On a Classical Limit of <Emphasis Type="Italic">q</Emphasis>-Deformed Whittaker Functions |
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Authors: | Anton Gerasimov Dimitri Lebedev Sergey Oblezin |
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Institution: | 1.Institute for Theoretical and Experimental Physics,Moscow,Russia;2.School of Mathematics,Trinity College Dublin,Dublin 2,Ireland;3.Hamilton Mathematics Institute,Trinity College Dublin,Dublin 2,Ireland |
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Abstract: | Previously, we derive a representation of q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function as a sum over Gelfand–Zetlin patterns. This representation provides an analog of the Shintani–Casselman–Shalika formula for q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker functions. In this note, we provide a derivation of the Givental integral representation of the classical \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function as a limit q → 1 of the sum over the Gelfand–Zetlin patterns representation of the q-deformed \({\mathfrak{gl}_{\ell+1}}\) -Whittaker function. Thus, Givental representation provides an analog the Shintani–Casselman–Shalika formula for classical Whittaker functions. |
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