Locally compact abelian groups with symplectic self-duality

Abstract

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer).Comment: 23 page

Generalize Heisenberg Groups and Self-Duality

“…The connection between locally compact Mumford groups G with Z(G) ≅ T and self-dualities is clear comparing (1) (with Z(G) ≅ T) and (2) (with L = G Z(G)), as pointed out already in [20]. For more details about the relation between generalized Heisenberg groups, Mackey -Weil groups and Mumford groups, on one hand, and symplectic self-dualities on the other hand, see Theorem 6.5.…”

“…This gives rise to a canonical map: In case G carries a locally compact group topology and Z(G) ≅ T, the target group in (1) is nothing else but the Pontryagin dual of G Z(G) (see Remark 4.2 for more detail). In order to describe the properties of such a group G related to its irreducible unitary representations, Mumford [19] imposed additionally the condition that M is a topological isomorphism (these groups were used also in [20,21]). Clearly, every Mackey -Weil group has this property.…”

“…We only keep the condition on the Mumford map to be a topological isomorphism and we call these groups Mumford groups (Definition 5.1). Clearly, the locally compact Mumford groups G with center topologically isomorphic to T are precisely the above mentioned groups isolated in [19,20,21]. We give criteria for a generalized Heisenberg group to be a Mumford group, and we give criteria for a Mumford group to be a generalized Heisenberg group.…”

“…The "classical line" of study of the self-dual locally compact abelian groups imposes eventual additional restraints on the structure of the groups ( [12,13,22,23,29]). A somewhat more recent line prefers to avoid imposing structural restraints on the groups involved, but makes heavy use of homological algebra [10,11], or uses essentially properties on the bilinear form b ∇ ∶ L × L → T associated to the self-duality (L, ∇) ( [20,21]), defined as follows: b ∇ (x, y) = ∇(x)(y) for x, y ∈ L. In the opposite direction, self-dualities are often built via appropriate bilinear forms ω ∶ L × L → T. In the case of a locally compact field L, this is simply the field multiplication in L and this form ω is symmetric (Definition 2.1(a 3 )), although in other cases the bilinear form of a self-duality can be alternating (Definition 2.1(a 2 )), and in such a case one speaks of symplectic self-dualities ( [20]).…”

Combinatorics of finite abelian groups and Weil representations

“…It is worth noting that every Heisenberg group that is finite modulo centre is isomorphic to one of the groups considered here (for the precise statement, see Prasad-Shapiro-Vemuri [20], particularly, Section 3 and Corollary 5.7). For example, the seemingly different Heisenberg groups used by Tanaka [28] to construct representations in the principal series and cuspidal series of finite SL 2 are isomorphic.…”