Symmetry properties of the zero sets of niltheta functions
Author:
Sharon Goodman
Journal:
Trans. Amer. Math. Soc. 266 (1981), 441460
MSC:
Primary 22E25; Secondary 14K25, 33A75
MathSciNet review:
617544
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Abstract: Let denote the three dimensional Heisenberg group, and let be a discrete twogenerator subgroup of such that is compact. Then we may decompose into primary summands with respect to the right regular representation of on as follows: . It can be shown that for is a multiplicity space for the representation of multiplicity . The distinguished subspace theory of . Auslander and . Brezin singles out a finite number of the decompositions of , which are in some ways nicer than the others. They define algebraically an integer valued function, called the index, on the set of irreducible closed invariant subspaces of such that the distinguished subspaces have index one. In this paper, we give an analyticgeometric interpretation of the index. Every space in contains a unique (up to constant multiple) special function, called a niltheta function, that arises as a solution of a certain differential operator on . These niltheta functions have been shown to be closely related to the classical theta functions. Since the classical theta functions are determined (up to constant multiple) by their zero sets, it is natural to attempt to classify the spaces in using various properties of the zero sets of the niltheta functions lying in these spaces. We define the index of a niltheta function in using the symmetry properties of its zero set. Our main theorem asserts that the algebraic index of a space in equals the index of the unique niltheta function lying in that space. We have thus an analyticgeometric characterization of the index. We then use these results to give a complete description of the zero sets of those niltheta functions of a fixed index. We also investigate the behavior of the index under the multiplication of niltheta functions; i.e. we discuss how the index of the niltheta function relates to the indices of the niltheta functions and .
 [1]
Louis
Auslander, An exposition of the structure of
solvmanifolds. I. Algebraic theory, Bull. Amer.
Math. Soc. 79 (1973), no. 2, 227–261. MR 0486307
(58 #6066a), http://dx.doi.org/10.1090/S000299041973131349
 [2]
Louis
Auslander, Lecture notes on niltheta functions, American
Mathematical Society, Providence, R.I., 1977. Regional Conference Series in
Mathematics, No. 34. MR 0466409
(57 #6289)
 [3]
L.
Auslander and J.
Brezin, Translationinvariant subspaces in 𝐿² of a
compact nilmanifold. I, Invent. Math. 20 (1973),
1–14. MR
0322100 (48 #464)
 [4]
Louis
Auslander and Richard
Tolimieri, Abelian harmonic analysis, theta functions and function
algebras on a nilmanifold, Lecture Notes in Mathematics, Vol. 436,
SpringerVerlag, BerlinNew York, 1975. MR 0414785
(54 #2877)
 [5]
L.
Auslander and R.
Tolimieri, Algebraic structures for
⨁∑_{𝑛≥1}𝐿²(𝑍/𝑛)\
compatible with the finite Fourier transform, Trans. Amer. Math. Soc. 244 (1978), 263–272. MR 506619
(80b:22012), http://dx.doi.org/10.1090/S00029947197805066194
 [6]
S. Goodman, Thesis, City University of New York, 1978.
 [7]
Serge
Lang, Introduction to algebraic and abelian functions,
AddisonWesley Publishing Co., Inc., Reading, Mass., 1972. MR 0327780
(48 #6122)
 [8]
L. Pukansky, Leçons sur les représentations des groupes, Monographies de la Sociéte Mathématique de France, No. 2, Dunod, Paris, 1967.
 [9]
Harry
E. Rauch and Aaron
Lebowitz, Elliptic functions, theta functions, and Riemann
surfaces, The Williams\thinspace&\thinspace Wilkins Co.,
Baltimore, Md., 1973. MR 0349993
(50 #2486)
 [10]
H.
P. F. SwinnertonDyer, Analytic theory of abelian varieties,
Cambridge University Press, LondonNew York, 1974. London Mathematical
Society Lecture Note Series, No. 14. MR 0366934
(51 #3180)
 [11]
R.
Tolimieri, Heisenberg manifolds and theta
functions, Trans. Amer. Math. Soc. 239 (1978), 293–319. MR 487050
(81d:22012), http://dx.doi.org/10.1090/S00029947197804870507
 [1]
 L. Auslander, An exposition of the structure of solvmanifolds. I, Bull. Amer. Math. Soc. 79 (1973), 227261. MR 0486307 (58:6066a)
 [2]
 , Lecture notes on niltheta functions, Conf. Board Math. Sci. Regional Conf. Series, no. 34, Amer. Math. Soc., Providence, R.I., 1977. MR 0466409 (57:6289)
 [3]
 L. Auslander and J. Brezin, Translation invariant subspaces in of a compact nilmanifold. I, Invent. Math. 20 (1973). MR 0322100 (48:464)
 [4]
 L. Auslander and R. Tolimieri, with assistance from H. Rauch, Abelian harmonic analysis, theta functions, and function algebras on a nilmanifold, Lecture Notes in Math., vol. 436, SpringerVerlag, Berlin and New York, 1975. MR 0414785 (54:2877)
 [5]
 , Algebraic structures for compatible with the finite Fourier transform, Trans. Amer. Math. Soc. 244 (1978), 263272. MR 506619 (80b:22012)
 [6]
 S. Goodman, Thesis, City University of New York, 1978.
 [7]
 S. Lang, Introduction to algebraic and abelian functions, AddisonWesley, Reading, Mass., 1972. MR 0327780 (48:6122)
 [8]
 L. Pukansky, Leçons sur les représentations des groupes, Monographies de la Sociéte Mathématique de France, No. 2, Dunod, Paris, 1967.
 [9]
 H. Rauch and A. Lebowitz, Elliptic functions, theta functions, and Riemann surfaces, Williams and Wilkins, Baltimore, Md., 1973. MR 0349993 (50:2486)
 [10]
 H. P. F. SwinnertonDyer, Analytic theory of abelian varieties, London Math. Soc. Lecture Note Series, no. 14, Cambridge Univ. Press, New York, 1974. MR 0366934 (51:3180)
 [11]
 R. Tolimieri, Heisenberg manifolds and theta functions, Trans. Amer. Math. Soc. 239 (1978), 293319. MR 487050 (81d:22012)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198106175442
PII:
S 00029947(1981)06175442
Article copyright:
© Copyright 1981
American Mathematical Society
