Kac-Moody Lie algebra and Hilbert modular forms

已知通过将SL_2(Z)的模形式与多项式不定二次形式的theta级数的核函数积分，可以得到格拉斯曼流形上具有奇点的自同构形式。 R.E.Borcherds 就该主题发表了一系列论文。设A为theta级数变量实部对应的非奇异积分对称矩阵，A_为虚部对应的正定实对称矩阵。矩阵A_满足条件A_ A^<-1>A_ =A，整个A_构成格拉斯曼流形。在 A 的某些条件下，自守形式的奇点决定了格拉斯曼方程上的 Weyl 室。它遵循一些 Kac-Moody 李代数的自同构形式和 Weyl 群及其分母函数之间的关系。设 K 为全实代数数域，设 O_K 为整数环。在本研究中，我们尝试将上述所有论证扩展到希尔伯特模群 SL_2(O_K) 的情况。我们给出了系数为 K 的对称矩阵 A 的具有多项式的 θ 级数的反演公式和变换公式。进一步，在 A 具有各向异性向量的情况下，我们将 θ 级数扩展到涉及低次二次形式的 θ 级数的级数。这个结果对应了Borcherds的论文“Automorphic forms with oddities oh Grassmanns”的定理5.2，这是在Grassmanns上获得自同构的关键。这个结果可能有助于研究与 Weyl 室相关的自同构形式，Weyl 室对 K 有理数。

Its is known that by integrating modular forms for SL_2(Z) with the kernel function which is theta series of indefinite quadratic form with polynomial, one obtain automorphic forms with singularities on Grassmann manifolds. There is a series of papers by R.E.Borcherds on this topic. Let A be the nonsingular integral symmetric matrix corresponding to the real part of variable of theta series, and let A_+ be the positive definite real symmetric matrix corresponding to the imaginary part. The matrix A_+ salifies the condition A_+A^<-1>A_+=A, and the whole of A_+ form the Grassmann manifold. Under some conditions of A, the singularities of the automorphic form determines Weyl chamber on the Grassmannians. It follows the relation between the automorphic forms and the Weyl groups of some Kac-Moody Lie algebras, and their denominator functions.Let K be a totally real algebraic number field, and let O_K be the ring of integers. In this research, we try to extend the all of the above argument to the case of Hilbert modular group SL_2(O_K). We show the inversion formula and transformation formulas for theta series with polynomial, of symmetric matrix A with coefficients in K. Further in the case that A has an anisotropic vector, we extend the theta series to the series involving theta series of quadratic forms of lower degree. This result is corresponding to Theorem 5.2 of Borcherds' paper "Automorphic forms with singularities oh Grassmanns", which is the key to obtaining automorphic form on Grassmanns. This result may be useful to study automorphic forms associated with Weyl chamber which is rational over K.