Weil representations via abstract data and Heisenberg groups: A comparison

Lattice Representations of Heisenberg Groups

This Heisenberg group is a 2-step nilpotent Lie group and is important in the study of toroidal compactifications of Siegel moduli spaces. In fact, H (g,h) R is obtained as the unipotent radical of the parabolic subgroup of Sp(g+h,R) associated with the rational boundary component Fg ( cf. [F-C] p. 123 or [N] p. 21 ). For the motivation of the study of this Heisenberg group we refer to [Y4]-[Y8...

Classification of irreducible representations of Heisenberg groups and algebras

Definition. A Heisenberg group for a symplectic vector space (V, ω) is the Lie group with the underlying manifold V ×R and the multiplication (u, s)(v, t) = (u+ v, s+ t+ ω(u, v)/2) where u, v ∈ V and s, t ∈ R. The map t 7→ (0, t) is a Lie group homomorphism from R to the Heisenberg group. Its image coincides with the center of the Heisenberg group. The dimension of the Heisenberg group equals 2...

M ay 2 00 9 HEISENBERG GROUPS , THETA FUNCTIONS AND THE WEIL REPRESENTATION

φ-factorable operators and Weyl-Heisenberg frames on LCA groups

Self-dual Representations of Division Algebras and Weil Groups: a Contrast

If ρ is a selfdual representation of a group G on a vector space V over C, we will say that ρ is orthogonal, resp. symplectic, if G leaves a nondegenerate symmetric, resp. alternating, bilinear form B : V ×V → C invariant. If ρ is irreducible, exactly one of these possibilities will occur, and we may define a sign c(ρ) ∈ {±1}, taken to be +1, resp. −1, in the orthogonal, resp. symplectic, case....