arXiv subjects
Publications and source records attributed to Taeuk Nam.
We construct three tamely ramified local equivalences of factorization module categories. The first is a factorization version of the Arkhipov-Bezrukavnikov equivalence at a point. The second is a factorization version of the Bezrukavnikov equivalence at a point. The third is an Iwahori-ramified version of the factorizable Fundamental Local Equivalence.
Drinfeld and Gaitsgory proved that $\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G)$ is compactly generated. Let $\mathrm{Bun}_G^{\mathrm{I}}$ be the algebraic stack of principal $G$-bundles on $X$ together with Iwahori level structure at a fixed point $x \in X$. More generally, for a finite collection of points $x_1, ..., x_k \in X$, let $\mathrm{Bun}_G^{(\mathrm{I}; x_1, ..., x_k)}$ be the algebraic stack of principal $G$-bundles on $X$ together with Iwahori level structure at each point $x_j$. We will show that $\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G^{\mathrm{I}})$ and $\mathrm{D}-\mathrm{mod}(\mathrm{Bun}_G^{(\mathrm{I}; x_1, ..., x_k)})$ are compactly generated.
Let $k$ be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra $\operatorname{Mat}_{n \times n}(k)$ can be endowed with a $k$-linear involution in one way if $n$ is odd and in two ways if $n$ is even. In this paper, we consider $r$-tuples $A_\bullet \in \operatorname{Mat}_{n\times n}(k)^r$ such that the entries of $A_\bullet$ fail to generate $\operatorname{Mat}_{n\times n}(k)$ as an algebra with involution. We show that the locus of such $r$-tuples forms a closed subvariety $Z(r;V)$ of $\operatorname{Mat}_{n\times n}(k)^r$ that is not irreducible. We describe the irreducible components and we calculate the dimension of the largest component of $Z(r;V)$ in all cases. This gives a numerical answer to the question of how generic it is for an $r$-tuple $(a_1, \dots, a_r)$ of elements in $\operatorname{Mat}_{n\times n}(k)$ to generate it as an algebra with involution.
In this paper, we show that, under certain assumptions, a parabolic induction $Ind_B^Gλ$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \[ Ind_B^Gλ= \left(Ind_P^G St_M\otimes χ_0\right) \oplus \left(Ind_P^G \mathbf{1}_M\otimes χ_0\right), \] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\mathbf{1}_M$ is the trivial representation of $M$, $St_M$ is the Steinberg representation of $M$ and $χ_0$ is a certain character of $M$. We construct examples of this phenomenon for all simply-connected simple groups of rank at least $2$.