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Register a new userSome properties for morphism of representations
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Let $psi : Gto GL(V)$ and $varphi :G to GL (W)$ be representations of finite group $G$. A linear map $T: Vto W$ is called a morphism from $psi$ to $varphi$ if it satisfys $Tpsi_g= varphi_g T$ for each $gin G$ and let $mathrm{Hom}_G (psi ,varphi)$ denote the set of all morphisms. In this paper, we make full stufy of the subspace $mathrm{Hom}_G(psi, varphi)$. As byproducts, we include the proof of the first orthogonality relation and Schurs orthogonality relation.
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We give an explicit construction of test vectors for $T$-equivariant linear functionals on representations $Pi$ of $GL_2$ of a $p$-adic field $F$, where $T$ is a non-split torus. Of particular interest is the case when both the representations are ramified; we completely solve this problem for principal series and Steinberg representations of $GL_2$, as well as for depth zero supercuspidals over $mathbf{Q}_p$. A key ingredient is a theorem of Casselman and Silberger, which allows us to quickly reduce almost all cases to that of the principal series, which can be analyzed directly. Our method shows that the only genuinely difficult cases are the characters of $T$ which occur in the primitive part (or type) of $Pi$ when $Pi$ is supercuspidal. The method to handle the depth zero case is based on modular representation theory, motivated by considerations from Deligne-Lusztig theory and the de Rham cohomology of Deligne-Lusztig-Drinfeld curves. The proof also reveals some interesting features related to the Langlands correspondence in characteristic $p$. We show in particular that the test vector problem has an obstruction in characteristic $p$ beyond the root number criterion of Waldspurger and Tunnell, and exhibits an unexpected dichotomy related to the weights in Serres conjecture and the signs of standard Gauss sums.
Let $K/F$ be a quadratic extension of $p$-adic fields, $sigma$ the nontrivial element of the Galois group of $K$ over $F$, and $pi$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $pi^{vee}$ the smooth contragredient of $pi$, and by $pi^{sigma}$ the representation $picirc sigma$, we show that the representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $pi^vee otimes pi^sigma$ is distinguished with respect to $GL(2n,F)$. This is a step towards the classification of distinguished generic representations of general linear groups over $p$-adic fields.
This paper explores various homological regularity phenomena (in the sense of Auslander) in category $mathcal{O}$ and its several variations and generalizations. Additionally, we address the problem of determining projective dimension of twisted and shuffled projective and tilting modules.
Let $mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $mathfrak{g}$ over $mathbb C$, induced from a Levi subalgebra whose 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 $S mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broers result for the subregular nilpotent orbit. Along the way we give another proof of Broers result that $overline{ mathcal{O}}$ is normal. We also prove a result connecting a property of invariants related to flat bases to the question of when one copy of the adjoint representation is in the ideal in $S mathfrak{g}^*$ generated by another copy of the adjoint representation and the fundamental invariants.
Let $F$ be either $mathbb{R}$ or a finite extension of $mathbb{Q}_p$, and let $G$ be a finite central extension of the group of $F$-points of a reductive group defined over $F$. Also let $pi$ be a smooth representation of $G$ (Frechet of moderate growth if $F=mathbb{R}$). For each nilpotent orbit $mathcal{O}$ we consider a certain Whittaker quotient $pi_{mathcal{O}}$ of $pi$. We define the Whittaker support WS$(pi)$ to be the set of maximal $mathcal{O}$ among those for which $pi_{mathcal{O}} eq 0$. In this paper we prove that all $mathcal{O}inmathrm{WS}(pi)$ are quasi-admissible nilpotent orbits, generalizing some of the results in [Moe96,JLS16]. If $F$ is $p$-adic and $pi$ is quasi-cuspidal then we show that all $mathcal{O}inmathrm{WS}(pi)$ are $F$-distinguished, i.e. do not intersect the Lie algebra of any proper Levi subgroup of $G$ defined over $F$. We also give an adaptation of our argument to automorphic representations, generalizing some results from [GRS03,Shen16,JLS16,Cai] and confirming some conjectures from [Ginz06]. Our methods are a synergy of the methods of the above-mentioned papers, and of our preceding paper [GGS17].
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