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We show that relative Property (T) for the abelianization of a nilpotent normal subgroup implies relative Property (T) for the subgroup itself. This and other results are a consequence of a theorem of independent interest, which states that if $H$ is a closed subgroup of a locally compact group $G$, and $A$ is a closed subgroup of the center of $H$, such that $A$ is normal in $G$, and $(G/A, H/A)$ has relative Property (T), then $(G, H^{(1)})$ has relative Property (T), where $H^{(1)}$ is the closure of the commutator subgroup of $H$. In fact, the assumption that $A$ is in the center of $H$ can be replaced with the weaker assumption that $A$ is abelian and every $H$-invariant finite measure on the unitary dual of $A$ is supported on the set of fixed points.
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Suppose that $G$ is a finite group and $H$ is a nilpotent subgroup of $G$. If a character of $H$ induces an irreducible character of $G$, then the generalized Fitting subgroup of $G$ is nilpotent.
We prove Clifford theoretic results on the representations of finite groups which only hold in characteristic $2$. Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $varphi$ be an irreducible $2$-Brauer character of $N$ which is self-dual. We prove that there is a unique self-dual irreducible Brauer character $theta$ of $G$ such that $varphi$ occurs with odd multiplicity in the restriction of $theta$ to $N$. Moreover this multiplicity is $1$. Conversely if $theta$ is an irreducible $2$-Brauer character of $G$ which is self-dual but not of quadratic type, the restriction of $theta$ to $N$ is a sum of distinct self-dual irreducible Brauer character of $N$, none of which have quadratic type. Let $b$ be a real $2$-block of $N$. We show that there is a unique real $2$-block of $G$ covering $b$ which is weakly regular.
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $mathrm{F}_4$, where we observe that the parametrization is uniform over good primes $p > 3$, but differs for the bad prime $p = 3$. We also explain how it has been applied for all groups of rank $4$ or less.
We classify all triples $(G,V,H)$ such that $SL_n(q)leq Gleq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $Vdar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.
The notion of strong 1-boundedness for finite von Neumann algebras was introduced by Jung. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all Property (T) von Neumann algebras with finite dimensional center and group von Neumann algebras of Property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung, and Shlyakhtenko. We also give a new proof of a result of Shlyakhtenko which states that if $G$ is a sofic, finitely presented group with vanishing first $ell^2$-Betti number, then $L(G)$ is strongly 1-bounded. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.
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