let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $e/f$ be a galois extension‎. ‎we study the realization fields of‎ ‎finite subgroups $g$ of $gl_n(e)$ stable under the natural‎ ‎operation of the galois group of $e/f$‎. ‎though for sufficiently large $n$ and a fixed‎ algebraic number field $f$ every its finite extension $e$ is‎ ‎re...

The main purpose of this article is to supplement the authors’ results on degenerate principal series representations of real symplectic groups with the analogous results for metaplectic groups. The basic theme, as in the previous case, is that their structures are anticipated by certain natural subrepresentations constructed from Howe correspondence. This supplement is necessary as these repre...

This is an overview of our series of papers on the modular generalized Springer correspondence [AHJR2, AHJR3, AHJR4, AHJR5]. It is an expansion of a lecture given by the second author in the Fifth Conference of the Tsinghua Sanya International Mathematics Forum, Sanya, December 2014, as part of the Master Lecture ‘Algebraic Groups and their Representations’ Workshop honouring G. Lusztig. The ma...

Let E be an elliptic curve over Q, and let %[ and %] be odd two-dimensional Artin representations for which %[ ⊗ %] is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms f , g, and h of respective weights two, one, and one, giving rise to E , %[, and %] via the constructions of Eichler and Shimura, and of Deligne and Serre. This artic...

Consider the two dimensional symplectic torus (T, ω) and an hyperbolic automorphism A of T. The automorphism A is known to be ergodic. In 1980, using a non-trivial procedure called quantization, the physicists J. Hannay and M.V. Berry attached to this automorphism a quantum operator ρ ~ (A) acting on a Hilbert space H~. One of the central questions of ”Quantum Chaos Theory”, in this model, is w...