The Gelfand-Tsetlin basis for infinite-dimensional representations of $g{l}_n(\mathbb{C})$

Abstract

We consider the problem of determination of the Gelfand-Tsetlin basis for unitary principal series representations of the Lie algebra $g{l}_n(\mathbb{C})$. The Gelfand-Tsetlin basis for an infinite-dimensional representation can be defined as the basis of common eigenfunctions of corner quantum minors of the corresponding L-operator. The construction is based on the induction with respect to the rank of the algebra: an element of the basis for $g{l}_n(\mathbb{C})$ is expressed in terms of a Mellin-Barnes type integral of an element of the basis for $g{l}_{n-1}(\mathbb{C})$. The integration variables are the parameters (in other words, the quantum numbers) setting the eigenfunction. Explicit results are obtained for ranks $3$ and $4$, and the orthogonality of constructed sets of basis elements is demonstrated. For $g{l}_3(\mathbb{C})$ the kernel of the integral is expressed in terms of gamma-functions of the parameters of eigenfunctions, and in the case of $g{l}_4(\mathbb{C})$ -- in terms of a hypergeometric function of the complex field at unity. The formulas presented for an arbitrary rank make it possible to obtain the system of finite-difference equations for the kernel. They include expressions for the quantum minors of $g{l}_n(\mathbb{C})$ L-operator via the minors of $g{l}_{n-1}(\mathbb{C})$ L-operator for the principal series representations, as well as formulas for action of some non-corner minors on the eigenfunctions of corner ones. The latter hold for any representation of $g{l}_n(\mathbb{C})$ (not only principal series) in which the corner minors of the L-operator can be diagonalized.

arXiv comment39 pages; v2: corrected the error in the numerical factor in formula (4.41); v3: minor corrections, references and comments added

References in corpus (23)

• Integrability of Conformal Fishnet Theory
• Quantum Spectral Curve and Structure Constants in N=4 SYM: Cusps in the Ladder Limit
• Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism
• Conformal algebra: R-matrix and star-triangle relation
• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Exactly solvable magnet of conformal spins in four dimensions
• Eigenfunctions of $GL(N,\RR)$ Toda chain: The Mellin-Barnes representation
• Correlation Functions in The Itzykson-Zuber Model
• Separation of variables and scalar products at any rank
• Separated variables and wave functions for rational gl(N) spin chains in the companion twist frame
• Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow
• Iterative construction of eigenfunctions of the monodromy matrix for SL(2,C) magnet
• On Complex Gamma-Function Integrals
• Barnes-Ismagilov integrals and hypergeometric functions of the complex field
• Separation of variables for the quantum SL(3,C) spin magnet: eigenfunctions of Sklyanin B-operator
• $\mathrm{SL}(2,\mathbb{C})$ Gustafson Integrals
• Gustafson integrals for $SL(2,\mathbb{C})$ spin magnet
• From Baxter Q-Operators to Local Charges
• An analog of the Dougall formula and of the de Branges--Wilson integral
• Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb{R})$ Spin Chains
• On representation theory of partition algebras for complex reflection groups
• A unified approach to construction of Gelfand-Tsetlin-Zhelobenko base vectors for series $A$, $B$, $C$, $D$
• Quantum Toda Lattice: a Challenge for Representation Theory