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In this article we study the principal block of the category of real Harish-Chandra modules for the group $\mathsf{SL}_2(\RR)$ and relate it to the category of finite dimensional modules over the so-called real Gelfand order. We describe…

For simply-connected Lie groups of type E over \( p \)-adic local field \( F \), we determine the degree of the cover required for a given \( F \)-split nilpotent orbit to be quasi-admissible or raisable, respectively. Combining this result…

Let $k$ be an algebraically closed field of a prime characteristic $p$. Let $G$ be a finite group. We investigate the Brauer indecomposability of Scott $kG$-modules in relation to the kernel of modules. We generalize a criterion for Brauer…

The Delannoy category is an interesting pre-Tannakian category associated to the oligomorphic group $\mathbb{G}$ of automorphisms of the totally ordered set $(\mathbf{R}, <)$. By construction, it admits some obvious simple commutative…

Let $p>0$ be a prime, $G$ be a finite $p$-group and $\Bbbk$ be an algebraically closed field of characteristic $p$. Dave Benson has conjectured that if $p=2$ and $V$ is an odd-dimensional indecomposable representation of $G$ then all…

We establish conditions on a family of coproduct-preserving tt-functors $f_i\colon \mathcal{T}\to \mathcal{T}_i$ between tt-categories with small coproducts, ensuring that the localizing tensor ideal generated by an object $x \in…

We review $\mathrm{Hom}$-infinite Frobenius categorification of cluster algebras with coefficients and use it to give two applications of Jensen--King--Su's Frobenius categorification of the Grassmannian: 1) we determine the $g$-vectors of…

Biserial algebras are a classical class in the representation theory of algebras, generalizing Nakayama algebras. They were further generalized by Green and Schroll to multiserial algebras, which share many structural properties with…

A LAnKe (also known as a Filippov algebra or a Lie algebra of the $n$-th kind) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown…

We extend the local Langlands conjectures to a certain class of disconnected groups, allowing non-abelian component groups, and recast in this language some aspects of twisted endoscopy. We further introduce normalized twisted transfer…

We prove that if the space of newforms is non-zero for every irreducible generic supercuspidal representation of ${\rm SO}_{2n+1}$ then it is also non-zero for all irreducible generic representations of ${\rm SO}_{2n+1}$.

We study the harmonic representation of $SU(p,q)$ in connection to the complex Weyl correspondence on the Fock space. In particular, we give explicit formulas for the complex Weyl symbols of the harmonic representation operators. Similar…

For a ring $A$ there is a well-known duality between definable subcategories of right $A$-modules and definable subcategories of left $A$ modules. This is a consequence of Auslander-Gruson-Jensen duality $\rm…

We introduce a one-skeleton path model for Mirkovic-Vilonen polytopes in type A_n. We prove that the Minkowski sum of (MV) polytopes corresponds to the concatenation of one-skeleton paths of this model. We show that MV polytopes induced by…

In this article, we examine the restriction of the metaplectic representation $\pi$ over a $p$-adic field $k$, $p\neq2$, of zero characteristic to an isotropic torus $S$ contained in the symplectic group. First we give necessary and…

Friezes patterns are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their…

We identify an interesting special class of prime ideals in the finitary infinite symmetric group algebra. We show that the set of such ideals carries a semiring structure. Over the complex numbers, we establish a connection with spherical…

Let $(\mathbb{G},\mathbb{H})$ be a symmetric pair of reductive groups over a $p$-adic field with $p\neq 2$, attached to the involution $\theta$. Under the assumption that there exists a maximally $\theta$-split torus in $\mathbb{G}$, which…

We present a formula for trigonometric orthosymplectic $R$-matrices associated with any parity sequence, and establish their factorization into the ordered product of $q$-exponents parametrized by positive roots in the corresponding reduced…

In this note, we establish the periodicity of chains of affine standard Lyndon words in all types and determine tight bounds on that periodicity, greatly generalizing the $A$-type results of arXiv:2305.16299. Our approach crucially utilizes…