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 reductivegroups. 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.