Related papers: Multiple rational normal forms in Lie theory

Let $GL_M$ be general linear Lie group over the complex field. The irreducible rational representations of the group $GL_M$ are labeled by pairs of partitions $\mu$ and $\tilde\mu$ such that the total number of non-zero parts of $\mu$ and…

We study representations of simply-laced Weyl groups which are equipped with canonical bases. Our main result is that for a large class of representations, the separable elements of the Weyl group $W$ act on these canonical bases by…

We identify the $G(\mathbb R)$-orbits of the real locus $X(\mathbb R)$ of any spherical complex variety $X$ defined over $\mathbb R$ and homogeneous under a split connected reductive group $G$ defined also over $\mathbb R$. This is done by…

Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm…

Let $F$ be an algebraically closed field and consider the Lie algebra ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, where $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Refer to a ${\mathfrak…

We define an action of the Weyl group W of a simple Lie algebra g on a completion of the ring Y, which is the codomain of the q-character homomorphism of the corresponding quantum affine algebra U_q(g^). We prove that the subring of…

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $B_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb…

We consider the finite $W$-algebra $U(\g,e)$ associated to a nilpotent element $e \in \g$ in a simple complex Lie algebra $\g$ of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a…

Let G be the group of rational points of a connected reductive group over a finite field. Based on work of Lusztig and Yun, we make the Jordan decomposition for irreducible G-representations canonical. It comes in the form of an equivalence…

If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $\tau$ is a unitary representation of $H$, then the authors give a sufficient condition on $\xi\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^G\tau$.…

The class of generalized Lie-type color algebras contains the ones of generalized Lie-type algebras, of $n$-Lie algebras and superalgebras, commutative Leibniz $n$-ary algebras and superalgebras, among others. We focus on the class of…

The paper studies the dimensions of irreducible components of commuting varieties of (restricted) nilpotent $r$-tuples in a classical Lie algebra $\mathfrak{g}$ defined over an algebraically closed field $k$. As applications, we obtain some…

Let $G$ be a real semisimple Lie group with trivial centre and no compact factors. Given a conjugate pair of either real hyperbolic elements or unipotent elements $a$ and $b$ in $G$ we find a conjugating element $g \in G$ such that…

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in…

We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $G=\text{GL}(n)$. The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition…

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$,…

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent…

We examine the structure of the Levi component $MA$ in a minimal parabolic subgroup $P = MAN$ of a real reductive Lie group $G$ and work out the cases where $M$ is metabelian, equivalently where $\mathfrak{p}$ is solvable. When $G$ is a…

We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier…

For the flag variety G/B of a reductive algebraic group G we define a certain (set-theoretical) cross-section phi from G/B to G, which depends on a choice of reduced expression for the longest element in the Weyl group. This cross-section…