We use a gluing procedure introduced by Ginzburg to describe the relative Langlands dual of the hyperspherical Hamiltonian $(\mathrm{GL}_n \times \mathrm{GL}_m)$-variety $T^*(\mathrm{Hom}(\mathbb{C}^m,\mathbb{C}^n))$, and in particular the Rankin-Selberg case $m=n$. We show that the dual is isomorphic to the triangle part of Cherkis-Nakajima-Takayama bow varieties, recovering a result of Nakajima. Following a suggestion of Ginzburg, we explain how to modify the gluing so that the dual Hamiltonian variety of $T^*\mathbf{N}$, for any finite-dimensional representation $\mathbf{N}$ of a complex reductive group $G$, is naturally equipped with an anti-symplectic involution, and give an explicit formula for this involution in the Rankin-Selberg case.
We study localized homological mirror symmetry associated to an immersed Lagrangian brane $\mathbb{L}$, possibly equipped with a higher rank flat bundle, of a symplectic manifold $X$. Under a certain finiteness assumption on the Floer theory of $\mathbb{L}$, we deduce a quasi-equivalence $\mathcal{D}\mathrm{Fuk}_\mathbb{L}(X) \cong \mathcal{D}_{\mathrm{fd}}(\tilde{\mathcal A}_\mathbb{L})$ using Koszul duality, where $\tilde{\mathcal{A}}_{\mathbb{L}}$ is the dual differential graded quiver algebra called the extended localized mirror. We apply this to obtain some HMS results for plumbings of cotangent bundles of spheres, and to reproduce known results of split-generation of compact objects. In the second part of the paper, we consider bulk deformation cycles of $X$ that have non-trivial intersections with $\mathbb{L}$. This gives rise to noncommutative deformations of the mirror. When applied to (framed) plumbings, we obtain mirror functors to deformed preprojective algebras (or Nakajima quiver varieties at a general complex moment-map level). For the ADHM and affine $ADE$-type immersions, our construction produces mirror functors to the noncommutative spaces studied by Kapustin-Kuznetsov-Orlov, Baranovsky-Ginzburg-Kuznetsov and Kawamata.
Let $X = \mathbb{P}^1 \smallsetminus \{0,1,\infty\}$ be the thrice-punctured over a ring of $S$-integers $\mathcal{O}_{K,S}$ in a number field~$K$. For any quotient $\pi_1^{\mathrm{mot}}(X,0) \twoheadrightarrow \Pi$ of Deligne--Goncharov's motivic fundamental group there is an associated Selmer scheme which parametrises $\Pi$-torsors with a mixed Tate motive structure. We give several descriptions of the motivic Selmer scheme which make it amenable to computations, using $\mathbb{G}_m$-equivariant cocycles of algebraic groups, Lie algebras, and complete Hopf algebras. We prove that the Selmer scheme is isomorphic to an affine space $\mathbb{A}^N_{\mathbb{Q}}$, and we construct coordinates realising this isomorphism. This is a key ingredient for making the motivic Chabauty--Kim method explicit in a general setting, without restrictions on the base field or the choice of fundamental group quotient.
Let $S$ be a closed oriented surface of genus $g\geq 2$. We consider an $n$-pleated representation $\rho: \pi_1(S) \to \mathrm{PSL}_n(\mathbb{C})$ obtained by bending a Hitchin representation $\rho_0:\pi_1(S) \to \mathrm{PSL}_n(\mathbb{R})$ along a maximal geodesic lamination. The space of such $n$-pleated representations was recently introduced by Maloni-Martone-Mazzoli-Zhang who provided a parametrization via shear-bend cocycles. Our first result is that $\rho_0$ dominates $\rho$ in the Hilbert length spectrum and the translation-length spectrum; this generalizes our earlier result for finite laminations on punctured surfaces. Using this, we prove entropy rigidity results: namely, the Hilbert entropy of a quasi-Hitchin representation in the bending fiber is strictly greater than that of $\rho_0$, and the same for the translation-length entropy in the case that $\rho_0$ is $n$-Fuchsian. The proof involves analyzing the weighted planar networks for finite approximants of the monodromy matrix, and establishing a strict domination for \emph{most} curves using the equidistribution of closed geodesics in the unit tangent bundle of $S$.
We extend the multisymplectic reduction theory of Blacker -- itself the extension of Marsden--Weinstein--Meyer reduction to $k$-plectic manifolds -- to the setting of \emph{relative} multisymplectic geometry, in which a smooth map $F\colon M\to N$ carries a closed nondegenerate relative $(k{+}1)$-form $\varpi=(\omega,\eta)$ in the mapping-cone complex of $F$. We introduce the Leibniz algebra of relative Hamiltonian pairs and the associated relative moment maps $\mu\in\Omega^{k-1}(F, \mathfrak{g}^{*})$, and prove a relative multisymplectic reduction theorem: for a closed equivariant level $\phi\in\Omega^{k-1}(F, \mathfrak{g}^{*})$, the level pair of $\mu$ descends to a reduced smooth map $F_\phi\colon M_\phi\to N_\phi$ carrying a unique reduced closed relative form $\varpi_\phi$; remarkably, the horizontality of the trivializing component is forced by the relative closedness of the level. We further prove relative analogues of the reduction of dynamics, of the structure theory of split moment maps $\mu=\nu\cdot\kappa$ -- for which the splitting datum is a one-step cocycle in the relative Cartan model, so that split relative Hamiltonian $G$-spaces carry canonical relative homotopy moment maps -- and of the Duistermaat--Heckman-type variation formula: with respect to suitable conjugate distributions, the reduced relative class satisfies $\partial_\lambda[\varpi_\psi]=\langle c, \lambda\rangle \cdot[\kappa_\psi]$, where $c$ is the Chern form of the target model bundle acting through the natural $\Omega(N)$-module structure of the mapping cone. Finally, we transport the exact stationary phase approximation to the target component and prove a genuinely relative localization constraint: the fixed-point contributions of the pulled-back split package on the source cancel identically.
In this paper, we construct a lifting operator from the Grothendieck group of admissible Harish-Chandra modules of $G =\mathrm{SO}_{2n}(\mathbb C)$ (resp. $\mathrm{Sp}_{2n}(\mathbb C)$) to that of genuine representations of $\mathrm{Spin}_{2n}(\mathbb C)$ (resp. $\mathrm{Spin}_{2n+1}(\mathbb C)$). We determine the lift of the sum of special unipotent representations attached to any ${}^{\vee}\mathcal O \subseteq {}^{\vee}\mathfrak g$ explicitly. In particular, the representations occurring in these lifts, if nonzero, are genuine unipotent representations of complex Spin groups and are unitary. As a consequence, the lifting operator preserves unitarity on a large class of unitary representations.
We prove a Haglund--Haiman--Loehr type combinatorial formula for the torus fixed point classes $I_{\mu,w}$ in the equivariant $K$-theory $K_{\mathbb{C}^{*}\times\mathbb{C}^{*}}(\operatorname{PFH}_{n,n-k})$ of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When $w=\emptyset$, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson--Gorsky--Mellit action of the Carlsson--Mellit algebra $\mathbb{A}_{q,t}$.
Younggwang Cho, Jaeseong Oh· 0 citations
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