Related papers: Matrix coefficients of intertwining operators and …

Given a $K$-vector space $V$, let $\sigma(V,K)$ denote the covering number, i.e. the smallest (cardinal) number of proper subspaces whose union covers $V$. Analogously, define $\sigma(M,R)$ for a module $M$ over a unital commutative ring…

We show that there are precisely two, up to conjugation, anti-involutions sigma_{\pm} of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight…

In this paper, we provide necessary and sufficient conditions on a triple of weights $(u,v,w)$ so that the $t$-Haar multipliers $T^t_{w,\sigma}$, $t\in \R$, %defined in \cite{P} when $\sigma=1$, are uniformly (on the choice of signs…

We suggest a two-matrix model depending on three (infinite) sets of parameters which interpolates between all the models proposed in arXiv:2206.13038, and defined there through $W$-representations. We also discuss further generalizations of…

For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path…

We extend the old formalism of cut-and-join operators in the theory of Hurwitz $\tau$-functions to description of a wide family of KP-integrable {\it skew} Hurwitz $\tau$-functions, which include, in particular, the newly discovered…

For $w$ in the symmetric group $S_n$, let $\widetilde C_w$ be the corresponding modified, signless Kazhdan--Lusztig basis element of the type-$A$ Hecke algebra $H_n(q)$. An extension [Ann. Comb. 25, no. 3 (2021) pp. 757--787] of a result of…

For an untwisted affine Kac-Moody Lie algebra $\tilde{\mathfrak g}$, and a given positive integer level $k$, vertex operators $x(z)=\sum x(n)z^{-n-1}$, $x\in\mathfrak g$, generate a vertex operator algebra $V$. For the maximal root $\theta$…

Convolution admits a natural formulation as a functional operation on matrices. Motivated by the functional and entrywise calculi, this leads to a framework in which convolution defines a matrix transform that preserves positivity. Within…

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix} 1 & 0 \\ u & 1 \end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}$. Let $S_{u,v}$ be the monoid generated by $L_u$ and $R_v$, and $G_{u,v}$ be the group generated…

We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the…

Let $F$ be locally compact field with residue characteristic $p$, and $\mathbf{G}$ a connected reductive $F$-group. Let $\mathcal{U}$ be a pro-$p$ Iwahori subgroup of $G = \mathbf{G}(F)$. Fix a commutative ring $R$. If $\pi$ is a smooth…

Let $\tilde{G}$ be a symplectic or even orthogonal group over a p-adic field $F$, and $M$ the Levi factor of a maximal parabolic subgroup of $\tilde{G}$. Suppose that $M$ has the shape of three blocks of the same size. Let $\pi$ be a…

We calculate the $(\mathfrak{g},K)$ module structure for the principal series representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a hypergeometric generating function together with an inverse Mellin transform technique as an…

We introduce ring theoretic constructions that are similar to the construction of wreath product of groups. In particular, for a given graph $\Gamma=(V,E)$ and an associate algebra $A,$ we construct an algebra $B=A\, wr\, L(\Gamma)$ with…

Given a sequence of deterministic matrices $A = A_N$ and a sequence of deterministic nonnegative matrices $\Sigma=\Sigma_N$ such that $A\to a$ and $\Sigma\to \sigma$ in $\ast$-distribution for some operators $a$ and $\sigma$ in a finite von…

In this paper we characterize the inequality \begin{equation*} \bigg( \int_0^{\infty} \bigg( \int_0^x \big[ T_{u,b}f^* (t)\big]^r\,dt\bigg)^{\frac{q}{r}} w(x)\,dx\bigg)^{\frac{1}{q}} \le C \, \bigg( \int_0^{\infty} \bigg( \int_0^x [f^*…

Let W be an irreducible, finitely generated Coxeter group. The geometric representation provides an discrete embedding in the orthogonal group of the so-called Tits form. One can look at the representation modulo the kernel of this form; we…

We view all smooth metrics $g$ on a closed surface $\Sigma$ through their Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}}, \tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We…

In this paper we derive new sufficient conditions for a linear system matrix $$S(\lambda):=\left[\begin{array}{ccc} T(\lambda) & -U(\lambda) \\ V(\lambda) & W(\lambda) \end{array}\right],$$ where $T(\lambda)$ is assumed regular, to be…