In the 1990s Moy and Prasad revolutionized the representation theory of p-adic groups by showing how to use Bruhat-Tits theory to assign invariants to representations of p-adic groups. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.

# Past Algebra Seminar

Roughly speaking, a commutative-by-finite Hopf algebra is a Hopf

algebra which is an extension of a commutative Hopf algebra by a

finite dimensional Hopf algebra.

There are many big and significant classes of such algebras

(beyond of course the commutative ones and the finite dimensional ones!).

I'll make the definition precise, discuss examples

and review results, some old and some new.

No previous knowledge of Hopf algebras is necessary.

For a polynomial $h(x)$ in $F[x]$, where $F$ is any field, let $A$ be the

$F$-algebra given by generators $x$ and $y$ and relation $[y, x]=h$.

This family of algebras include the Weyl algebra, enveloping algebras of

$2$-dimensional Lie algebras, the Jordan plane and several other

interesting subalgebras of the Weyl algebra.

In a joint work in progress with Samuel Lopes, we computed the Hochschild

cohomology $HH^*(A)$ of $A$ and determined explicitly the Gerstenhaber

structure of $HH^*(A)$, as a Lie module over the Lie algebra $HH^1(A)$.

In case $F$ has characteristic $0$, this study has revealed that $HH^*(A)$

has finite length as a Lie module over $HH^1(A)$ with pairwise

non-isomorphic composition factors and the latter can be naturally

extended into irreducible representations of the Virasoro algebra.

Moreover, the whole action can be understood in terms of the partition

formed by the multiplicities of the irreducible factors of the polynomial

$h$.

Given a stability condition defined over a category, every object in this category

is filtered by some distinguished objects called semistables. This

filtration, that is unique up-to-isomorphism, is know as the

Harder-Narasimhan filtration.

One less studied property of stability conditions, when defined over an

abelian category, is the fact that each of them induce a chain of torsion

classes that is naturally indexed.

In this talk we will study arbitrary indexed chain of torsion classes. Our

first result states that every indexed chain of torsion classes induce a

Harder-Narasimhan filtration. Following ideas from Bridgeland we

show that the set of all indexed chains of torsion classes satisfying a mild

technical condition forms a topological space. If time we

will characterise the neighbourhood or some distinguished points.

In combinatorics, the 'nicest' way to prove that two sets have the same size is to find a bijection between them, giving more structure to the seeming numerical coincidences. In representation theory, many of the outstanding conjectures seem to imply that the characteristic p of the ground field can be allowed to vary, and we can relate different groups and different primes, to say that they have 'the same' representation theory. In this talk I will try to make precise what we could mean by this

The classical Dirac operator is part of an osp(1|2) realisation inside the Weyl-Clifford algebra which is Pin-invariant. This leads to a multiplicity-free decomposition of the space of spinor-valued polynomials in irreducible modules for this Howe dual pair. In this talk we review an abstract generalisation A of the Weyl algebra that retains a realisation of osp(1|2) and we determine its centraliser algebra explicitly. For the special case where A is a rational Cherednik algebra, the centralizer algebra provides a refinement of the previous decomposition whose analogue was no longer irreducible in general. As an example, for the group S3 in specific, we will examine the finite-dimensional irreducible modules of the centraliser algebra.

The celebrated localisation theorem of Beilinson-Bernstein asserts that there is an equivalence between representations of a Lie algebra and modules over the sheaf of differential operators on the corresponding flag variety. In this talk we discuss certain analogues of this result in various contexts. Namely, there is a localisation theorem for quantum groups due to Backelin and Kremnizer and, more recently, Ardakov and Wadsley also proved a localisation theorem working with certain completed enveloping algebras of p-adic Lie algebras. We then explain how to combine the ideas involved in these results to construct

a p-adic analytic quantum flag variety and a category of D-modules on it, and we show that the global section functor on these D-modules yields an equivalence of categories.

Polyfree groups are defined as groups having a series of normal

subgroups such that each sucessive quotient is free. This property

imples locally indicability and therefore also right orderability. Right

angled Artin groups are known to be polyfree (a result shown

independently by Duchamp-Krob, Howie and Hermiller-Sunic). Here we show

that Artin FC-groups for which all the defining relation are of even

type are also polyfree. This is a joint work with Ruven Blasco and Luis

Paris.

I will explain how Dirac operators provide precious information about geometric and algebraic aspects of representations of real Lie groups. In particular, we obtain an explicit realisation of representations, leading terms in the asymptotics of characters and a precise connection with nilpotent orbits.