Ellipticity and discrete series

We explain by elementary means why the existence of a discrete series representation of a real reductive group G implies the existence of a compact Cartan subgroup of G. The presented approach has the potential to generalize to real spherical spaces.

On counting cuspidal automorphic representations for GSp(4)

We find the number s k ⁢ ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ⁢ ( 4 , A Q ) \mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ⁢ ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

Representations induced from the Zelevinsky segment and discrete series in the half-integral case

Let {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

Relative Trace Formula for Compact Quotient and Pseudo-Coefficients for Relative Discrete Series

We introduce the notion of relative pseudo-coefficient for relative discrete series representations of real spherical homogeneous spaces of reductive groups. We prove that $K$-finite relative pseudo-coefficient does not exist for semisimple symmetric spaces of type $G_{\mathbb{C}}/G_{\mathbb{R}}$, where $K$ is a maximal compact subgroup of $G_{\mathbb{C}}$, and construct strong relative pseudo-coefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H. Jacquet for compact discrete quotient $\Gamma \backslash G$. This allows us to prove that a relative discrete series representation, which admits strong pseudo-coefficients with sufficiently small support, occurs in the spectral decomposition of $L^2(\Gamma \backslash G)$ with a nonzero period.

Coherent states and geometry on the Siegel–Jacobi disk

The coherent state representation of the Jacobi group [Formula: see text] is indexed with two parameters, [Formula: see text], describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1,1). The Ricci form, the scalar curvature and the geodesics of the Siegel–Jacobi disk [Formula: see text] are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel–Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding and the Cauchy formula for the Siegel–Jacobi disk are presented.

L-Functions for GSp(2) × GL(2): Archimedean Theory and Applications

Let Π be a generic cuspidal automorphic representation of GSp(2) defined over a totally real algebraic number field k whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product L-functions L(s,Π × σ) for an arbitrary cuspidal automorphic representation σ of GL(2). We also give an application to the spinor L-function of Π.

GROUP THEORY CLASSIFICATION SCHEMES OF THE N-ELECTRON QUANTUM DOT

We develop a general framework for discussing collective behavior in confined many-electron systems. Our specific goal is the application to N-electron quantum dots, which are mesoscopic semiconductor systems of great current interest as possible ultra-small electronic devices. In view of its broad applicability, we are able to cast the discussion of the many-electron problem in general terms. We consider the general N-interacting particle system in d dimensions and study its bilinear dynamical symmetry group which is the noncompact symplectic group Sp(2Nd, R). Giving their explicit dependence on N and d, we focus on the classification of many-particle bound states which requires knowledge of the unitary discrete series representation theory of Sp(2Nd, R) and the corresponding character reductions. We also discuss matrix elements of the generators, the implementation of the Pauli principle, and a procedure to derive total angular momentum quantum numbers associated with a given total spin.