[Paper Review] New Integral Representations of Whittaker Functions for Classical Lie Groups

A. Gerasimov, D. V. Lebedev+1 more|arXiv (Cornell University)|May 20, 2007

Advanced Algebra and Geometry7 references4 citations

TL;DR

This paper introduces new integral representations of Whittaker functions for classical Lie algebras $τ\mathfrak{sp}_{2\ell}$, $τ\mathfrak{so}_{2\ell}$, and $τ\mathfrak{so}_{2\ell+1}$, generalizing Givental's integral formula for $τ\mathfrak{gl}_{\ell+1}$. The construction preserves the recursive structure over rank $τ\ell$, with integral kernels described combinatorially via graphs, and establishes a direct correspondence between recursion operators and degenerate $τ\mathcal{Q}$-operators for affine Lie algebras $τ\widehat{\mathfrak{so}}_{2\ell}$, $τ\widehat{\mathfrak{so}}_{2\ell+1}$, and a twisted $τ\widehat{\mathfrak{gl}}_{2\ell}$, extending a known link from the $τ\mathfrak{gl}_{\ell+1}$ case.

ABSTRACT

We propose integral representations of the Whittaker functions for the classical Lie algebras sp(2l), so(2l) and so(2l+1). These integral representations generalize the integral representation of gl(l+1)-Whittaker functions first introduced by Givental. One of the salient features of the Givental representation is its recursive structure with respect to the rank of the Lie algebra gl(l+1). The proposed generalization of the Givental representation to the classical Lie algebras retains this property. It was shown elsewhere that the integral recursion operator for gl(l+1)-Whittaker function in the Givental representation coincides with a degeneration of the Baxter Q-operator for $\hat{gl(l+1)}$-Toda chains. We construct Q-operator for affine Lie algebras $\hat{so(2l)}$, $\hat{so(2l+1)}$ and a twisted form of $\hat{gl(2l)}$. We demonstrate that the relation between recursion integral operators of the generalized Givental representation and degenerate Q-operators remains valid for all classical Lie algebras.

Motivation & Objective

• To extend Givental's integral representation of $τ\mathfrak{gl}_{\ell+1}$-Whittaker functions to classical Lie algebras $τ\mathfrak{sp}_{2\ell}$, $τ\mathfrak{so}_{2\ell}$, and $τ\mathfrak{so}_{2\ell+1}$.
• To preserve the recursive structure over the rank $τ\ell$ observed in the $τ\mathfrak{gl}_{\ell+1}$ case.
• To provide a combinatorial description of the integral kernels via graphs, generalizing the torification of flag manifolds.
• To construct $τ\mathcal{Q}$-operators for affine Lie algebras $τ\widehat{\mathfrak{so}}_{2\ell}$, $τ\widehat{\mathfrak{so}}_{2\ell+1}$, and a twisted form of $τ\widehat{\mathfrak{gl}}_{2\ell}$.
• To demonstrate that the correspondence between recursion integral operators and degenerate $τ\mathcal{Q}$-operators holds universally across all classical Lie algebras.

Proposed method

• The construction uses a modified factorized representation of maximal unipotent subgroups in classical Lie groups, realized as upper-triangular matrices.
• The integral representations are derived from a parametrization of open parts of flag manifolds associated with $τ\mathfrak{sp}_{2\ell}$, $τ\mathfrak{so}_{2\ell}$, and $τ\mathfrak{so}_{2\ell+1}$.
• The recursive structure is encoded in differential operators $e_i^{(\ell)}$, $f_i^{(\ell)}$, and $h_i^{(\ell)}$ acting on variables $y_{k,j}$, with coefficients involving ratios of $y$-variables.
• The recursion operators are shown to coincide with degenerate $τ\mathcal{Q}$-operators for the corresponding affine Lie algebras, generalizing a known result for $τ\widehat{\mathfrak{gl}}_{\ell+1}$.
• The integral kernels for the recursion operators are non-trivial integrals, suggesting composition of elementary operators, with explicit expressions derived for zero eigenvalues.
• The framework connects quantum integrable systems with geometric structures on flag manifolds via $τ\mathcal{Q}$-operators and torification.

Experimental results

Research questions

• RQ1 Can Givental's integral representation of $τ\mathfrak{gl}_{\ell+1}$-Whittaker functions be generalized to classical Lie algebras $τ\mathfrak{sp}_{2\ell}$, $τ\mathfrak{so}_{2\ell}$, and $τ\mathfrak{so}_{2\ell+1}$?
• RQ2 Does the recursive structure of the Givental representation persist in this generalization, and how is it encoded in the integral kernels?
• RQ3 Is there a correspondence between the recursion integral operators and degenerate $τ\mathcal{Q}$-operators for the classical affine Lie algebras $τ\widehat{\mathfrak{so}}_{2\ell}$, $τ\widehat{\mathfrak{so}}_{2\ell+1}$, and a twisted $τ\widehat{\mathfrak{gl}}_{2\ell}$?
• RQ4 How do the integral kernels of the recursion operators differ from the $τ\mathfrak{gl}_{\ell+1}$ case, and what does this imply about their composition?
• RQ5 Can the combinatorial graph description of the integrand be extended from $τ\mathfrak{gl}_{\ell+1}$ to the classical Lie algebras, reflecting flat degenerations to toric Fano varieties?

Key findings

• The paper constructs new integral representations of Whittaker functions for $τ\mathfrak{sp}_{2\ell}$, $τ\mathfrak{so}_{2\ell}$, and $τ\mathfrak{so}_{2\ell+1}$, generalizing Givental's formula for $τ\mathfrak{gl}_{\ell+1}$.
• The integral representations retain a recursive structure over the rank $τ\ell$, with recursion operators defined via differential operators $e_i^{(\ell)}$, $f_i^{(\ell)}$, and $h_i^{(\ell)}$ acting on $y_{k,j}$ variables.
• The integral kernels of the recursion operators are non-trivial integrals, indicating they may be composed of elementary operators, unlike the simple kernel in the $τ\mathfrak{gl}_{\ell+1}$ case.
• The paper constructs $τ\mathcal{Q}$-operators for $τ\widehat{\mathfrak{so}}_{2\ell}$, $τ\widehat{\mathfrak{so}}_{2\ell+1}$, and a twisted form of $τ\widehat{\mathfrak{gl}}_{2\ell}$, extending the known link between $τ\mathcal{Q}$-operators and Toda chain systems.
• The correspondence between recursion integral operators and degenerate $τ\mathcal{Q}$-operators is shown to hold for all classical Lie algebras, generalizing a known result for $τ\widehat{\mathfrak{gl}}_{\ell+1}$.
• The construction provides a conceptual explanation for the Givental representation via $τ\mathcal{Q}$-operators, linking quantum integrable systems with geometric structures on flag manifolds through torification and flat degenerations.