There are two different approaches to constructing Whittaker functions of metaplectic groups over non-archimedean local fields. One approach, due to Chinta and Offen for the general linear group and to McNamara in general, represents the spherical Whittaker function in terms of a sum over a Weyl group. The second approach, by Brubaker, Bump and Friedberg and separately by McNamara, expresses it as a sum over a highest weight crystal.;This work builds a direct, combinatorial connection between the two approaches. This is done by exploring both in terms of Demazure and Demazure-Lusztig operators associated to the Weyl group of an irreducible root system. The relevance of Demazure and Demazure-Lusztig operators is indicated by results in the non-metaplectic setting: the Demazure character formula, Tokuyama's theorem and the work of Brubaker, Bump and Licata in describing Iwahori-Whittaker functions.;The first set of results is joint work with Gautam Chinta and Paul E. Gunnells. We define metaplectic Demazure and Demazure-Lusztig operators for a root system of any type. We prove that they satisfy the same Braid relations and quadratic relations as their nonmetaplectic analogues. Then we prove two formulas for the long word in the Weyl group. One is a metaplectic generalization of Demazure's character formula, and the other connects the same expression to Demazure-Lusztig operators. Comparing the two results to McNamara's construction of metaplectic Whittaker functions results in a formula for the Whittaker functions in the spirit of the Demazure character formula.;The second set of results relates to Tokuyama's theorem about the crystal description of type A characters. We prove a metaplectic generalization of this theorem. This establishes a combinatorial link between the two approaches to constructing Whittaker functions for metaplectic covers of any degree. The metaplectic version of Tokuyama's theorem is proved as a special case of a stronger result: a crystal description of polynomials produced by sums of Demazure-Lusztig operators acting on a monomial. These results make use of the Demazure and Demazure-Lusztig formulas above, and the branching structure of highest weight crystals of type A. The polynomials produced by sums of Demazure-Lusztig operators acting on a monomial are related to Iwahori fixed Whittaker functions in the nonmetaplectic setting.
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机译：有两种不同的方法来构造非档案式本地域上的元压电群的Whittaker函数。由于Chinta和Offen适用于一般线性基团，而通常适用于McNamara，一种方法表示球形的Whittaker函数，表示为Weyl基团的总和。第二种方法是Brubaker，Bump和Friedberg以及McNamara分别将其表示为重量最大的晶体上的总和。这项工作在两种方法之间建立了直接的组合联系。这是通过探索与不可约根系统的Weyl群相关的Demazure和Demazure-Lusztig算子来完成的。 Demazure和Demazure-Lusztig算子的相关性在非拟定论环境中的结果表明：Demazure字符公式，Tokuyama定理以及Brubaker，Bump和Licata在描述Iwahori-Whittaker函数中的工作。与Gautam Chinta和Paul E. Gunnells合作。我们为任何类型的根系统定义元整数Demazure和Demazure-Lusztig运算符。我们证明，它们满足与它们的非触变类似物相同的编织关系和二次关系。然后，我们证明了Weyl组中长词的两个公式。一种是Demazure的字符公式的全元广义，另一种将相同的表达式连接到Demazure-Lusztig运算符。将这两个结果与McNamara的元压电Whittaker函数的构造相比较，得出了遵循Demazure字符公式的精神的Whittaker函数公式。第二组结果与德山关于A型字符的晶体描述定理有关。我们证明了该定理的一个元广义。这在构造任何程度的全辛覆盖层的Whittaker函数的两种方法之间建立了组合链接。证明德山定理的偏辛形式是获得更强结果的特例：对作用于多项式的Demazure-Lusztig算子的和产生的多项式的晶体描述。这些结果使用了上面的Demazure和Demazure-Lusztig公式，以及A型重量最大的晶体的分支结构。由Demazure-Lusztig算子的和作用于多项式上的多项式与Iwahori固定的Whittaker函数有关设置。
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