Representation theory seminar

Organized by the representation theory group at SIMIS

Mondays at 4pm

   

Lucien Hennecart, CNRS
06/23, SIMIS room 1210

Mackey formula for representations of reductive groups
In this talk, I will describe the construction of induction and restriction morphisms on the critical cohomology associated with a function on a representation of a reductive group. The induction morphisms play a key role in obtaining a cohomological integrality decomposition, which is a decomposition into finite-dimensional pieces with enumerative significance. The building blocks of this decomposition are given by the BPS cohomology. After discussing this decomposition and its geometric meaning, I will present a cohomological version of the Mackey formula that relates the induction and restriction operations. I will also present examples of the formalism in low dimensions. This can be seen as a generalization to more general stacks of the cohomological Hall algebra multiplication and its localized restriction.

   

Past talks

Weiping Li, HKUST
06/16, SIMIS room 1210

Infinite dimensional algebra and instanton moduli spaces
Given a projective smooth surface X  and its blowup surface Y, Yoshioka calculated the blowup formula relating Betti numbers of the moduli space of rank two sheaves on X with those on Y. Nakajima asked the question of a representation-theoretic interpretation of the blowup formula. In the joint work with Qingyuan Jiang and Yu Zhao, we studied the representation of an extended Clifford algebra on the cohomology of the moduli space of stable sheaves on Y and answered Nakajima’s question.

   

Mikhail Bershtein, SISSA Trieste
At 2pm! 06/09, SIMIS room 1210

Highest-weight vectors and three-point functions in coset decomposition
We revisit the classical Goddard Kent Olive coset construction. We find the formulas for the highest weight vectors in coset decomposition and calculate their norms. We also derive formulas for matrix elements of natural vertex operators between these vectors. This leads to relations on conformal blocks. Due to the AGT relation, these relations are equivalent to blowup relations on Nekrasov partition functions with the presence of the surface defect. These relations can be used to prove Kyiv formulas for the Painlevé tau-functions (following Nekrasov’s method).

   

Emile Bouaziz, Academia Sinica
05/19, SIMIS room 1210

Elliptic Constructions via DAG
I’ll talk about some recent work with Adeel Khan in which we construct elliptic versions of loop spaces from derived algebraic geometry. I’ll explain the construction, which passes through a Tannakian formalism, and sketch the resulting equivariant elliptic cohomology theory. Time permitting I will explain some expected applications to geometric representation theory.