User:Dapengzhang0/sandbox

Definition of nil-theta functions
Let $$\mathfrak{n}$$ be the complexified Lie algebra of the Heisenberg group $$N$$. A basis of $$\mathfrak{n}$$ is given by the left-invariant vector fields $$X, Y, T$$ on $$N$$:
 * $$X(x,y,t) = {\partial \over \partial x},$$
 * $$Y(x,y,t) = {\partial \over \partial y} + x {\partial \over \partial t},$$
 * $$T(x,y,t) = {\partial \over \partial t}.$$

These vector fields are well-defined on the Heisenberg manifold $$\Gamma \backslash N$$.

Let $$V_{-i} = X-i Y$$. For each $$n>0$$, the vector field $$V_{-i}$$ on the Heisenberg manifold can be thought of as a differential operator on $$C^{\infty} (\Gamma \backslash N) \cap H_{n,m}$$ with the kernel generated by $$\boldsymbol{e}_{n,m}$$.

We call
 * $$\ker(V_{-i}: C^{\infty} (\Gamma \backslash N) \cap H_n \to H_n) = \left\{ \begin{array}{lr} K_n, & n>0 \\ \mathbb{C}, & n=0 \end{array} \right. $$

the space of nil-theta functions of degree $$n$$.

Algebra structure of nil-theta functions
The nil-theta functions with pointwise multiplication on $$\Gamma \backslash N$$ form a graded algebra $$\oplus_{n\ge 0} K_n$$ (here $$K_0 = \mathbb{C}$$).

Auslander and Tolimieri showed that this graded algebra is isomorphic to
 * $$\mathbb{C}[x_1, x_2^2, x_3^3]/(x_3^6 + x_1^4 x_2^2 + x_2^6)$$,

and that the finite Fourier transform is an automorphism of the graded algebra.

Relation to Jacobi theta functions
Let $$\vartheta(z; \tau) = \sum_{l =-\infty}^\infty \exp (\pi i l^2 \tau + 2 \pi i l z) $$ be the Jacobi theta function. Then
 * $$\vartheta(n(x+iy); ni) = (2n)^{1/4} e^{\pi n y^2} \boldsymbol{e}_{n,0}(\Gamma\langle y, x, 0 \rangle)$$.

Higher order theta functions with characteristics
An entire function $$f$$ on $$\mathbb{C}$$ is called a theta function of order $$n$$, period $$\tau$$ ($$\mathrm{Im}(\tau)>0$$) and characteristic $$[^a_b]$$ if it satisfies the following equations: The space of theta functions of order $$n$$, period $$\tau$$ and characteristic $$[^a_b]$$ is denoted by $$\Theta_n[^a_b](\tau, A)$$.
 * 1) $$f(z+1) = \exp(\pi i a ) f(z)$$,
 * 2) $$f(z+\tau) =\exp(\pi i b) \exp(-\pi i n (2z +\tau)) f(z)$$.
 * $$\dim \Theta_n[^a_b](\tau, A) =n$$.

A basis of $$\Theta_n[^0_0](i, A)$$ is
 * $$\theta_{n,m}(z) = \sum_{l\in \mathbb{Z}} \exp [ -\pi n (l+{m \over n} )^2 + 2 \pi i (l n+ m) z ) ]$$.

These higher order theta functions are related to the nil-theta functions by
 * $$\theta_{n,m}(x+ iy) = (2n)^{1/4} e^{\pi n y^2} \boldsymbol{e}_{n,m} (\Gamma \langle y, x, 0 \rangle )$$