arXiv subjects
Publications and source records attributed to Swarnendu Datta.
Given a countable abelian group $A$, we construct a row finite directed graph $\Gamma(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$ is canonically isomorphic to $A$. Moreover, each element of $\textrm {Aut}(A)$ is a lift of an automorphism of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$.
Let $G$ be a finite pseudoreflection group, $Ω\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(Ω)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbolθ}$ on $\mathcal H,$ where $\{θ_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbolθ= (θ_1, \ldots, θ_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of non-trivial joint reducing subspaces, where $\widehat G$ is the set of all equivalence classes of irreducible representations of $G.$ We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra $\mathcal O(Ω)$ of holomorphic functions on $Ω$. As a consequence, we show that for each $\varrho\in \widehat G,$ the multiplication operator $\mathbf M_{\boldsymbolθ}$ on the reducing subspace $\mathbb P_\varrho \mathcal H$ can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of $\mathbb C^{(\mathrm{deg}\,\varrho)^2}$-valued holomorphic functions on $\boldsymbolθ(Ω)$. This, in turn, provides a description of the structure of joint reducing subspaces of the multiplication operator induced by a representative of a proper holomorphic map from a domain $Ω$ in $\mathbb C^n$ which is factored by automorphisms $G\subseteq {\rm Aut}(Ω).$
Let $G$ be a unipotent group over a field of characteristic $p > 0$. The theory of character sheaves on $G$ was initiated by V. Drinfeld and developed jointly with D. Boyarchenko. They also introduced the notion of $\mathbb{L}$-packets of character sheaves. Each $\mathbb{L}$-packet can be described in terms of a modular category. Now suppose that the nilpotence class of $G$ is less than $p$. Then the $\mathbb{L}$-packets are in bijection with the set $\mathfrak{g}^*/G$ of coadjoint orbits, where $\mathfrak{g}$ is the Lie ring scheme obtained from $G$ using the Lazard correspondence and $\mathfrak{g}^*$ is the Serre dual of $G$. If $Ω$ is a coadjoint orbit, then the corresponding modular category can be identified with the category of $G$-equivariant local systems on $Ω$. This in turn is equivalent to the category of finite dimensional representations of a finite group. However, the associativity, braiding and ribbon constraints are nontrivial. Drinfeld gave a conjectural description of these constraints in 2006. In this article, we prove the formula describing the ribbon structure when $\dim(Ω)$ is even.
Let $G$ be a connnected, unipotent, perfect group scheme over an algebraically closed field of characteristic p > 0. V. Drinfeld has defined a certain metric group associated to biextensions of $G \times G$ by the discrete group $Q_p/Z_p$. We prove a certain conjecture of Drinfeld regarding the class of this metric group in the Witt group.