Representation Theory Papers of the Week

We use an automorphic L-function over one number field as a probe for the functorial structure of another automorphic L-function over a possibly different field.

We therefore adjoin a specified cuspidal L-function as a probe and study the resulting raw mixed correlations. Assuming a factorization into cuspidal L-functions over Q, we show that a mixed pair statistic detects the probe's occurrence and recovers its multiplicity.

We prove that every infinite-dimensional admissible smooth representation of $\mathrm{GL}_2(K)$ or $D^\times$ with a central character and satisfying a Casimir annihilation condition has Gelfand--Kirillov dimension $n$.

We prove the Díaz$\unicode[Arial]{x2013}$Park conjecture for sharpness of saturated fusion systems.

We prove signed positivity for the expansion of a parabolic hybrid basis in the Kazhdan-Lusztig basis of a Hecke algebra.

We construct a non-admissible smooth irreducible representation of $\mathrm{GL}_2(F)$ over $\overline{\mathbb{F}}_p$.

We offer a simple explanation in terms of self-duality under asymmetric orbifolds.

Representations of quivers over the virtual field $\mathbb{F}_1$ can be naturally extended to bound quivers with monomial and commutativity relations. In this paper, we introduce and study the Hall algebra $\mathcal{H}_{\mathbb{F}_1}(Q,I)$ for a bound quiver $(Q,I)$ equipped with such relations.

We prove that $\mathbf D^+_{\mathrm{fg},W}(R)=\operatorname{Loc}_{\mathrm{ex}}(\mathbf D^b_{\mathrm{fg},W}(R))$ whenever $W\subseteq\operatorname{Spec} R$ is specialization closed and has finite dimension.

For each pair of integers $m\ge 1$ and $n\ge 2$, we construct a finite inverse semigroup $S_{m,n}\subseteq M_m(\mathbb{C})$ and explicitly determine its marginal inverse subsemigroup $M_{x^n}(S_{m,n})$ associated with the word $x^n$.

For unipotent representations, we describe the maximal rational wavefront sets in terms of Lusztig's canonical quotient and their behavior under Alvis--Curtis duality.

In general, we show that $K_0$-finite holomorphic functions near $M$ are regular functions on $K$-orbits, and that $M$ carries non-constant holomorphic germs if some $K$-stable prime divisor of $Z$ misses $\kappa$, and, under a connectedness hypothesis, only constants otherwise.

We investigate the structure of locally analytic representations of $p$-adic groups, and establish a sufficient condition for the irreducibility of locally analytic vectors in certain Banach representations of $D^\times$, where $D$ is the quaternion division algebra over $\mathbb{Q}_p$.

For each integer $l\ge1$, we give closed formulas for the local energies on tensor squares of the level-$l$ perfect crystals of types $D_4^{(3)}$ and $G_2^{(1)}$, normalized to vanish at the all-zero coordinate tensors.

We characterize the weights of minimal singular vectors of $V^\kappa(\mathfrak{g})$ with respect to the generalized Cartan matrix, when $V^\kappa(\mathfrak{g})$ is not simple and $\kappa$ is not critical.

We study how Koszul duality relates two kinds of boundary constructions associated with a smooth category and a proper category.

Three constructions of vertex operator algebras realizing the Haagerup center were announced independently and almost simultaneously. The relations between these constructions are studied in this paper.

We classify the connected components of support $\tau$-tilting graphs of surface rigid algebras associated with partial tagged triangulations of punctured marked surfaces with nonempty boundary.

In this paper, we study quasi-projective dimensions of modules over rings connected by homological ring epimorphisms.