Kac-Moody Lie algebra and Hilbert modular forms

已知通过对SL_2(Z)的模形式与带多项式的不定二次形式的级数的核函数积分，可以得到在Grassmann流形上具有奇点的自同构形式。R.E.Borcherds发表了一系列关于这个主题的论文。设A为对应于级数变量实部的非奇异积分对称矩阵，设A_+为对应于虚部的正定实对称矩阵。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，自同构形式的奇异性决定了格拉斯曼算子上的Weyl室。讨论了一些Kac-Moody李代数的自同构形式与Weyl群的关系，以及它们的分母函数。设K为全实数代数域，设O_K为整数环。在本研究中，我们尝试将上述所有论证推广到Hilbert模群SL_2（O_K）的情况。我们给出了系数为k的对称矩阵A的多项式级数的反演公式和变换公式。进一步，在A具有各向异性矢量的情况下，我们将级数推广到包含较低次二次形式的级数。这一结果对应于Borcherds论文“带奇点的自同构形式oh Grassmanns”的定理5.2，该定理是获得Grassmanns上自同构形式的关键。这一结果对研究与K上有理的Weyl室相关的自同构形式有一定的指导意义。

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.