Related papers: Decompositions of Generalized Wavelet Representati…

It is shown that each linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitary…

A unitary finite dimensional quandle representation is decomposable into a direct sum of irreducible represenations. Not all quandle representations satisfy this property. We prove that a finite dimensional quandle represenation $\rho :Q…

$W$-representation realizes partition functions by an action of a cut-and-join-like operator on the vacuum state with a zero-mode background. We provide explicit formulas of this kind for $\beta$- and $q,t$-deformations of the simplest…

Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their…

Let $\bold G$ be a reductive algebraic group defined over $\Q$, and let $\Gamma$ be an arithmetic subgroup of $\bold G(\Q)$. Let $X$ be the symmetric space for $\bold G(\R)$, and assume $X$ is contractible. Then the cohomology (mod torsion)…

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ and $G$ be the corresponding simply connected algebraic group. Consider a nilpotent element $e\in \mathfrak{g}$, the corresponding element $\chi=(e, \bullet)$ in $\mathfrak{g}^*$,…

We obtain a decomposition formula of a representation of Sp(p,q) and SO^\ast(2n) unitarily induced from a derived functor module, which enables us to reduce the problem of irreducible decompositions to the study of derived functor modules.…

Let G be the group of points of a quasi-split reductive algebraic group over a local field F. It follows from the local Langlands conjectures that to every non-trivial additive character of F and every representation of the Langlands dual…

We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine…

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where…

The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra…

We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group ${\mathbb R {}}^*_+=\{r:r>0\}$ of…

We present a unitary approach to the construction of representations and intertwining operators. We apply it to the $C^*$-algebras, groups, Gabor type unitary systems and wavelets. We give an application of our method to the theory of…

Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special…

By analogy with the real and complex affine groups, whose unitary irreducible representations are used to define the one and two-dimensional continuous wavelet transforms, we study here the quaternionic affine group and construct its…

We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the…

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil…

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…

In these lectures we describe the construction of a gauge invariant renormalization group equation for pure non-Abelian gauge theory. In the process, a non-perturbative gauge invariant continuum Wilsonian effective action is precisely…

If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $\tau$ is a unitary representation of $H$, then the authors give a sufficient condition on $\xi\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^G\tau$.…