Symmetry properties of the zero sets of nil-theta functions

Author:
Sharon Goodman

Journal:
Trans. Amer. Math. Soc. **266** (1981), 441-460

MSC:
Primary 22E25; Secondary 14K25, 33A75

DOI:
https://doi.org/10.1090/S0002-9947-1981-0617544-2

MathSciNet review:
617544

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Abstract: Let denote the three dimensional Heisenberg group, and let be a discrete two-generator 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 analytic-geometric interpretation of the index. Every space in contains a unique (up to constant multiple) special function, called a nil-theta function, that arises as a solution of a certain differential operator on . These nil-theta 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 nil-theta functions lying in these spaces. We define the index of a nil-theta 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 nil-theta function lying in that space. We have thus an analytic-geometric characterization of the index.

We then use these results to give a complete description of the zero sets of those nil-theta functions of a fixed index. We also investigate the behavior of the index under the multiplication of nil-theta functions; i.e. we discuss how the index of the nil-theta function relates to the indices of the nil-theta 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**, https://doi.org/10.1090/S0002-9904-1973-13134-9

Louis Auslander,*An exposition of the structure of solvmanifolds. II. 𝐺-induced flows*, Bull. Amer. Math. Soc.**79**(1973), no. 2, 262–285. MR**0486308**, https://doi.org/10.1090/S0002-9904-1973-13139-8**[2]**Louis Auslander,*Lecture notes on nil-theta functions*, American Mathematical Society, Providence, R.I., 1977. Regional Conference Series in Mathematics, No. 34. MR**0466409****[3]**L. Auslander and J. Brezin,*Translation-invariant subspaces in 𝐿² of a compact nilmanifold. I*, Invent. Math.**20**(1973), 1–14. MR**0322100**, https://doi.org/10.1007/BF01405260**[4]**Louis Auslander and Richard Tolimieri,*Abelian harmonic analysis, theta functions and function algebras on a nilmanifold*, Lecture Notes in Mathematics, Vol. 436, Springer-Verlag, Berlin-New York, 1975. MR**0414785****[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**, https://doi.org/10.1090/S0002-9947-1978-0506619-4**[6]**S. Goodman, Thesis, City University of New York, 1978.**[7]**Serge Lang,*Introduction to algebraic and abelian functions*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1972. MR**0327780****[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 & Wilkins Co., Baltimore, Md., 1973. MR**0349993****[10]**H. P. F. Swinnerton-Dyer,*Analytic theory of abelian varieties*, Cambridge University Press, London-New York, 1974. London Mathematical Society Lecture Note Series, No. 14. MR**0366934****[11]**R. Tolimieri,*Heisenberg manifolds and theta functions*, Trans. Amer. Math. Soc.**239**(1978), 293–319. MR**487050**, https://doi.org/10.1090/S0002-9947-1978-0487050-7

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DOI:
https://doi.org/10.1090/S0002-9947-1981-0617544-2

Article copyright:
© Copyright 1981
American Mathematical Society