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We show that any stack $\mathfrak{X}$ of finite type over a Noetherian scheme has a presentation $X \rightarrow \mathfrak{X}$ by a scheme of finite type such that $X(F) \rightarrow \mathfrak{X}(F)$ is onto, for every finite or real closed field $F$. Under some additional conditions on $\mathfrak{X}$, we show the same for all perfect fields. We prove similar results for (some) Henselian rings. We give two applications of the main result. One is to counting isomorphism classes of stacks over the rings $\mathbb{Z}/p^n$; the other is about the relation between real algebraic and Nash stacks., Comment: 20 pages

We study equivariant morphisms from zero dimensional schemes to varieties and show that, under suitable assumptions, all such morphisms factor via a canonical one. We relate the above to Algebraic Representations of Ergodic Actions.

We prove that a quotient of a Nash manifold $X$ by a closed equivalence relation $R\subset X\times X$, which is submersive over $X$, yields a Nash manifold $X/R$., Comment: 18 pages, comments are welcome

Let a complex algebraic reductive group $\bf G$ act on a complex algebraic manifold $\bf X$. For a $\bf G$-invariant subvariety $\Xi$ of the nilpotent cone $\mathcal{N}(\mathfrak{g}^*)\subset \mathfrak{g}^*$ we define a notion of $\Xi$-symplectic complexity of $\bf X$. This notion generalizes the notion of complexity defined in [Vin86]. We prove several properties of this notion, and relate it to the notion of $\Xi$-complexity defined in [AG] motivated by its relation with representation theory., Comment: 10 p

For a Nash submersion $\phi\colon X\to Y$, we study the complex $\mathcal{SDR}(\phi)$ of Schwartz sections of the relative de Rham complex of $\phi$. We define the notion of Schwartz sections of constructible sheaves on Nash manifolds and prove that $\mathcal{SDR}(\phi)$ is homotopy equivalent to the Schwartz sections of the proper push-forward $\phi_!\mathbb{R}_X$ of the constant sheaf $\mathbb{R}_X$. Using this equivalence, we show that $\mathcal{SDR}(\phi)$ depends (up to homotopy equivalence) only on the homology type of the map $\phi$. We also deduce that $\mathcal{SDR}(\phi)$ has Hausdorff homology spaces., Comment: 70 pages

International audience; In the context of geometry and analysis on non-archimedean local fields, we study two recent notions, $C^{\mathrm exp}$-class distributions from [11] and WF-holonomicity from [1], and we show that any distribution of $C^{\mathrm exp}$-class is WF-holonomic. Thus we answer a question from [1] by providing a framework of WF-holonomic distributions for non-archimedean local fields which is stable under taking Fourier transforms and which contains many natural distributions, in particular, the distributions studied in [1]. We further show that one can regularize distributions without leaving the $C^{\mathrm exp}$-class. Finally, we show a close link between zero loci and smooth loci for functions and distributions of $C^{\mathrm exp}$-class, by proving a converse to a result of [11]. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.

For a finite group $G$, Frobenius found a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s}$ for even integers $s$, where $\mathrm{Irr} G$ is the set of irreducible representations of $G$. We generalize this formula to the relative case: for a subgroup $H$, we find a formula for the values of the function $\sum_{\mathrm{Irr} G} (\dim\, \pi)^{-s} (\dim\, \pi ^H)^{-t}$. We apply our results to compute the E-polynomials of Fock--Goncharov spaces and to relate the Gelfand property to the geometry of generalized Fock--Goncharov spaces.

In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over arbitrary local field F of characteristic zero. Our main tool is Luna slice theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pair (GL(n,C),GL(n,R)) is a Gelfand pair. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-archimedean F the later is a classical theorem of Gelfand and Kazhdan. We use the techniques developed here in our subsequent work [AG3] where we prove an archimedean analog of the theorem on uniqueness of linear periods by H. Jacquet and S. Rallis., 34 pages, 1 figure. v2: A proof of a version of localization principle inserted. v3: minor changes. v4: definition of symmetric pair slightly changed. v5: minor changes + lemma D.0.3 added for clarification. v6: minor changes - see Theorem 4.0.5. v7:minor changes. Appendix A by Avraham Aizenbud, Dmitry Gourevitch and Eitan Sayag

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 de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute cohomologies of the Lie algebra 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 Shapiro lemma., 21 pages, v2: minor corrections