We investigate the action of the affine Yangian $Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1)$ on the singular cohomology of the moduli space of stable sheaves $\mathcal{M}$ on a smooth projective surface $S$. Through an intersection-theoretic analysis of nested moduli spaces, we obtain a proof of the representation $$Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1) \curvearrowright H_{\mathcal{M}}.$$ This intersection-theoretic approach extends classical constructions to the full Yangian structure and circumvents the $K$-theoretic machinery traditionally required to establish this action.
Let $F=\mathbb{F}_q((u))((t))$ and $H=\mathrm{GL}_n(F)_L\times\mathrm{GL}_n(F)_R$. An ordered tuple $\alpha=(\alpha_1,\dots, \alpha_n) $ defines a cross-$K_2$ multiplier on the doubled Borel. Twisted equivariantization then yields an exact uniformly smooth categorical principal series $2$-representation of $H$. For uni...
Nearly $\mathrm{G}_2$ manifolds are positive Einstein seven-dimensional manifolds whose associated cone metric has special holonomy equal to, or contained in, $\mathrm{Spin}(7)$. We study nearly parallel $\mathrm{G}_2$-structures invariant under a cohomogeneity-one action of $\mathrm{SO}(4)$, and construct the first in...
We compute the homological Euler characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces compu...
We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to \[ \pi:(\mathbb{S}^{2n+1},g_\epsilon)\longrightarrow(\mathbb{C}P^n,h), \] where $g_\epsilon$ is a Berger metri...
We study the $\mathbb{P}^1$-local motivic homotopy theory under the Nisnevich topology on $Sm_S$. We show that although the homotopy category $\mathcal{H}^{\mathbb{P}^1}(-)$ admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On the unstable side, we...
We determine the branching of irreducible unitary representations of the basic classical Lie superalgebras $\mathfrak{sl}(m|n)$, $\mathfrak{osp}(m|2n)$, and $F(4)$ under restriction to their even subalgebras. The proof combines Dirac inequalities for the Huang-Pand\v{z}i\'{c}-Dirac operator $\operatorname{D}_{\mathfrak...
Steffen Schmidt· 0 citations
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