A generalization of the Hardy theorem to semisimple Lie groups. By Mitsuhiko Ebata,*1) Masaaki Eguchi,*2),*5) Shin Koizumi,*3),*5) and Keisaku Kumahara*4) ...
No. 7]
Proc. Japan Acad., 75, Ser. A (1999)
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A generalization of the Hardy theorem to semisimple Lie groups By Mitsuhiko Ebata,∗1) Masaaki Eguchi,∗2),∗5) Shin Koizumi,∗3),∗5) and Keisaku Kumahara∗4),∗6) (Communicated by Heisuke Hioronaka, m. j. a., Sept. 13, 1999)
1. Introduction. The classical uncertainty principle asserts that a function and its Fourier transform cannot both be concentrated on intervals of small measure. In the case of the Euclidean space, various forms of the uncertainty principle are known. One of them is the following Hardy theorem (cf. [1, pp. 155-158]). If a measurable function f on R satisfies |f | ≤ C exp{−ax2 } and |fˆ| ≤ C exp{−by 2 } and ab > then f = 0 √ (a.e.). Here we take fˆ(y) = √ 1/4,R ∞ (1/ 2π) −∞ f (x) exp{ −1xy}dx as the definition of the Fourier transform of f . Recently A. Sitaram and M. Sundari [6] generalized this theorem to the cases of the semisimple Lie groups with one conjugacy class of Cartan subgroups, the Riemannian symmetric spaces and SL(2, R). On the other hand M. Sundari [7] showed the Hardy theorem for the Euclidean motion group. And also M. Eguchi, S. Koizumi and K. Kumahara generalized the results to Cartan motion group [2] and gave an Lp version for motion groups [3]. In this paper we give an analogue of the Hardy theorem to noncompact semisimple Lie groups. 2. Notation and preliminaries. The standard symbols Z, R and C shall be used for the set of the integers, the real numbers and the complex numbers. If V is a vector space over R, Vc , V ∗ and Vc∗ denote its complexification, its real dual and its comˆ denotes plex dual, respectively. For a Lie group L, L the set of all equivalence classes of irreducible unitary representations of L. If L is a reductive Lie group, ∗1) Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8521. ∗2) Faculty of Integrated Arts and Sciences, Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8521. ∗3) Faculty of Information sciences and Management, Onomichi Junior College, 1600 Hisayamada-cho, Onomichi, Hiroshima 722-8506. ∗4) The University of the Air, 2-11 Wakaba, Mihama-ku, Chiba 261-8586. ∗5) Partly supported by the Grants-in-Aid for Scientific Research (No. 09640190), the Ministry of Education, Science, Sports and Culture of Japan. ∗6) Partly supported by the Grants-in-Aid for Scientific Research (No. 10640171), the Ministry of Education, Science, Sports and Culture of Japan.
ˆ disc denotes the subset comprised of all equivalence L classes of discrete series. As usual for a Lie group, we use lower case German letters to denote its Lie algebras. If H is a complex separable Hilbert space, the operator norm on H will be denoted by k · k∞ . Let G be a connected semisimple Lie group with finite center. We fix a maximal compact subgroup K of G and denote by θ the corresponding Cartan involution. Let g = k + p be the Cartan decomposition of g defined by θ. Let ap be a maximal abelian subspace of p and denote by Σ the set of all restricted roots of g relative to ap . We fix an order in a∗p and denote by Σ+ the set of all positive restricted roots. Let {α1 , · · · , αl } be the set of simple roots of (g, ap ) o nP l n α ; n ∈ Z (i = 1, · · · , l) and put L = i i i o nP i=1 l + (i = 1, · · · , l) . and L = i=1 ni αi ; ni ∈ Z≥0 Pl Pl For each λ = i=1 λi αi ∈ L+ , put |λ| = i=1 λi . For q = (q1 , . . . , ql ) ∈ Zl≥0 , we also put α(H)q = Pl α1 (H)q1 · · · αl (H)ql (H ∈ a) and |q| = i=1 qi . Let a− a choice of negative Weyl chamber. We p be P set np = λ∈Σ+ gλ and put Ap = exp ap , Np = exp np and Mp = ZK (ap ). Then P0 = Mp Ap Np is a minimal parabolic subgroup of G. When g = k exp X (k ∈ K, X ∈ p), we set σ(g) = kXk, where k · k denoting the norm coming from the Killing form of g. We write Car(G) for the set of all θ-stable Cartan subgroups and denote by Car0 (G) the subset of Car(G) comprised of all noncompact ones. For J ∈ Car(G), let PJ = MJ AJ NJ be the Langlands decomposition of the cuspidal parabolic subgroup PJ associated to J. We remark that if J ∈ Car0 (G) then AJ 6= {e}. Under the decomposition G=KPJ =KMJ AJ NJ , each g∈G can be written as g=κ(g)µJ (g) exp HJ (g) nJ (g), where κ(g) ∈ K, µJ (g) ∈ MJ , HJ (g) ∈ aJ ˆ J )disc and ν ∈ a∗ . We and nJ (g) ∈ NJ . Let σ ∈ (M J denote by (πJ,σ,ν , HJ,σ,ν ) the representation induced from σ ⊗ ν ⊗ 1 of PJ to G. Then it is known that (πJ,σ,ν , HJ,σ,ν ) is unitary.
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3. The main theorem. As the Hardy theˆ disc = ∅ is obtained in [6], we orem in the case G ˆ disc 6= ∅. Let suppose rankG = rankK, that is, G 0 J ∈ Car (G) and PJ = MJ AJ NJ be the corresponding cuspidal parabolic subgroup of G. For ˆ J )disc and ν ∈ a∗ , we define the Fourier transσ ∈ (M J form fˆJ (σ, ν) of f ∈ L1 (G) by Z fˆJ (σ, ν) = f (g)πJ,σ,ν (g) dg. G
We write πλ for the discrete series representation of G with Harish-Chandra parameter λ. We define the Fourier transform fˆd (λ) of f ∈ L1 (G) by Z fˆd (λ) = f (g)πλ (g) dg. The following is an analogue of Hardy theorem for semisimple Lie group. Theorem 3.1. Let f be a measurable function on G such that (1) (2)
|f (g)| ≤ Ce−aσ(g)
Φ(a) =
r X X
α(log a)q e(µj +ρ)(log a)
j=1 q∈M
X
Cµj +λ,q eλ(log a) ,
a ∈ A− p.
λ∈L+
Here the series on the right hand side is absolutely and uniformly convergent as long as a ∈ A− p, −αj (log a) ≥ εj > 0.
2
kfˆJ (σ, ν)k∞ ≤ CJ,σ e−bkνk
2
(ν ∈ a∗J )
for each J ∈ Car0 (G), some constants a > 0, b > 0, C > 0 and CJ,σ > 0. If ab > 1/4 then f = 0 (a.e.). We give here a sketch of the proof. From the assumption for the function f , we see that f ∈ L1 (G)∩ ˆ J )disc × (a∗ )c . L2 (G) and fˆJ (σ, ν) makes sense on (M J It is shown that the assumption of f and fˆJ in Theorem 3.1 implies fˆJ (σ, ν) = 0 for ν ∈ a∗J . This is accomplished by a similar argument to that of [6]. By the Plancherel formula and the above argument for continuous series, we can suppose f ∈ L2d (G), L2d (G) denoting the closed subspace of L2 (G) spanned by the set of K-finite matrix elements of discrete series of G. Decomposing f into the sum X X f= χτ1 ∗ f ∗ χτ2 = fτ1 ,τ2 and using the assumptions (1) and (2), we can prove that fτ1 ,τ2 satisfies also the assumptions (1) and (2). Therefore, we can assume X f= Ck Φk , 1≤k≤s
where Φk is a matrix element of a discrete series. We apply the asymptotic expansion to each Φk and pay attention that the leading exponents are of first order. Then we can see the assumption (2) on f leads to a contradiction by the following theorem. Theorem 3.2 ([4,5]). Let Φ be a matrix element of a discrete series. Then there exist mutually integrally non-equivalent elements µ1 , · · · , µr ∈ a∗J , a finite set M ⊂ Zl≥0 and constants Cµj +λ,q ∈ C (j = 1, · · · , r, λ ∈ L+ , q ∈ M) with Cµj ,q 6= 0 (j = 1, · · · , r, q ∈ M), such that
×
G
[Vol. 75(A),
References [ 1 ] H. Dym and H. P. McKean: Fourier series and Integral. Academic Press, New York, pp. 155–158 (1972). [ 2 ] M. Eguchi, S. Koizumi and K. Kumahara: An analogue of the Hardy theorem for the Cartan motion group. Proc. Japan Acad., 74A, 149–151 (1998). [ 3 ] M. Eguchi, S. Koizumi and K. Kumahara: An Lp version of the Hardy theorem for motion groups. J. Austral. Math. Soc. (in print). [ 4 ] Harish-Chandra: Some results on differential equations (1960) (manuscript). [ 5 ] D. Miliˇci´c: Asymptotic behaviour of matrix coefficients of the discrete series. Duke Math. J., 44, 59–88 (1977). [ 6 ] A. Sitaram and M. Sundari: An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups. Pacific J. Math., 177, 187–200 (1997). [ 7 ] M. Sundari: Hardy’s theorem for the n-dimensional Euclidean motion group. Proc. Amer. Math. Soc., 126, 1199–1204 (1998).
