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We explain and correct a mistake in Section 2.6 and Appendix C of the first and second author's paper "Representation Growth and Rational Singularities of the Moduli Space of Local Systems" [1]. [ABSTRACT FROM AUTHOR]

Regrettably, the paper [1] had two incorrect lemmas. Nevertheless, the statements of the theorems in [1] are correct with no modification required. Specifically, [1, Lemma 6.4] does not hold for spherical spaces, and also the conclusion of [1, Lemma B.2] is not valid in the generality stated there. The effect of the error in [1, Lemma 6.4] on the proof of [1, Theorem 6.1] is significant. In § 1 one we provide a proof of [1, Theorem 6.1] that does not relay upon [1, Lemma 6.4]. The effect of the error in [1, Lemma B.2] is minor and we explain the nessecary modifications in §2. [ABSTRACT FROM AUTHOR]

For a spherical variety X of a reductive group G over a non-archimedean local field F, and for a smooth representation π of G we study homological multiplicities dim Ext G ∗ (S (X) , π) . Based on Bernstein's decomposition of the category of smooth representations of G, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities dim Hom G (S (X) , π) are finite. The latter are known to be finite for symmetric varieties and for many other spherical varieties and conjectured to be finite for all spherical varieties. Furthermore, we show that the Euler–Poincaré characteristic is constant in families parabolically induced from finite length representations of a Levi subgroup M < G. In the case when M = G we compute these multiplicities more explicitly. [ABSTRACT FROM AUTHOR]

We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of GL3(C). Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently in [Sak16]. [ABSTRACT FROM AUTHOR]

We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group $${{G}}$$ defined over $$\mathbb {R}$$ , the space of $${{G(\mathbb {R})}}$$ -equivariant distributions on the manifold of real points of any algebraic $${{G}}$$ -manifold $${{X}}$$ is finite-dimensional if $${{G}}$$ has finitely many orbits on $${{X}}$$ . [ABSTRACT FROM AUTHOR]

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 $$V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}$$ 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. [ABSTRACT FROM AUTHOR]

Let G be a semisimple algebraic group defined over $$\mathbb {Q}_p$$ , and let $$\Gamma $$ be a compact open subgroup of $$G(\mathbb {Q}_p)$$ . We relate the asymptotic representation theory of $$\Gamma $$ and the singularities of the moduli space of G-local systems on a smooth projective curve, proving new theorems about both: For the proof, we study the analytic properties of push forwards of smooth measures under algebraic maps. More precisely, we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities. [ABSTRACT FROM AUTHOR]

We study the Fourier transform of the absolute value of a polynomial on a finite-dimensional vector space over a local field of characteristic 0. We prove that this transform is smooth on an open dense set. We prove this result for the Archimedean and the non-Archimedean case in a uniform way. The Archimedean case was proved in . The non-Archimedean case was proved in and , . Our method is different from those described in , , , . It is based on Hironaka's desingularization theorem, unlike which is based on the theory of D-modules and , , which is based on model theory. Our method also gives bounds on the open dense set where the Fourier transform is smooth and, moreover, on the wave front set of the Fourier transform. These bounds are explicit in terms of resolution of singularities and field-independent. We also prove the same results on the Fourier transform of more general measures of algebraic origins. [ABSTRACT FROM AUTHOR]

We prove vanishing of $$\mathfrak {z}$$ -eigen distributions on a split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup. This is a generalization of a result by Shalika, that concerned the group case. Shalika's result was crucial in the proof of his multiplicity one theorem. We view our result as a step in the study of multiplicities of quasi-regular representations on spherical varieties. As an application we prove non-vanishing of spherical Bessel functions. [ABSTRACT FROM AUTHOR]

The notion of derivatives for smooth representations of GL( n, ℚ) was defined in . In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in and called the 'adduced' representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In the companion paper we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [, , , ]. [ABSTRACT FROM AUTHOR]

The notion of derivatives for smooth representations of GL( n, ℚ) was defined in . In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in and called the 'adduced' representation. In derivatives of all orders were defined for smooth admissible Fréchet representations (of moderate growth). A key ingredient of this definition is the functor of twisted coinvariants with respect to the nilradical of the mirabolic subgroup. In this paper we prove exactness of this functor and compute it on a certain class of representations. This implies exactness of the highest derivative functor, and allows to compute highest derivatives of all monomial representations. In these results are applied to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations. [ABSTRACT FROM AUTHOR]

Let X be a smooth real algebraic variety. Let ξ be a distribution on it. One can define the singular support of ξ to be the singular support of the D-module generated by ξ (sometimes it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of T*X. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see, e.g., [AG4, AS, Say]). The aim of this paper is to give an analog of this theorem to the non-Archimedean case. The theory of D-modules is not available to us so we need a different definition of the singular support. We use the notion wave front set from [Hef] and define the singular support to be its Zariski closure. Then we prove that the singular support satisfies some property that we call weakly coisotropic, which is weaker than being coisotropic but is enough for some applications. We also prove some other properties of the singular support that were trivial in the Archimedean case (using the algebraic definition) but not obvious in the non-Archimedean case. We provide two applications of those results: [ABSTRACT FROM AUTHOR]

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma. [ABSTRACT FROM AUTHOR]