...This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated...

... a natural box-circle duality: a bijection between the bᵢ and the lᵢ in which bᵢ is circled if and only if the corresponding lᵢ is boxed. It also considers a striking property of the crystal graph and proceeds by obtaining two BZL patterns...

...This chapter translates Statements A and B into Statements A′ and B′ in the language of crystal bases, and explains in this language how Statement B′ implies Statement A′. It first introduces the relevant definition, which is provisional since it assumes that we can give an appropriate definition...

...This chapter describes the properties of Kashiwara's crystal and its role in unipotent p-adic integrations related to Whittaker functions. In many cases, integrations of representation theoretic import over the maximal unipotent subgroup of a p-adic group can...

... and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals...

...This chapter shows that Weyl group multiple Dirichlet series are expected to be Whittaker coefficients of metaplectic Eisenstein series. The fact that Whittaker coefficients of Eisenstein series reduce to the crystal description that was given in Chapter 2 is proved for Type A. On the adele group...

...This chapter recalls the use of the Schützenberger involution on Gelfand-Tsetlin patterns to prove that Statement B implies Statement A. These statements will be discussed two more times in the later chapters of the book. Chapter 18 reinterprets both Statements A and B in terms of crystals...