On \(q\)-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker function. I
type: Item
instance of: scholarly article
title: On \(q\)-deformed \({\mathfrak{gl}_{\ell+1}}\)-Whittaker function. I
zbMATH Open document ID: 1222.17015
published in: Communications in Mathematical Physics
publication date: 2011-02-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0803.0970
review text: 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).
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 33D80
zbMATH DE Number: 5852650
zbMATH Keywords: \(q\)-deformed Whittaker function
zbMATH Keywords: Toda chain
zbMATH Keywords: Gelfand-Zetlin basis
zbMATH Keywords: character
zbMATH Keywords: representation
author: Anton Gerasimov
author: Dimitri R. Lebedev
author: Sergey Oblezin
MaRDI profile type: Publication
OpenAlex ID: W2146301993
arXiv ID: 0803.0970
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DOI: 10.1007/S00220-009-0917-Y
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description: scientific article