arXiv math/06022292006-02-10math.AGmath.ATnlin.SI

Toda格的奇异结构与某些紧李群的上同调

Singular structure of Toda lattices and cohomology of certain compact Lie groups

L. Casian, Y. Kodama

备注：: 23 pages, 12 figures, To appear in the proceedings "Topics in Integrable Systems, Special Functions, Orthogonal Polynomials and Random Matrices: Special Volume, Journal of Computational and Applied Mathematics"

摘要：

研究了实分裂半单李代数$\mathfrak g$的Toda格的奇点（爆破）.证明了Toda格沿轨道的爆破点总数由Chevalley群$K({\mathbb F}_q)$的点数给出，该Chevalley群$K({\mathbb F}_q)$与有限域${\mathbb F}_q$上具有$\check{\mathfrak g}={\rm Lie}(\check G)$的群$\check G$的极大紧子群$K$相关。这里$\check{\mathfrak g}$是${\mathfrak g}$的朗兰兹对偶。Toda格的爆破由$τ$ -函数的零点集给出。例如，A型Toda晶格的爆破由与矩形Young图相关的Schur多项式的零点决定。这些Schur多项式是幂零Toda格的$τ$ -函数。然后，我们猜想，爆破的数量也给出了这些舒尔多项式的实根的数量为一个特定的变量。我们还讨论了与仿射Kac-Moody代数相关的旗流形的实上同调的周期Toda格的情况。

英文摘要：

We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra $\mathfrak g$. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group $K({\mathbb F}_q)$ related to the maximal compact subgroup $K$ of the group $\check G$ with $\check{\mathfrak g}={\rm Lie}(\check G)$ over the finite field ${\mathbb F}_q$. Here $\check{\mathfrak g}$ is the Langlands dual of ${\mathfrak g}$. The blow-ups of the Toda lattice are given by the zero set of the $τ$-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the $τ$-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra.

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摘要：

英文摘要：

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