AI 中文总结
研究可对称化卡-穆迪根系的扭曲外尔群多重狄利克雷级数，用钦塔-冈内尔斯方法构建\(p\)部分，给出函数分解定理，证明其对有理函数域上的级数成立，还得到仿射\(\widetilde{A}_1\)情形下的额外函数方程及显式公式。
AI 中文摘要
我们研究与可对称化卡-穆迪根系相关的扭曲外尔群多重狄利克雷级数，利用钦塔-冈内尔斯方法构建其\(p\)部分。主要结果是关于在扭曲钦塔-冈内尔斯作用下不变函数的分解定理：在自然解析假设下，此类函数具有由扭曲参数确定的最高权模中的优势权索引的移位钦塔-冈内尔斯平均的唯一展开式。特别地，证明了该分解对有理函数域上的扭曲多重狄利克雷级数成立。还表明相关钦塔-冈内尔斯平均可解析延拓到复化蒂茨锥内部。在仿射\(\widetilde{A}_1\)情形下，证明了未扭曲平均和由基本权扭曲的平均的额外函数方程，得到了具有无平方因子扭曲参数的多重狄利克雷级数的显式公式，并表明其也满足额外函数方程。
英文摘要
We study twisted Weyl group multiple Dirichlet series attached to symmetrizable Kac-Moody root systems, using the Chinta-Gunnells method to construct their $p$-parts. Our main result is a decomposition theorem for functions invariant under the twisted Chinta-Gunnells action: under natural analytic hypotheses, such a function has a unique expansion in terms of shifted Chinta-Gunnells averages, indexed by the dominant weights in the highest weight module determined by the twisting parameter. In particular, we show that this decomposition holds for twisted multiple Dirichlet series over rational function fields. For finite root systems, these results were proved by Friedlander. We also show that the relevant Chinta-Gunnells averages admit analytic continuation to the interior of the complexified Tits cone. In the affine $\widetilde{A}_1$ case, we prove extra functional equations, not arising from the Weyl group, for the untwisted average and for averages twisted by fundamental weights. As a consequence, we obtain an explicit formula for the multiple Dirichlet series with square-free twisting parameters, and show that it also satisfies an extra functional equation.
Comments40 pages. Substantially revised version, with improved convergence results and proofs, a more precise treatment of q-Weil coefficients, and improved exposition
