(work in progress with R. Rouquier) I will present a
computational (yet conjectural) method to determine some
decomposition matrices for finite groups of Lie type. These
matrices encode how ordinary representations decompose when they
are reduced to...
The problem of classification of perverse sheaves on the
quotient h/W for a semisimple Lie algebra g has an explicit answer
which turns out to be related to the algebraic properties of
induction and restriction operations for parabolic
subalgebras...
The Hecke algebra admits an involution which preserves the
standard basis and exchanges the canonical basis with its dual.
This involution is categorified by "monoidal Koszul duality" for
Hecke categories, studied in positive characteristic in my...
I'll present joint work with Tsao-Hsien Chen on the geometry of
real and symmetric matrices. For classical groups, we use
hyperkahler geometry to lift the Kostant-Sekiguchi correspondence
to an equivariant homeomorphism. As an application, we show...
Given a representation of a reductive group,
Braverman-Finkelberg-Nakajima defined a Poisson variety called the
Coulomb branch, using a convolution algebra construction. This
variety comes with a natural deformation quantization, called a
Coulomb...
For a finite group G one has a process of modular reduction
which takes a KG-module, over a field K of characteristic zero, and
produces a kG-module, over a field k of positive characteristic.
Starting with a simple KG-module its modular reduction...
Reverse plane partitions - or RPPs for short - are order
reversing maps of minuscule posets in types ADE. We report on joint
work in progress with Elek, Kamnitzer, Libman, and Morton-Ferguson
in which we give a type independent proof that RPPs form...
I will review certain stabilization phenomena in the
characteristic zero representation theory of general linear and
symmetric groups as the rank tends to infinity. Then I will give a
survey of some results and conjectures about analogs of these
in...
Let G be a semi-simple algebraic group over an algebraically
closed field k of positive characteristic and let B be a Borel
subgroup. The cohomology of line bundles on the flag variety G/B
induced by characters of B are important objects in the...
