Graded affine Hecke algebras were introduced by Lusztig for studying the representation theory of p-adic groups. In particular, some problems about extensions of representations of p-adic groups can be transferred to problems in the graded Hecke algebra setting. The study of extensions gives insight to the structure of various reducible modules. In this talk, I shall discuss some methods of computing Ext-groups for graded Hecke algebras.

The talk is based on arXiv:1410.1495, arXiv:1510.05410 and forthcoming work.

# Past Algebra Seminar

Abstract: This is a joint work with E. Breuillard.

A conjecture of Breuillard asserts that for every positive integer d, there is a positive constant c such that the following holds. Let S be a finite subset of GL(d,C) that generates a group, which is not virtually nilpotent. Then |S^n|>exp(cn) for all n.

Considering an algebraic number a that is not a root of unity and the semigroup generated by the affine transformations x-> ax+1, x-> ax+1, the above conjecture implies that the Mahler measure of a is at least 1+c' for some c'>0 depending on c. This property is known as Lehmer's conjecture.

I will talk about the converse of this implication, namely that Lehmer's conjecture implies the uniform growth conjecture of

Breuillard.

Our work (which is joint with Simon Smith) began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations.

One is the generalisation in which point stabilisers are merely assumed to satisfy min-{\sc N}, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on the socle of~$M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

Our work (which is joint with Simon Smith) began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-{\sc N}, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on the socle of~$M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result is based on work of Crawley-Bouvey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.

If $R = F_q[t]$ is the polynomial ring over a finite field

then the group $SL_2(R)$ is not finitely generated. The group $SL_3(R)$ is

finitely generated but not finitely presented, while $SL_4(R)$ is

finitely presented. These examples are facets of a larger picture that

I will talk about.

I will report on recent progress towards understanding the growth of the torsion of the homology of subgroups of finite index in a given residually finite group G.

The cases I will consider are when G is amenable (joint work with P, Kropholler and A. Kar) and when G is right angled (joint work with M. Abert and T. Gelander).