Complex Fourier analysis on a nilpotent Lie group

Goodman, Roe

doi:10.1090/S0002-9947-1971-0417334-3

Complex Fourier analysis on a nilpotent Lie group
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Trans. Amer. Math. Soc. 160 (1971), 373-391 Request permission

Abstract:

Let $G$ be a simply-connected nilpotent Lie group, with complexification ${G_c}$. The functions on $G$ which are analytic vectors for the left regular representation of $G$ on ${L_2}(G)$ are determined in this paper, via a dual characterization in terms of their analytic continuation to ${G_c}$, and by properties of their ${L_2}$ Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case $G = R$ this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip.

References

J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. I, Amer. J. Math. 81 (1959), 160–170 (French). MR 103943, DOI 10.2307/2372853
Jacques Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. IV, Canadian J. Math. 11 (1959), 321–344 (French). MR 106963, DOI 10.4153/CJM-1959-034-8
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173

Generalized functions

Spaces of fundamental functions

21

37

Roe W. Goodman, Analytic and entire vectors for representations of Lie groups, Trans. Amer. Math. Soc. 143 (1969), 55–76. MR 248285, DOI 10.1090/S0002-9947-1969-0248285-6
Roe W. Goodman, Differential operators of infinite order on a Lie group. I, J. Math. Mech. 19 (1969/1970), 879–894. MR 0255736
Roe W. Goodman, One-parameter groups generated by operators in an enveloping algebra. , J. Functional Analysis 6 (1970), 218–236. MR 0268330, DOI 10.1016/0022-1236(70)90059-5
G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco-London-Amsterdam, 1965. MR 0207883
Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968, DOI 10.1007/978-3-662-11761-3
Edward Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615. MR 107176, DOI 10.2307/1970331
Edward Nelson and W. Forrest Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math. 81 (1959), 547–560. MR 110024, DOI 10.2307/2372913
Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019

Regularity aspects of the theory of infinite dimensional representations of Lie groups

Leçons sur les représentations des groupes

36

W. Forrest Stinespring, Integrability of Fourier transforms for unimodular Lie groups, Duke Math. J. 26 (1959), 123–131. MR 104161
Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824