Let $\mathfrak g$ be a semisimple Lie algebra. A $\mathfrak g$-algebra is an associative algebra $R$ on which $\mathfrak g$ acts by derivations. There are several significant examples. Let $V$ a finite dimensional $\mathfrak g$ module and take $R=\mathrm{End} V$ or $R=D(V)$ being the ring of derivations on $V$ . Again take $R=U(\mathfrak g)$. In all these cases $ S=U(\mathfrak g)\otimes R$ is again a $\mathfrak g$-algebra. Finally let $T$ denote the subalgebra of invariants of $S$.

For the first choice of $R$ above the representation theory of $T$ can be rather explicitly described in terms of Kazhdan-Lusztig polynomials. In the second case the simple $T$ modules can be described in terms of the simple $D(V)$ modules. In the third case it is shown that all simple $T$ modules are finite dimensional, despite the fact that $T$ is not a PI ring, except for the case $\mathfrak g =\mathfrak {sl}(2)$.

# Past Algebra Seminar

A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

The Peter-Weyl idempotent of a parahoric subgroup is the sum of the idempotents of irreducible representations which have a nonzero Iwahori fixed vector. The associated convolution algebra is called a Peter-Weyl Iwahori algebra. We show any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra. Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algebra have a natural C*-algebra structure, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution denoted as •, and the Morita equivalence preserves irreducible and unitary modules for the • involution. This work is joint with Dan Barbasch.

Let $F$ be a non-Archimedean local field with ring of integers $\mathcal O$ and maximal ideal $\mathfrak p$. T. Shintani and G. Hill independently introduced a large class of smooth representations of $GL_N(\mathcal O)$, called regular representations. Roughly speaking they correspond to elements in the Lie algebra $M_N(\mathcal O)$ which are regular mod $\mathfrak p$ (i.e, having centraliser of dimension $N$). The study of regular representations of $GL_N(\mathcal O)$ goes back to Shintani in the 1960s, and independently and later, Hill, who both constructed the regular representations with even conductor, but left the much harder case of odd conductor open. In recent simultaneous and independent work, Krakovski, Onn and Singla gave a construction of the regular representations of $GL_N(\mathcal O)$ when the residue characteristic of $\mathcal O$ is not $2$.

In this talk I will present a complete construction of all the regular representations of $GL_N(\mathcal O)$. The approach is analogous to, and motivated by, the construction of supercuspidal representations of $GL_N(F)$ due to Bushnell and Kutzko. This is joint work with Shaun Stevens.

It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb C)$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak g$ is any simple Lie algebra over $\mathbb C$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak g$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak g^L$.

It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal O_1<\mathcal O_2$ are adjacent in the partial order then so are their duals $\mathcal O_1^t>\mathcal O_2^t$, and the isolated singularity attached to the pair $(\mathcal O_1,\mathcal O_2)$ is dual to the singularity attached to $(\mathcal O_2^t,\mathcal O_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.

In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal O_1<\mathcal O_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.

In this joint work with D. Ciubotaru, we introduce the notion of local and global indices of Dirac operators for a rational Cherednik algebra H, with underlying reflection group G. In the local theory, I will report on some relations between the (local) Dirac index of a simple module in category O, the graded G-character and the composition series polynomials for standard modules. In the global theory, we introduce an "integral-reflection" module over which we define and compute the index of a (global) Dirac operator and show that the index is independent of the parameters. If time permits, I will discuss some local-global relations.

Pseudo-reductive groups are smooth connected linear algebraic groups over a field k whose k-defined unipotent radical is trivial. If k is perfect then all pseudo-reductive groups are reductive, but if k is imperfect (hence of characteristic p) then one gets a strictly larger collection of groups. They come up in a number of natural situations, not least when one wishes to say something about the simple representations of all smooth connected linear algebraic groups. Recent work by Conrad-Gabber-Prasad has made it possible to reduce the classification of the simple representations of pseudo-reductive groups to the split reductive case. I’ll explain how. This is joint work with Mike Bate.