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D. R. Lebedev

Publications and source records attributed to D. R. Lebedev.

6 recordsLinked to original sources
arXiv2024

The Hecke-Baxter operators via Heisenberg group extensions

The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator was introduced as an element of the $O_{\ell+1}$-spherical Hecke algebra associated with the Gelfand pair $O_{\ell+1}\subset GL_{\ell+1}(\mathbb{R})$. It was specified by the property to act on an $O_{\ell+1}$-fixed vector in a $GL_{\ell+1}(\mathbb{R})$-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 $GL_{\ell+1}(\mathbb{R})\times GL_{\ell+1}(\mathbb{R})$ 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 $Sp_{2\ell+2}(\mathbb{R})\times Sp_{2\ell+2}(\mathbb{R})$ by a Heisenberg Lie group.

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arXiv2024

On equivalence of the Mellin-Barnes and the Givental integral representations of the Whittaker functions

We construct an integral transformation intertwining the Gelfand-Tsetlin and the (modified) Gauss-Givental realizations of principle series representations of gl(3). This provides a direct identification of the corresponding integral representations for the gl(3)-Whittaker function. The construction essentially uses integral identities due to Barnes and Gustafson thus providing a basis for their representation theory interpretation. The result of this paper might be useful for constructing the explicit analytic realization of the mirror symmetry map in the case of the flag manifold GL(3)/B.

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arXiv2022

On a matrix element representation of the GKZ hypergeometric functions

We develop a representation theory approach to the study of generalized hypergeometric functions of Gelfand, Kapranov and Zelevisnky (GKZ). We show that the GKZ hypergeometric functions may be identified with matrix elements of non-reductive Lie algebras $\mathfrak{L}_N$ of oscillator type. The Whittaker functions associated with principal series representations of $\mathfrak{gl}_{\ell+1}(\mathbb{R})$ being special cases of GKZ hypergeometric functions, thus admit along with a standard matrix element representations associated with reductive Lie algebra $\mathfrak{gl}_{\ell+1}(\mathbb{R})$, another matrix element representation in terms of $\mathfrak{L}_{\ell(\ell+1)}$.

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arXiv2020

On quantum $\mathfrak{osp}(1|2\ell)$-Toda chain

The orthosymplectic super Lie algebra $\mathfrak{osp}(1|\,2\ell)$ is the closest analog of standard Lie algebras in the world of super Lie algebras. We demonstrate that the corresponding $\mathfrak{osp}(1|\,2\ell)$-Toda chain turns out to be an instance of a $BC_\ell$-Toda chain. The underlying reason for this relation is discussed.

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arXiv2019

On normalizers of maximal tori in classical Lie groups

The normalizer $N_G(H_G)$ of a maximal torus $H_G$ in a semisimple complex Lie group $G$ does not in general allow a presentation as a semidirect product of $H_G$ and the corresponding Weyl group $W_G$. Meanwhile, splitting holds for classical groups corresponding to the root systems $A_\ell$, $B_\ell$, $D_\ell$. For the remaining classical groups corresponding to the root systems $C_\ell$ there still exists an embedding of the Tits extension of $W_G$ into normalizer $N_G(H_G)$. We provide explicit unified construction of the lifts of the Weyl groups into normalizers of maximal tori for classical Lie groups corresponding to the root systems $A_\ell$, $B_\ell$, $D_\ell$ using embeddings into general linear Lie groups. For symplectic series of classical Lie groups we provide an explanation of impossibility of embedding of the Weyl group into the symplectic group. The explicit formula for adjoint action of the lifts of the Weyl groups on $\mathfrak{g}={\rm Lie}(G)$ are given. Finally some examples of the groups closely associated with classical Lie groups are considered.

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