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1. Non-vanishing of geometric Whittaker coefficients for reductive groups

   https://doi.org/10.1090/jams/1051  

   Færgeman, Joakim; Raskin, Sam (April 2025, Journal of the American Mathematical Society)

   We prove that cuspidal automorphic $D$-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from $G{L}_n$to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are $t$-exact on the category of tempered $D$-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for $G{L}_n$. We also show that Whittaker coefficient functors are $t$-exact for sheaves with nilpotent singular support. An additional consequence of our results is that the tempered, restricted geometric Langlands conjecture must be $t$-exact. We apply our results to show that for suitably irreducible local systems, Whittaker-normalized Hecke eigensheaves are perverse sheaves that are irreducible on each connected component of ${\mathrm{Bun}}_G$. 

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2. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter; Zeitlin, Anton M. (September 2023, Communications in Mathematical Physics)

   Abstract We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$ $C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$ $C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$ $X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$ $l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ $ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$ $P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. 

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3. A support theorem for\break the Hitchin fibration: The case of GL _n and K _C

   https://doi.org/10.1515/crelle-2021-0045  

   de Cataldo, Mark Andrea; Heinloth, Jochen; Migliorini, Luca (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for ${\mathrm{GL}}_n${\mathrm{GL}_{n}}over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every $n\ge 2${n\geq{2}}. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare the fibration to the one defined over versal deformations of spectral curves. 

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4. Irregular loci in the Emerton–Gee stack for GL ₂

   https://doi.org/10.1515/crelle-2024-0045  

   Bellovin, Rebecca; Borade, Neelima; Hilado, Anton; Kansal, Kalyani; Lee, Heejong; Levin, Brandon; Savitt, David; Wiersema, Hanneke (July 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let $K/{\mathbf{Q}}_p$K/\mathbf{Q}_{p}be unramified.Inside the Emerton–Gee stack ${\mathcal{X}}_2$\mathcal{X}_{2}, one can consider the locus of two-dimensional mod 𝑝 representations of $\mathrm{Gal}\left(\bar{K}/K\right)$\mathrm{Gal}(\overline{K}/K)having a crystalline lift with specified Hodge–Tate weights.We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms.We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible.We also establish various inclusion relations between these loci. 

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5. The immersion poset on partitions

   https://doi.org/10.1007/s10801-025-01380-z  

   Johnston, Lisa; Kenepp, David; Nguyen, Evuilynn; Paul, Digjoy; Schilling, Anne; Simone, Mary_Claire; Zhou, Regina (February 2025, Journal of Algebraic Combinatorics)

   Abstract We introduce the immersion poset$$({\mathcal {P}}(n), \leqslant _I)$$ $(P\left(n\right),{⩽}_I)$on partitions, defined by$$\lambda \leqslant _I \mu $$ $\lambda {⩽}_I\mu$if and only if$$s_\mu (x_1, \ldots , x_N) - s_\lambda (x_1, \ldots , x_N)$$ $s_{\mu }(x_1,\dots ,x_N)-s_{\lambda }(x_1,\dots ,x_N)$is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of$$GL_N({\mathbb {C}})$$ $G{L}_N\left(C\right)$form an immersion pair, as defined by Prasad and Raghunathan [7]. We develop injections$$\textsf{SSYT}(\lambda , \nu ) \hookrightarrow \textsf{SSYT}(\mu , \nu )$$ $\mathrm{SSYT}(\lambda ,\nu )\hookrightarrow \mathrm{SSYT}(\mu ,\nu )$on semistandard Young tableaux given constraints on the shape of$$\lambda $$ $\lambda$, and present results on immersion relations among hook and two column partitions. The standard immersion poset$$({\mathcal {P}}(n), \leqslant _{std})$$ $(P\left(n\right),{⩽}_{\mathrm{std}})$is a refinement of the immersion poset, defined by$$\lambda \leqslant _{std} \mu $$ $\lambda {⩽}_{\mathrm{std}}\mu$if and only if$$\lambda \leqslant _D \mu $$ $\lambda {⩽}_D\mu$in dominance order and$$f^\lambda \leqslant f^\mu $$ $f^{\lambda }⩽f^{\mu }$, where$$f^\nu $$ $f^{\nu }$is the number of standard Young tableaux of shape$$\nu $$ $\nu$. We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram [12]. 

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