2024/12/16 by А. А. Герасимов, Gerasimov, A. A., D. R. Lebedev +3
Mathematics · Computer Science · #Advanced Topics in Algebra #advanced mathematical theories #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2412.11604
The GLℓ+1(ℝ) Hecke-Baxter operator was introduced as an element of the Oℓ+1-spherical Hecke algebra associated with the Gelfand pair Oℓ+1⊂ GLℓ+1(ℝ). It was specified by the property to act on an Oℓ+1-fixed vector in a GLℓ+1(ℝ)-principal series representation via multiplication by the local Archimedean L-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group GLℓ+1(ℝ)× GLℓ+1(ℝ) by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group Sp2ℓ+2(ℝ)× Sp2ℓ+2(ℝ) by a Heisenberg Lie group.