Related papers: On parabolic Whittaker functions II

We derive a Mellin-Barnes integral representation for solution to generalized (parabolic) quantum Toda lattice introduced in \cite{GLO}, which presumably describes the $(S^1\times U_N)$-equivariant Gromov-Witten invariants of Grassmann…

Let F be a p-adic field, and Gn one of the groups GL(n, F), GSO(2n-1, F), GSp(2n, F), or GSO(2(n - 1), F). Using the mirabolic subgroup or analogues of it, and related "derivative" functors, we give an asymptotic expansion of functions in…

Let $F$ be a $p$-adic field, and $G_n$ one of the groups $GL(n,F)$, $GSO(2n-1,F)$, $GSp(2n,F)$, or $GSO(2(n-1),F)$. Using the mirabolic subgroup or analogues of it, and related "derivative" functors, we give an asymptotic expansion of…

We give a characterization of a generalized Whittaker model of a degenerate principal series representation of $GL(n,\R)$ as the kernel of some differential operators. By this characterization, we investigate some examples on $GL(4,\R)$. We…

A representation of a specialization of a q-deformed class one lattice gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed. For…

In this paper, a construction to imitate the Bernstein-Zelevinsky derivative for automorphic representations on $GL_n(\mathbb{A})$ is introduced. We will later consider the induced representation \[I(\tau_1,\tau_2;\underline{s}) =…

We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns' birational sequences and explain how they give rise to points in $\text{trop}(\text{Gr}(k,\mathbb C^n))$, Speyer-Sturmfels' tropical Grassmannian.…

We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of $U((\mathfrak{gl}(N))$ in terms of first order differential…

Let $\mathfrak{o}_l$ be a finite principal ideal local ring of length $l$. The degenerate Whittaker space associated with a representation of ${\rm GL}_{2n}(\mathfrak{o}_l)$ is a representation of ${\rm GL}_n(\mathfrak{o}_l)$. For strongly…

In this paper we investigate the tau-functions for the stationary sector of Gromov-Witten theory of the complex projective line and its version, relative to one point. In particular, we construct the integral representation for the points…

In this note we discuss an integral representation for the vertex function of the cotangent bundle over the Grassmannian, $X=T^{*} Gr(k,n)$. This integral representation can be used to compute the $\hbar\to \infty$ limit of the vertex…

The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker…

Inspired by recent formul\ae\ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov-Witten invariants of $\mathbb{P}^1$ as an isomonodromic tau function associated with a difference equation. As a…

Let F be a p-adic field and let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over F. We show here that if n is odd then all the genuine…

We establish integral analogues of results of Bushnell and Henniart for spaces of Whittaker functions arising from the groups GL_n(F) for F a p-adic field. We apply the resulting theory to the existence of representations arising from the…

Generalized $Z_k$-graded Grassmann variables are used to label coherent states related to the nilpotent representation of the q-oscillator of Biedenharn and Macfarlane when the deformation parameter is a root of unity. These states are then…

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…

We provide some experimental results on the decomposition of the parabolic induction of $\pi\otimes\pi$ in the Grothendieck group where $\pi$ is an irreducible representation of $GL_n$ over a local non-archimedean field.

Let $\pi$ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $\pi = \pi_{\theta}$, corresponds to a regular character $\theta$ of $\mathbb{F}_{q^{kn}}^{*}$. We consider the twisted Jacquet…

Exploring the concept of the extended Galilei group $\mathcal{G}$, a representation for the symplectic quantum mechanics in the manifold of $\mathcal{G}$, written in the light-cone of a five-dimensional De Sitter space-time, is derived…