞X∈aǫsupq∈Q|Fq(eM,X)| = 0 .Finally we choose an appropriate set Q. Let ǫ > 0 be arbitrary.There exists X1, . . . , XN ∈ a such that(3.9) a =N⋃j=1(Xj + aǫ) .Set aj = exp(Xj) ∈ A and define a compact subset of G byQ :=N⋃j=1Kaj .Then, for every X ∈ a we have X − Xj ∈ aǫ for some j. Hence withq = k exp(Xj)limX→∞F (kM,X) = limX→∞Fq(eM,X −Xj) = 0,as was to be shown. Remark. Let f ∈ L2(Y ) be a K-finite function which is also finite forthe center of U(g). Then it follows from [4] that f vanishes at infinity.The present result is more general, since such a function necessarilybelongs to E∞2 .8 BERNHARD KRO¨TZ AND HENRIK SCHLICHTKRULLReferences[1] E. van den Ban, The principal series for a reductive symmetric space, II, Jour.Funct. Anal. 109 (1992), 331–441.[2] J. Dixmier and P. Malliavin, Factorisations de fonctions et de vecteursinde´finiment diffe´rentiables, Bull. Sci. Math. 102 (1978), 307–330.[3] N. S. Poulsen, On C∞-vectors and intertwining bilinear forms for representa-tions of Lie groups, J. Funct. Anal. 9 (1972), 87–120.[4] Z. Rudnick and H. Schlichtkrull, Decay of eigenfunctions on semisimple sym-metric spaces, Duke Math. J. 64 (1991), no. 3, 445–450.Max-Planck-Institut fu¨r Mathematik, Vivatsgasse 7, D-53111 Bonn,email: kroetz@mpim-bonn.mpg.deDepartment of Mathematics, University of Copenhagen, Univer-sitetsparken 5, DK-2100 København, email: schlicht@math.ku.dk

On function spaces on symmetric spaces

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