Eisenstein series on reductive symmetric spaces and representations of Hecke algebras.

1993 ◽  
Vol 1993 (445) ◽  
pp. 45-108
Keyword(s):  
Eisenstein Series ◽  
Symmetric Spaces ◽  
Hecke Algebras

scholarly journals Quasisplit Hecke algebras and symmetric spaces

2014 ◽  
Vol 163 (5) ◽  
pp. 983-1034 ◽  
Author(s):  
George Lusztig ◽  
David A. Vogan Jr.
Keyword(s):  
Symmetric Spaces ◽  
Hecke Algebras

scholarly journals Shimura varieties at level and Galois representations

2020 ◽  
Vol 156 (6) ◽  
pp. 1152-1230 ◽  
Author(s):  
Ana Caraiani ◽  
Daniel R. Gulotta ◽  
Chi-Yun Hsu ◽  
Christian Johansson ◽  
Lucia Mocz ◽  
...  
Keyword(s):  
Symmetric Spaces ◽  
Galois Representations ◽  
Hecke Algebras ◽  
Derived Category ◽  
Shimura Varieties ◽  
Nilpotent Ideal ◽  
Locally Symmetric Spaces ◽  
Compactly Supported ◽  
Symmetric Varieties

We show that the compactly supported cohomology of certain $\text{U}(n,n)$- or $\text{Sp}(2n)$-Shimura varieties with $\unicode[STIX]{x1D6E4}_{1}(p^{\infty })$-level vanishes above the middle degree. The only assumption is that we work over a CM field $F$ in which the prime $p$ splits completely. We also give an application to Galois representations for torsion in the cohomology of the locally symmetric spaces for $\text{GL}_{n}/F$. More precisely, we use the vanishing result for Shimura varieties to eliminate the nilpotent ideal in the construction of these Galois representations. This strengthens recent results of Scholze [On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2) 182 (2015), 945–1066; MR 3418533] and Newton–Thorne [Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), e21; MR 3528275].

Eisenstein Cohomology for GL and the Special Values of Rankin-Selberg L-Functions

2019 ◽  
Author(s):  
Anantharam Raghuram ◽  
Günter Harder
Keyword(s):  
Eisenstein Series ◽  
Symmetric Spaces ◽  
Constant Term ◽  
Local System ◽  
Arithmetic Groups ◽  
Special Values ◽  
Cohomology Of Arithmetic Groups ◽  
Finite Dimensional ◽  
Eisenstein Cohomology ◽  
Cohomological Interpretation

This book studies the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel–Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin–Selberg L-functions for GL(n) × GL(m), where n + m = N. The book carries through the entire program with an eye toward generalizations. The book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.

Mean ergodic theorem in function symmetric spaces for infinite measure

2018 ◽  
Vol 2018 (1) ◽  
pp. 35-46
Author(s):  
Vladimir Chilin ◽  
◽  
Aleksandr Veksler ◽  
Keyword(s):  
Ergodic Theorem ◽  
Symmetric Spaces ◽  
Infinite Measure ◽  
Mean Ergodic Theorem
Sign in / Sign up

Export Citation Format

Share Document