Skip to main content
European Commission logo print header

Stacks in Representation Theory, Relative Trace Formula and analysis of Automorphic Periods

Final Report Summary - STACKSRTFPERIODS (Stacks in Representation Theory, Relative Trace Formula and analysis of Automorphic Periods)

ERC project. Stacks in Representation Theory, Relative Trace Formula and analysis of Automorphic Periods.

In the original proposal Bernstein suggested a method to establish convexity and subconvexity bounds for several
kinds of automorphic periods. The main idea was that one can generalise the space of densities, originally defined
for a smooth manifold M, to any smooth stack X, and thus define the space of densities Dens(X).
Using this construction Bernstein suggested to apply standard methods of studying the spaces Dens(M)
(mostly methods related to the Relative Trace Formula — RTF) to study the spaces Dens(X). This had to lead to
new estimates for automorphic periods.

Working on this project with his group Bernstein has realised that in general the notion of stacks plays the central
role in Representation Theory. In paper “Stacks in Representation Theory” he explained that many problems and
basic notions of Representation Theory have to be reinterpreted in the language of stacks. This language gives
completely new point of view and provides new language to study problems in Representation Theory. Many previously
known results that had quite awkward formulation become quite natural after adopting this new point of view.

One of the results of switching to the stack language has been a realisation that the standard approach to the Local
Langlands Correspondence (LLC) is slightly incorrect and has to be modified. Namely this approach is based on the
notion of the Langlands dual group that in “correct” theory should be slightly modified.
One of the side effects of this new formulation of LLC was a new approach to the definition of global L-functions of
automorphic representations. In particular, now it became clear how to think about special values of these L-functions.
Bernstein described this new approach in the paper “How to modify the Langlands dual group”.

Bernstein and Reznikov developed a micro local approach to representation spaces for representations of real reductive
groups. This approach allows to describe operators in a representation space V in terms of their symbols that are functions
on some symplectic variety S ( this variety is an analogue of the cotangent bundle in the usual micro-local calculus).
They hope to use this description to analyse the identities arising in the calculus of automorphic periods. This should lead
to new strong sub-convexity bounds for automorphic periods.

Bernstein and Reznikov developed a new procedure for defining global invariants of automorphic representations.
This allowed to give concise formulation of several fundamental results in the theory of automorphic representations and
to formulate several generalisations of these results.
In many cases this procedure allows to relate bounds on automorphic periods with bounds on special values of L-functions.
This approach was described in the paper “Periods and Global Invariants of Automorphic Representations” by Bernstein
and Reznikov.