Hecke indefinite modular forms

introduction

Let $\mathfrak{g} = A_1^{(1)}$. Fix a dominant integral weight $\Lambda$ of $\mathfrak{g}$ of level $m \geq 1$, and let $\lambda$ be a maximal dominant weight of $L(\Lambda)$.

Let $N$ denote the quadratic form defined on $\mathbb{R}^2$ by: $$ N(x,y):= 2 (m+2)x^2 - 2m y^2 \;\;\;\; (x,y \in \mathbb{R})$$ and let $(\cdot|\cdot)$ denote the corresponding symmetric bilinear form. Let $M:=\mathbb{Z}^2$ and let $M^*$ denote the lattice dual to $M$ with respect to this form.

Let $O(N)$ denote the group of invertible linear operators on $\mathbb{R}^2$ preserving $N$, and $SO_0(N)$ be the connected component of $O(N)$ containing the identity. We then have the groups $G := \{g \in SO_0(N): g M =M \}$ and $G_0 := \{g \in G: g \text{ fixes } M^*/M \text{ pointwise}\}$. The set $U^+:=\{(x,y) \in \mathbb{R}^2: N(x,y) >0\}$ is preserved under the action of $O(N)$ on $\mathbb{R}^2$. We let $A:=\frac{\langle{\Lambda + \rho,\check{\alpha}_1}\rangle}{2(m+2)}$ and $B:= \frac{\langle{\lambda, \check{\alpha}_1}\rangle}{2m}$ where $\check{\alpha}_1$ is the coroot corresponding to the underlying finite type diagram ($\mathfrak{sl}_2$ in this case), and $\rho$ is the Weyl vector. Then, $(A,B) \in M^*$, and we set $L:= (A,B) + M$.

The Hecke indefinite modular form is the following sum: $$\theta_L(\tau) := \sum_{\substack{(x,y) \in L \cap U^+ \\ (x,y) \text{ mod } G_0}} \mathrm{sign}(x,y) \, e^{\pi i \tau N(x,y)},$$ where $\mathrm{sign}(x,y) = 1$ for $x \geq 0$ and $-1$ for $x<0$. This is an absolutely convergent sum for $\tau$ in the upper half plane $\mathbb{H}$, and defines a cusp form of weight 1.

examples

Hecke

$$ \prod_{n=1}^{\infty}(1-q^n)^2=\sum_{\substack{m,n=-\infty \\ n\geq 2|m|}}^{\infty}(-1)^{n+m}q^{(n^2-3m^2)/2+(n+m)/2} $$

Kac-Peterson

$$ \prod_{n=1}^{\infty}(1-q^n)(1-q^{2n})=\sum_{\substack{m,n=-\infty \\ n\geq 3|m|}}^{\infty}(-1)^{n+m}q^{(n^2-8m^2)/2+n/2} $$

string functions

thm (Kac-Peterson)

Let $\mathfrak{g} = A_1^{(1)}$. Let $\Lambda$ be a dominant integral weight of $\mathfrak{g}$, and $\lambda$ be a maximal dominant weight of $L(\Lambda)$. Then $$c^{\Lambda}_{\lambda}(\tau) = \theta_L(\tau) \, \eta(\tau)^{\scriptstyle{s} -3}.$$ Here $\theta_L(\tau)$ is a Hecke indefinite modular form and $\eta(\tau)$ is the Dedekind eta function.

memo

see Appendix of Jimbo, Miwa and Okado, 1986

related items

articles

Westerholt-Raum, Martin. “H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Indefinite Theta Series.” arXiv:1207.5603 [math], July 24, 2012. http://arxiv.org/abs/1207.5603.
Sharma, Sachin S., and Sankaran Viswanath. ‘The $t$-Analogs of String Functions for $A_1^{(1)}$ and Hecke Indefinite Modular Forms’. arXiv:1302.6200 [math], 25 February 2013. http://arxiv.org/abs/1302.6200.
Polishchuk, Alexander. ‘A New Look at Hecke’s Indefinite Theta Series’. arXiv:math/0012005, 1 December 2000. http://arxiv.org/abs/math/0012005.
Hiramatsu, Toyokazu, Noburo Ishii, and Yoshio Mimura. ‘On Indefinite Modular Forms of Weight One’. Journal of the Mathematical Society of Japan 38, no. 1 (January 1986): 67–83. doi:10.2969/jmsj/03810067.
Jimbo, Michio, Tetsuji Miwa, and Masato Okado. ‘Solvable Lattice Models with Broken ZN Symmetry and Hecke’s Indefinite Modular Forms’. Nuclear Physics B 275, no. 3 (24 November 1986): 517–45. doi:10.1016/0550-3213(86)90611-5.
Jimbo, Michio, and Tetsuji Miwa. ‘A Solvable Lattice Model and Related Rogers-Ramanujan Type Identities’. Physica D: Nonlinear Phenomena 15, no. 3 (April 1985): 335–53. doi:10.1016/S0167-2789(85)80003-8.
Andrews, George E. "Hecke modular forms and the Kac-Peterson identities." Transactions of the American Mathematical Society (1984): 451-458.
Kac, V. G., and D. H. Peterson. ‘Affine Lie Algebras and Hecke Modular Forms’. Bulletin (New Series) of the American Mathematical Society 3, no. 3 (November 1980): 1057–61. http://projecteuclid.org/euclid.bams/1183547694
Über einen Zusammenhang zwischen elliptischen Modulfunktionen und indefiniten quadratischen Formen
- E. Hecke, Mathematische Werke, Vandenhoeck and Ruprecht, Góttingen, 1959, pp. 418-427