We provide a family of representations of GL2n over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp2ndistinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, relies on the theory of automorphic forms. The results of this paper together with...

An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel’fand-Zetlin basis is given. Particular attention is paid to the so-called star type I representations (“unitary representations”), and to a simple class of representations V (p), with p any positive integer. Then, the notion of Wigner Quantum Oscillators (WQOs) is recalled. In...

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators b i . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V (p) of osp(1|2n). Apart from the s...

In this paper we collect some facts about the topology on the space of irreducible unitary representations of a real reductive group. The main goal is Theorem 10, which asserts that most of the “cohomological” unitary representations for real reductive groups (see [VZ]) are isolated. Many of the intermediate results can be extended to groups over any local field, but we will discuss these gener...

A spin system is a sequence of self-adjoint unitary operators U1, U2, . . . acting on a Hilbert space H which either commute or anticommute, UiUj = ±UjUi for all i, j; it is is called irreducible when {U1, U2, . . . } is an irreducible set of operators. There is a unique infinite matrix (cij) with 0, 1 entries satisfying UiUj = (−1) cij UjUi, i, j = 1, 2, . . . . Every matrix (cij ) with 0, 1 e...

Let F be a local non-Archimedean field of characteristic zero with finite residue field. Based on Tadić’s classification the unitary dual $${\mathrm {GL}}_{2n}(F)$$ , we classify irreducible representations that have nonzero linear periods, in terms Speh periods. We also give necessary and sufficient condition for existence period representation.