On a classical limit of \(q\)-deformed Whittaker functions
The note under review forms a part of a programme by the authors to develop a unified geometric understanding of archimedean and non-archimedean arithmetic geometry. In previous papers, the authors defined and studied \(q\)-deformed Whittaker functions, whose role is to interpolate non-Archimedean and archimedean Whittaker functions in the \(q\to 0\) limit and in the \(q\to 1\) limit correspondingly. The Shintani-Casselman-Shalika (SCS) formula expresses non-archimedean spherical \(p\)-adic Whittaker functions on reductive algebraic groups as characters of finite-dimensional representations of the Langlands dual group. In previous work, the authors expressed class one \(q\)-deformed \(\mathfrak{gl}_{\ell+1}\)-Whittaker functions as sums over sets of Gelfand-Zetlin tableaux, leading to a \(q\)-analogue of the SCS formula for class one \(q\)-deformed \(\mathfrak{gl}_{\ell+1}\)-Whittaker functions. The note under review explicitly computes the \(q\to 1\) limit of class one \(q\)-deformed \(\mathfrak{gl}_{\ell+1}\)-Whittaker functions, showing that these are indeed archimedean Whittaker functions (Theorem 3.1). This proves that the \(q\to 1\) limit of the sum over Gelfand-Zetlin patterns gives rise to the Givental integral representation of class one \(\mathfrak{gl}_{\ell+1}\)-Whittaker functions, providing an archimedean analogue of the SCS formula as suggested in Remark 2.1 in [Commun. Math. Phys. 294, No. 1, 97--119 (2010; Zbl 1222.17015)]. The generalization of Theorem 3.1 from \(\mathfrak{gl}_{\ell+1}\) to all classical series of Lie algebras, suggested in [Russ. Math. Surv. 67, No. 1, 1--92 (2012); translation from Usp. Mat. Nauk 67, No. 1, 3--96 (2012; Zbl 1267.17007)], is deferred to a future paper.
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