Let V be a symplectic vector space over a finite or local field. We compute the character of the Weil representation of the metaplectic group Mp(V ). The final formulas are overtly free of choices (e.g. they do not involve the usual choice of a Lagrangian subspace of V ). Along the way, in results similar to those of K. Maktouf, we relate the character to the Weil index of a certain quadratic form, which may be understood as a Maslov index. This relation also expresses the character as the pullback of a certain simple function from Mp(V ⊕ V ).

Abstract. Kashiwara defined the Maslov index (associated to a collection of Lagrangian subspaces of a symplectic vector space over a field F ) as a class in the Witt group W (F ) of quadratic forms. We construct a canonical quadratic vector space in this class and show how to understand the basic properties of the Maslov index without passing to W (F )-that is, more or less, how to upgrade Kashiwara's equalities in W (F ) to canonical isomorphisms between quadratic spaces. We also show how our canonical quadratic form occurs naturally in the context of the Weil representation. The quadratic space is defined using elementary linear algebra. On the other hand, it has a nice interpretation in terms of sheaf cohomology, due to A. Beilinson.

It is notoriously difficult to find an intuitively satisfactory rule for evaluating populations based on the welfare of the people in them. Standard examples, like total utilitarianism, either entail the Repugnant Conclusion or in some other way contradict common intuitions about the relative value of populations. Several philosophers have presented formal arguments that seem to show that this happens of necessity: our core intuitions stand in contradiction. This paper assesses the state of play, focusing on the most powerful of these 'impossibility theorems', as developed by Gustaf Arrhenius. I highlight two ways in which these theorems fall short of their goal: some appeal to a supposedly egalitarian condition which, however, does not properly reflect egalitarian intuitions; the others rely on a background assumption about the structure of welfare which cannot be taken for granted. Nonetheless, the theorems remain important: they give insight into the difficulty, if perhaps not the impossibility, of constructing a satisfactory population axiology. We should aim for reflective equilibrium between intuitions and more theoretical considerations. I conclude by highlighting one possible ingredient in this equilibrium, which, I argue, leaves open a still wider range of acceptable theories: the possibility of vague or otherwise indeterminate value relations.

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