In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is: what are the minimal models within a birational class? It is not even clear a priori what the correct definition is of a minimal model in this context.

We show that a generic Sklyanin algebra (a noncommutative analogue of P^2) satisfies the surprising property that it has no birational connected graded noetherian overrings, and explain why this is a reasonable definition of 'minimal model.' We show also that the noncommutative versions of P^1xP^1 and of the Hirzebruch surface F_2 are minimal.

This is joint work in progress with Dan Rogalski and Toby Stafford.

# Past Algebra Seminar

I will discuss recent results in finite simple groups. These include growth, generation (with a number theoretic flavour), and conjectures of Gowers and Viola on mixing and complexity whose proof requires representation theory as a main tool.

We give an overview of joint work with Lewis Topley on modular W-algebras. In particular, we outline the classification 1-dimensional modules for modular W-algebras for gl_n, which in turn this leads to a classification of minimal dimensional modules for reduced enveloping algebras for gl_n.

Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an

Euler-Poincare formula for the r-depth Bernstein projector. We establish an Euler-Poincare formula for the projector to an individual depth zero Bernstein component in terms of an equivariant system of Peter-Weyl idempotents of parahoric subgroups P associated to a block of the reductive quotient of P. This work is joint with Dan Barbasch and Dan Ciubotaru.

The local Langlands correspondence for classical groups gives a natural finite-to-one map between certain representations of p-adic classical groups and certain self-dual representations of the absolute Weil group of a p-adic field (and more). On both sides of the correspondence, the description of the representations involves a ``wild part'' of more arithmetic nature and a ``tame part'' of more geometric nature, and the notion of endo-parameter (due to Bushnell--Henniart for general linear groups) is designed to describe the ``wild part'' of the Langlands correspondence. I will explain what this means and the connection with representations of affine Hecke algebras. This is joint work with Blondel--Henniart, with Lust, and with Kurinczuk--Skodlerack.

For a group algebra over a self-injective ring

there are two stable categories: the usual one modulo projectives

and a relative one where one works modulo representations

which are free over the coefficient ring.

I'll describe the connection between these two stable categories,

which are "birational" in an appropriate sense.

I'll then make some comments on the specific case

where the coefficient ring is Z/nZ and give a more

precise description of the relative stable category.

What are the irreducible constituents of a smooth representation of a p-adic group that is constructed through parabolic induction? In the case of GL_n this is the study of the multiplicative behaviour of irreducible representations in the Bernstein-Zelevinski ring. Strikingly, the same decomposition problem can be reformulated through various Lie-theoretic settings of type A, such as canonical bases in quantum groups, representations of affine Hecke algebras, quantum affine Lie algebras, or more recently, KLR algebras. While partially touching on some of these phenomena, I will present new results on the problem using mostly classical tools. In particular, we will see how introducing a width invariant to an irreducible representation can circumvent the complexity involved in computations of Kazhdan-Lusztig polynomials.

It has long been expected, and is now proved in many important cases,

that quantum algebras are more rigid than their classical limits. That is, they

have much smaller automorphism groups. This begs the question of whether this

broken symmetry can be recovered.

I will outline an approach to this question using the ideas of noncommutative

projective geometry, from which we see that the correct object to study is a

groupoid, rather than a group, and maps in this groupoid are the replacement

for automorphisms. I will illustrate this with the example of quantum

projective space.

This is joint work with Nicholas Cooney (Clermont-Ferrand).

In this talk, we consider a split connected semisimple group G defined over a global field F. Let A denote the ring of adèles of F and K a maximal compact subgroup of G(A) with the property that the local factors of K are hyperspecial at every non-archimedian place. Our interest is to study a certain subspace of the space of square-integrable functions on the adelic quotient G(F)\G(A). Namely, we want to study functions coming from induced representations from an unramified character of a Borel subgroup and which are K-invariant.

Our goal is to describe how the decomposition of such space can be related with the Plancherel decomposition of a graded affine Hecke algebra (GAHA).

The main ingredients are standard analytic properties of the Dedekind zeta-function as well as known properties of the so-called residue distributions, introduced by Heckman-Opdam in their study of the Plancherel decomposition of a GAHA and a result by M. Reeder on the support of the weight spaces of

the anti-spherical discrete series representations of affine Hecke algebras. These last ingredients are of a purely local nature.

This talk is based on joint work with V. Heiermann and E. Opdam.

Given a complex vector space $V$, consider the ring $R_{a,b}(V)$ of polynomial functions on the space of configurations of $a$ vectors and $b$ covectors which are invariant under the natural action of $SL(V)$. Rings of this type play a central role in representation theory, and their study dates back to Hilbert. Over the last three decades, different bases of these spaces with remarkable properties were found. To explicitly construct, as well as to compare, some of these bases remains a challenging problem, already open when $V$ is 3-dimensional.

In this talk, I report on recent developments in the 3-dimensional setting of this theory.