Related papers: Scattered representations of complex classical Lie…

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…

There is a well-known correspondence between the symplectic variety of representations of the fundamental group of a punctured Riemann surface into a compact Lie group G, with fixed conjugacy classes at the punctures, and a complex variety…

By a theorem of D. Wigner, an irreducible unitary representation with non-zero $(\frak{g},K)$-cohomology has trivial infinitesimal character, and hence up to unitary equivalence, these are finite in number. We have determined the number of…

We classify the irreducible unitary modules in category O for the rational Cherednik algebras of type G(r,1,n) and give explicit combinatorial formulas for their graded characters. More precisely, we produce a combinatorial algorithm…

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 $G=\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $\Pi$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $\Pi$ occurs inside a principal series representation of…

In Part 1 of this paper we construct a spectral sequence converging to the relative Lie algebra cohomology associated to the action of any subgroup $G$ of the symplectic group on the polynomial Fock model of the Weil representation, see…

We classify non-polar irreducible representations of connected compact Lie groups whose orbit space is isometric to that of a representation of a finite extension of $Sp(1)^k$ for some $k>0$. It follows that they are obtained from isotropy…

We study corners and fundamental corners of the irreducible representations of the groups G=Spin(n,1) that are not elementary, i.e. that are equivalent to subquotients of reducible nonunitary principal series representations. For even n we…

A real form $G_0$ of a complex semisimple Lie group $G$ has only finitely many orbits in any given compact $G$-homogeneous projective algebraic manifold $Z=G/Q$. A maximal compact subgroup $K_0$ of $G_0$ has special orbits $C$ which are…

We show that spin-flip probabilities emerge in the relativistic regime for scalar potentials, absent in the standard Dirac representation. We examine 1D scattering for the Dirac equation employing an alternate matrix representation…

We study a spectral problem related to the finite-dimensional characters of the groups $Sp(2N)$, $SO(2N+1)$, and $SO(2N)$, which form the classical series $C$, $B$, and $D$, respectively. The irreducible characters of these three series are…

This paper gives a complete classification of the unitary irreducible spherical representations of split real and p-adic groups. The results were obtained around 2000, the changes to the new version are expository.

We consider compact locally symmetric spaces $\Gamma\backslash G/H$ where $G/H$ is a non-compact semisimple symmetric space and $\Gamma$ is a discrete subgroup of $G$. We discuss some features of the joint spectrum of the (commutative)…

The tempered representations of a real reductive Lie group $G$ are naturally partitioned into series associated with conjugacy classes of Cartan subgroups $H$ of $G$. We define partial Dirac cohomology, apply it for geometric construction…

We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In…

The decomposition of the spinor bundle of the spin Grassmann manifolds $G_{m,n}=SO(m+n)/SO(m)\times SO(n)$ into irreducible representations of $\mathfrak{so}(m)\oplus\mathfrak{so}(n)$ is presented. A universal construction is developed and…

Let $G$ be a simple real Lie group with maximal parabolic subgroup $P$ whose nilradical is abelian. Then $X=G/P$ is called a symmetric $R$-space. We study the degenerate principal series representations of $G$ on $C^\infty(X)$ in the case…

We classify the finite dimensional representations of the quantum symmetric pair coideal subalgebra $B_{\mathbf c}$ of type $DII$ corresponding to the symmetric pair $(so(2N),so(2N-1))$. For $B_{\mathbf c}$ defined over an arbitrary field…

In this paper we study the analytic realisation of the discrete series representations for the group $G=Sp(1,1)$ as a subspace of the space of square integrable sections in a homogeneous vector bundle over the symmetric space $G/K:=Sp(1,1)…