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We aim to introduce a new extension of Mittag-Leffler function via q-analogue and obtained their significant properties including integral representation, q-differentiation, q-Laplace transform, image formula under q-derivative operators.…
In this paper we develop a method of constructing Hilbert spaces and the representation of the formal algebra of quantum observables in deformation quantization which is an analog of the well-known GNS construction for complex…
After introducing the notion of uniform integrality of critical values of the Rankin-Selberg $L$-functions for $\mathrm{GL}_{n}\times \mathrm{GL}_{n-1}$, we study it when the base field is totally imaginary. For this purpose, we adopt…
For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra U_q(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Z_q(gl(n+1),gl(n)) is given in terms of the generators and their defining relations.…
We calculate the dg algebra of global functions on commuting stacks of complex reductive groups using tools from Betti Geometric Langlands. In particular, we prove that the ring of invariant functions on the commuting scheme is reduced. Our…
Gelfand-Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand-Tsetlin polytope $\mathrm{GT}(\lambda)$ is equal to the dimension of the corresponding irreducible representation of…
In this paper, certain generalized fractional derivative formulae are introduced involving the k-Mittag-Leffler function. Then their image formulae (using Beta transform, Laplace transform and Whittaker transform) are also established. The…
The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. We study the structure of the twisted homology and cohomology…
In this paper we construct a generalization of the classical Steinberg section for the quotient map of a semisimple group with respect to the conjugation action. We then give various applications of our construction including the…
We give explicit real-analytic functions whose zero sets characterize the images of the standard maps of wave-front singularities. Such functions are realizations of the main-analytic sets in the sense of Ishikawa-Koike-Shiota (1984). More…
In this note, we study irreducible unitary representations of special linear groups of lower ranks, in terms of the matrix models of Gelfand-Naimark and Gelfand-Graev. Review of existing literature is provided. We also add some new…
In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal…
We develop a $\mathtt{q}$-analogue of the theory of conjugation equivariant $\mathcal D$-modules on a complex reductive group $G$. In particular, we define quantum Hotta-Kashiwara modules and compute their endomorphism algebras. We use the…
We study degenerate Whittaker vectors in scalar type holomorphic discrete series representations of tube type Hermitian Lie groups and their analytic continuation. In four different realizations, the bounded domain picture, the tube domain…
For the discrete series representations of ${\rm GL}(n)$ over a non-archimedean local field $F$, we define a notion of functions similar to "zonal spherical functions" for unramified principal series. We prove the existence of such…
It is shown that Mellin transforms of p-adic Whittaker functions exist for generic characters. For good choices of vectors they are rational functions. For class one vectors they can be calculated explicitly. It turns out that they are…
Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a…
A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include…
We construct a vertex representation for the quantum toroidal algebra through the quantum general linear algebra. Using a new realization of the quantum general linear algebra we construct vertex operators for root vectors on the basic…
We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein's theory of analytic continuation of holonomic…