On a classical limit of \(q\)-deformed Whittaker functions
type: Item
instance of: scholarly article
title: On a classical limit of \(q\)-deformed Whittaker functions
zbMATH Open document ID: 1260.17012
published in: Letters in Mathematical Physics
publication date: 2012-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.4567
review text: 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.
reviewed by: Daniel Moskovich
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 22E57
Mathematics Subject Classification ID: 33D80
zbMATH DE Number: 6088833
zbMATH Keywords: \(q\)-deformed Whittaker function
zbMATH Keywords: Toda chain
zbMATH Keywords: Givental integral representation
zbMATH Keywords: Gelfand-Zetlin patterns
zbMATH Keywords: Shintani-Casselman-Shalika formula
zbMATH Keywords: archimedean arithmetic geometry
author: Anton Gerasimov
author: Sergey Oblezin
author: Dimitri Lebedev
MaRDI profile type: Publication
OpenAlex ID: W3106028399
arXiv ID: 1101.4567
cites work: Q3926803
cites work: Q4952803
cites work: On a Gauss-Givental representation of quantum Toda chain wave function
cites work: On \(q\)-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker function. I
cites work: On \(q\)-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker function. II
cites work: On \(q\)-deformed \(\mathfrak{gl}_{\ell+1}\)-Whittaker function. III
cites work: Parabolic Whittaker functions and topological field theories. I.
cites work: From archimedean \(L\)-factors to topological field theories
cites work: New integral representations of Whittaker functions for classical Lie groups
cites work: Baxter operator and archimedean Hecke algebra
cites work: Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture
cites work: Relativistic Toda systems
cites work: On an explicit formula for class-$1$ “Whittaker functions” on $GL_n $ over $\beta $-adic fields
DOI: 10.1007/S11005-012-0545-X
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description: scientific article; zbMATH DE number 6088833