Delaboring Republicanism

This article criticizes radical labor republicanism on republican grounds. I show that its demand for universal workplace democracy via workers’ cooperatives conflicts with republican freedom along three different dimensions: first, freedom to choose an occupation…and not to choose one; second, freedom within the very cooperatives that workers are to democratically govern; and third, freedom within the newly proletarian state. In the conclusion, I ask whether these criticisms apply, at least in part, to the more modest, incrementalist strand of labor republicanism. To the extent that they do, delaboring republicanism might be the best response

Existence of Klyachko Models for GL(n;R) and GL(n;C)

We prove that any unitary representation of GL(n;R) and GL(n;C) admits an equivariant linear form with respect to one of the subgroups considered by Klyachko

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let $H$ be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold $X$ and a real algebraic bundle $\mathcal{E}$ on $X$. Let $\mathfrak{h}$ be the Lie algebra of $H$. Let $\mathcal{S}(X,\mathcal{E})$ be the space of Schwartz sections of $\mathcal{E}$. We prove that $\mathfrak{h}\mathcal{S}(X,\mathcal{E})$ is a closed subspace of $\mathcal{S}(X,\mathcal{E})$ of finite codimension. We give an application of this result in the case when $H$ is a real spherical subgroup of a real reductive group $G$. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $\pi$ be a Casselman-Wallach representation of $G$ and $V$ be the corresponding Harish-Chandra module. Then the natural morphism of coinvariants $V_{\mathfrak{h}}\to \pi_{\mathfrak{h}}$ is an isomorphism if and only if any linear $\mathfrak{h}$-invariant functional on $V$ is continuous in the topology induced from $\pi$. The latter statement is known to hold in two important special cases: if $H$ includes a symmetric subgroup, and if $H$ includes the nilradical of a minimal parabolic subgroup of $G$.Comment: v4: version appearing in Math. Z + erratum added in the en

Invariant Functionals on the Speh representation

We study Sp(2n,R)-invariant functionals on the spaces of smooth vectors in Speh representations of GL(2n,R). For even n we give expressions for such invariant functionals using an explicit realization of the space of smooth vectors in the Speh representations. Furthermore, we show that the functional we construct is, up to a constant, the unique functional on the Speh representation which is invariant under the Siegel parabolic subgroup of Sp(2n,R). For odd n we show that the Speh representations do not admit an invariant functional with respect to the subgroup U(n) of Sp(2n,R) consisting of unitary matrices. Our construction, combined with the argument in [GOSS12], gives a purely local and explicit construction of Klyachko models for all unitary representations of GL(2n,R).Comment: 14 pages. v4: minor corrections in Theorem 2.2, Lemma 2.9 and section

Multiplicity free Jacquet modules

Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let P=MU be the corresponding parabolic subgroup. Let J denote the Jacquet functor from representations of G to representations of M (i.e. the functor of coinvariants w.r.t. U). In this paper we prove that J is a multiplicity free functor, i.e. dim Hom(J(\pi),\rho)<= 1, for any irreducible representations \pi of G and \rho of M. To do that we adapt the classical method of Gelfand and Kazhdan that proves "multiplicity free" property of certain representations to prove "multiplicity free" property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.Comment: 12 pages; Canadian Mathematical Bulletin, Published electronically on June 29, 201

A reduction principle for Fourier coefficients of automorphic forms

In this paper we analyze a general class of Fourier coefficients of automorphic forms on reductive adelic groups $\mathbf{G}(\mathbb{A}_\mathbb{K})$ and their covers. We prove that any such Fourier coefficient is expressible through integrals and sums involving 'Levi-distinguished' Fourier coefficients. By the latter we mean the class of Fourier coefficients obtained by first taking the constant term along the nilradical of a parabolic subgroup, and then further taking a Fourier coefficient corresponding to a $\mathbb{K}$-distinguished nilpotent orbit in the Levi quotient. In a follow-up paper we use this result to establish explicit formulas for Fourier expansions of automorphic forms attached to minimal and next-to-minimal representations of simply-laced reductive groups.Comment: 35 pages. v2: Extended results and paper split into two parts with second part appearing soon. New title to reflect new focus of this part. v3: Minor corrections and updated reference to the second part that has appeared as arXiv:1908.08296. v4: Minor corrections and reformulation