Invariant analysis on, and discrete series for real reductive groups

Author(s):  
V. S. Varadarajan
Author(s):  
Patrick Delorme ◽  
Pascale Harinck

Abstract We introduce the notion of relative pseudo-coefficient for relative discrete series representations of real spherical homogeneous spaces of reductive groups. We prove that $K$-finite relative pseudo-coefficient does not exist for semisimple symmetric spaces of type $G_{\mathbb{C}}/G_{\mathbb{R}}$, where $K$ is a maximal compact subgroup of $G_{\mathbb{C}}$, and construct strong relative pseudo-coefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H. Jacquet for compact discrete quotient $\Gamma \backslash G$. This allows us to prove that a relative discrete series representation, which admits strong pseudo-coefficients with sufficiently small support, occurs in the spectral decomposition of $L^2(\Gamma \backslash G)$ with a nonzero period.


2018 ◽  
Vol 13 (4) ◽  
pp. 496-517
Author(s):  
Ned Hercock

This essay examines the objects in George Oppen's Discrete Series (1934). It considers their primary property to be their hardness – many of them have distinctively uniform and impenetrable surfaces. This hardness and uniformity is contrasted with 19th century organicism (Gerard Manley Hopkins and John Ruskin). Taking my cue from Kirsten Blythe Painter I show how in their work with hard objects these poems participate within a wider cultural and philosophical turn towards hardness in the early twentieth century (Marcel Duchamp, Adolf Loos, Ludwig Wittgenstein and others). I describe the thinking these poems do with regard to industrialization and to human experience of a resolutely object world – I argue that the presentation of these objects bears witness to the production history of the type of objects which in this era are becoming preponderant in parts of the world. Finally, I suggest that the objects’ impenetrability offers a kind of anti-aesthetic relief: perception without conception. If ‘philosophy recognizes the Concept in everything’ it is still possible, these poems show, to experience resistance to this imperious process of conceptualization. Within thinking objects (poems) these are objects which do not think.


2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ivan Matić

AbstractLet {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.


Author(s):  
Federico Scavia

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .


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