On \(q\)-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker function. I
In the paper under review the authors study \(q\)-deformed \(\mathfrak{gl}_{l+1}\)-Whittaker functions. These functions can be identified with common eigenvalues of a set of commuting \(q\)-deformed Toda chain Hamiltonians. The main result of the paper gives a new expression for the \(q\)-deformed \(\mathfrak{gl}_{l+1}\)-Whittaker functions. It follows that a \(q\)-deformed \(\mathfrak{gl}_{l+1}\)-Whittaker function can be represented as a character of \(\mathbb{C}^*\times \text{GL}_{l+1}\). Taking the limit \(q\rightarrow 1\) leads to a similar representation of the corresponding classical \(\mathfrak{gl}_{l+1}\)-Whittaker function. In the limit \(q\rightarrow 0\) the constructed \(q\)-deformed \(\mathfrak{gl}_{l+1}\)-Whittaker function reduces to a \(p\)-adic Whittaker function. This means that the representation of a \(q\)-deformed \(\mathfrak{gl}_{l+1}\)-Whittaker function can be considered as a \(q\)-version of the Shintani-Casselman-Shalika representation of \(\text{GL}_{l+1}\) (expressed in the Gelfand-Zetlin basis). For Part II, see Commun. Math. Phys. 294, No. 1, 121--143 (2010; Zbl 1222.17016).
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