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The purpose of this paper is to find explicit formulae for the total residue of some interesting rational functions with poles on hyperplanes determined by roots of type Ar = {(ei−ej)|1 ≤ i, j ≤ (r+1), i ≠ j}. As pointed out by Zeilberger [Z], these calculations are mere reformulations of Morris...
This article is a contribution to the domain of (convergent) deformation quantization of symmetric spaces using the representation theory of Lie groups. We realize the regularrepresentation of SL(2, ℝ) on the space of smooth functions on the Poincare disc as a subrepresentation of SL(2, ℝ) in the Weyl–Moyal star product algebra on ℝ2. We indicate how it is possible to extend our construction to the...
This article is a report on a work which will be published elsewhere with complete proofs. It deals with some local zeta functions associated to a family of symmetric spaces arising from 3-gradings of reductive Lie algebras. Let $$ \tilde{\mathfrak{g}} = {V^{ - }} \oplus \mathfrak{g} \oplus {V^{ + }} $$ be a 3-graded real reductive Lie algebra. Let $$ \tilde{G} $$ be the adjoint group of ...
The question of unitarity of representations in the analytic continuation of discrete series from a Borel-de Siebenthal chamber is considered for those linear equal-rank classical simple Lie groups G that have not been treated fully before. Groups treated earlier by other authors include those for which G has real rank one or has a symmetric space with an invariant complex structure . Thus the groups...
Let g be a complex semisimple Lie algebra and let t be the subalgebra of fixed elements in g under the action of an involutory automorphism of g. Any such involution is the complexification of the Cartan involution of a real form of g. If Vλ is an irreducible finite-dimensional representation of g, the Iwasawa decomposition implies that Vλ is a cyclic U(t)module where the cyclic...
In 1958 Nagata [5] gave an ingenious argument that demonstrated the existence of counterexamples to Hilbert’s Fourteenth Problem. Recall that the original problem is the following: Let K = F(x1, x2,..., xn) be the function field of affine n-space V = Fn over an algebraically closed field F, and suppose L...
In this article I describe my recent geometric localization argument dealing with actions of noncompact groups which provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed in [Sch]. A corresponding problem in the compact group setting was solved by N. Berline, E. Getzler and M. Vergne in [BGV] by an application of...
The positive spin ladder representations for G = SU(p, q)may be realized in a Fock space, in Dolbeault cohomology over G/S(U(p, q−1) × U(1)), and as certain holomorphic sections of a vector bundle over G/S(U(p) × U(q)). A Penrose transform, also referred to as a double fibration transform, intertwines the Dolbeault model into the vector bundle model. By passing through the Fock space realization of...
Summation formulas have played a very important role in analysis and number theory, dating back to the Poisson summation formula. The modern formulation of Poisson summation asserts the equality 1.1 $$\sum\limits_{{n \in \mathbb{Z}}} {f(n) = \sum\limits_{{n \in \mathbb{Z}}} {\widehat{f}(n)} } \left( {\widehat{f}(t) = \int_{\mathbb{R}} {f(x){{e}^{{ - 2\pi ixt}}}dx} } \right),$$ valid (at least)...
According to MacKay [1980] the irreducible characters of finite sub-groups of SU(2) are in a natural 1-1 correspondence with the extended Coxeter-Dynkin graphs of type ADE. We show that the character values themselves can be given by an uniform formula, as special values of polynomials which arise naturally as numerators of Poincaré series associated to finite subgroups of SU(2) acting on polynomials...
A survey of joint work with Henrik Schlichtkrull on the induction of certain relations among (partial) Eisenstein integrals for the minimal principal series of a reductive symmetric space is given. The application of this principle of induction to the proof of the Fourier inversion formula in [11] and to the proof of the Paley-Wiener theorem in [15] is explained. Finally, the relation with the Plancherel...
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