覆叠群的主列的局部系数和Gamma因子
Local coefficients and gamma factors for principal series of covering groups
摘要：
考虑了$p$ -adic域上约化群的$n$ -fold Brylinski-Deligne覆盖.由于这种覆盖的不可约真表示的惠特克泛函空间不是一维的，因此可以考虑由交织算子产生的局部系数矩阵，这是线性情况下局部系数的自然模拟。本文主要研究真主列，建立了局部系数矩阵的一些基本性质，包括算术不变量的研究。因此，我们证明了一种形式的Casselman-Shalika公式，它可以被看作是一个自然的模拟线性代数群。我们还深入研究了局部系数矩阵的行为，真正的主列的限制，从覆盖${\rm GL}_2$到${\rm SL}_2$。特别地，揭示了局部系数矩阵和伽玛因子或次伽玛因子之间的一些进一步的关系。
英文摘要：
We consider an $n$-fold Brylinski-Deligne cover of a reductive group over a $p$-adic field. Since the space of Whittaker functionals of an irreducible genuine representation of such a cover is not one-dimensional, one can consider a local coefficients matrix arising from an intertwining operator, which is the natural analogue of the local coefficients in the linear case. In this paper, we concentrate on genuine principal series and establish some fundamental properties of such a local coefficients matrix, including the investigation of its arithmetic invariants. As a consequence, we prove a form of the Casselman-Shalika formula which could be viewed as a natural analogue for linear algebraic groups. We also investigate in some depth the behaviour of the local coefficients matrix with respect to the restriction of genuine principal series from covers of ${\rm GL}_2$ to ${\rm SL}_2$. In particular, some further relations are unveiled between local coefficients matrices and gamma factors or metaplectic-gamma factors.
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