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