Self-dual modules in characteristic two and normal subgroups

Hecke algebras of normalizers of parabolic subgroups

In the context of Hecke algebras of complex reflection groups, we prove that the generalized Hecke algebras of normalizers of parabolic subgroups are semidirect products, under suitable conditions on the parameters involved in their definition.

Covers for self-dual supercuspidal representations of the Siegel Levi subgroup of classical p-adic groups

We study components of the Bernstein category for a p-adic classical group (with p odd) with inertial support a self-dual positive level supercuspidal representation of a Siegel Levi subgroup. More precisely, we use the method of covers to construct a Bushnell-Kutzko type for such a component. A detailed knowledge of the Hecke algebra of the type should have number-theoretic implications.

Remarks on quantum unipotent subgroup and dual canonical basis

We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.

Self-dual representations with vectors fixed under an Iwahori subgroup

Let $G$ be the group of $F$-points of a split connected reductive $F$-group over a non-Archimedean local field $F$ of characteristic 0. Let $π$ be an irreducible smooth self-dual representation of $G$. The space $W$ of $π$ carries a non-degenerate $G$-invariant bilinear form $(\,,\,)$ which is unique up to scaling. The form is easily seen to be symmetric or skew-symmetric and we set $\varepsilon(π)=\pm 1$ accordingly. In this article, we show that $\varepsilon{(π)}=1$ when $π$ is a generic representation of $G$ with non-zero vectors fixed under an Iwahori subgroup $I$.

Character correspondeces in solvable groups with a self-normalizing Sylow subgroup

In 1973, I. M. Isaacs described a correspondence between characters of degree not divisible by a fixed prime $p$ of a finite solvable group $G$ and those of the normalizer of Sylow $p$-subgroup of $G$, whenever the index of the normalizer in $G$ is odd. This correspondence is natural in the sense that an algorithm is provided to compute it, and the result of the application of the algorithm does not depend on choices made. Later on, for $p$-solvable groups with self-normalizing Sylow $p$-subgroup, G. Navarro showed that every irreducible character of degree not divisible by $p$ has a unique li…

Representations induced from a normal subgroup of prime index

Let $G$ be a finite group, $H$ be a normal subgroup of prime index $p$. Let $F$ be a field of either characteristic $0$ or prime to $|G|$. Let $η$ be an irreducible $F$-representation of $H$. If $F$ is an algebraically closed field of characteristic either $0$ or prime to $|G|$, then the induced representation $η\uparrow^{G}_{H}$ is either irreducible or a direct sum of $p$ pairwise inequivalent irreducible representations. In this paper, we show that if $F$ is not assumed algebraically closed field, then there are five possibilities in the decomposition of induced representation into irreduci…

Normal Reflection Subgroups

We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalized exponents. Our refinement gives a uniform proof and generalization of a recent theorem of the second author.

On Self-Normalising Sylow $2$-Subgroups in Type A

Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalising Sylow $2$-subgroup, which is given in terms of the ordinary irreducible characters of $G$. The first-named author has reduced the proof of this conjecture to showing that certain related statements hold when $G$ is quasisimple. In this article we show that these conditions are satisfied when $G/Z(G)$ is $\mathrm{PSL}_n(q)$, $\mathrm{PSU}_n(q)$, or a simple group of Lie type defined over a finite field of characteristic $2$.

Quantum Unipotent Subgroup and dual canonical basis

Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical basis $S^{*}$. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras to the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$, which contains quantum flag minors and satisfies a factorizatio…

复反射群的正规反射子群

Normal Reflection Subgroups of Complex Reflection Groups

本研究聚焦复反射群的正规反射子群，通过新方法细化Orlik-Solomon定理，给出第二作者近期定理的统一证明及推广。