Unipotent representations attached to the principal nilpotent orbit

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the K \mathbf {K} -types, associated varieties, and Langlands parameters of all such representations.

W algebras, cosets and VOAs for 4d $$ \mathcal{N} $$ = 2 SCFTs from M5 branes

We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the W algebras defined using nilpotent orbit with partition [qm, 1s]. Gauging above AD matters, we can find VOAs for more general $$ \mathcal{N} $$ N = 2 SCFTs engineered from 6d (2, 0) theories. For example, the VOA for general (AN − 1, Ak − 1) theory is found as the coset of a collection of above W algebras. Various new interesting properties of 2d VOAs such as level-rank duality, conformal embedding, collapsing levels, coset constructions for known VOAs can be derived from 4d theory.

Support varieties of baby Verma modules

Let [Formula: see text] be a connected reductive algebraic group over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text]. For a given nilpotent [Formula: see text]-character [Formula: see text], let [Formula: see text] be a baby Verma module associated with a restricted weight [Formula: see text]. A conjecture describing the support variety of [Formula: see text] via that of its restricted counterpart is given: [Formula: see text]. Under the assumption of [Formula: see text](the Coxeter number) and [Formula: see text] [Formula: see text]-regular, this conjecture is proved when [Formula: see text] falls in the regular nilpotent orbit for any [Formula: see text] and the subregular nilpotent orbit for [Formula: see text] being of type [Formula: see text]. We also verify this conjecture whenever [Formula: see text] is of type [Formula: see text] and [Formula: see text] falls in the minimal nilpotent orbit.

Equations for Some Nilpotent Varieties

Let ${\mathcal{O}}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ of rank $n$ over $\mathbb C$, induced from a Levi subalgebra whose $s$ simple roots are orthogonal, short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{\mathcal{O}}$ in $S \mathfrak{g}^{\ast }$. In such cases, the ideal is generated by bases of either one or two copies of the representation whose highest weight is the dominant short root, along with $n-s$ fundamental invariants of $S \mathfrak{g}^{\ast }$. This extends Broer’s result for the subregular nilpotent orbit, which is the case of $s=1$. Along the way we give another proof of Broer’s result that $\overline{\mathcal{O}}$ is normal. We also prove a result relating a property of the invariants of $S \mathfrak{g}^{\ast }$ to the following question: when does a copy of the adjoint representation in $S \mathfrak{g}^{\ast }$ belong to the ideal in $S \mathfrak{g}^{\ast }$ generated by another copy of the adjoint representation together with the invariants of $S \mathfrak{g}^{\ast }$?

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.