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Michael Finkelberg

Publications and source records attributed to Michael Finkelberg.

At least 19 recordsLinked to original sources
arXiv2026

Relative Langlands duality and Koszul duality

Consider a pair of $S$-dual hyperspherical varieties $G\circlearrowright X$ and $G^\vee\circlearrowright X^\vee$ equipped with equivariant quantizations $Q(X)$, $Q(X^\vee)$. Assume that the local conjecture of Ben-Zvi, Sakellaridis and Venkatesh holds for this pair, and also that $X\simeq T^*_\psi(Y)$ is polarized, so that $Q(X)=D_\psi(Y)$. Let $B\subset G$ (resp. $B^\vee\subset G^\vee$) be Borel subgroups. Then using a variant of the $S^1$-equivariant localization of arxiv:0706.0322, we deduce an equivalence between the ${\mathbb Z}/2$-graded $B$-equivariant category $(D_\psi(Y)\operatorname{-mod}^B)^{{\mathbb Z}/2}$ and the ${\mathbb Z}/2$-graded unipotent $B^\vee$-monodromic category $(Q(X^\vee)\operatorname{-mod}^{B^\vee,\operatorname{mon}})^{{\mathbb Z}/2}$.

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arXiv2025

Modular reduction of complex representations of finite reductive groups

The main result describes the Brauer-Nesbitt reduction of unipotent representations of a finite group of Lie type, expressing it as an explicit linear combination of the restriction of Weyl modules from the algebraic group to the group of $\mathbb{F}_q$ points. This partly confirms Lusztig's conjecture (2021), which was the main source of motivation for this work. The explicit virtual representations of the algebraic group come from a certain endomorphism of the space ${\mathbb Z}[T]$ of regular functions on the torus which approximates pullback under Frobenius and is linear over the ring ${\mathbb Z}[T]^W$ of $W$-invariant functions. This endomorphism is constructed from a new basis for ${\mathbb Z}[T]$ over ${\mathbb Z}[T]^W$ which we call the Kazhdan-Lusztig-Steinberg basis. We compare this basis to the canonical basis appearing in the study of modular representations of the algebraic group and the related noncommutative Springer resolution. This leads to canonically defined objects in the derived category of $G$-modules representing the above virtual representations and to a geometric interpretation for the resulting lift of the principal series representation $\overline{\mathbb{F}_q} [G/P(\mathbb{F}_q)]$ to a virtual representation of the algebraic group, which comes from a decomposition of diagonal in the equivariant Grothendieck group of the partial flag variety.

math.RT↗
arXiv2024

Relative Langlands duality for $\mathfrak{osp}(2n + 1|2n)$

We establish an $S$-duality converse to the one studied by the 1st, 2nd and 4th authors; this is also a case of a twisted version of the relative Langlands duality of Ben Zvi, Sakellaridis and Venkatesh.. Namely, we prove that the $S$-dual of $\text{SO}(2n+1)\times \text{Sp}(2n)$ acting on the tensor product of their tautological representations is the symplectic mirabolic space $\text{Sp}(2n)\times\text{Sp}(2n)$ acting on the product $T^* \text{Sp}(2n)$ and the tautological representations of $\text{Sp}(2n)$. (Note that due to the anomaly, the dual of the second factor $\text{Sp}(2n)$ is the metaplectic dual, i.e. $\text{Sp}(2n)$). We also formulate the corresponding global conjecture, which describes explicitly the categorical theta-correspondence on the Langlands dual side.

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arXiv2024

Gaiotto Conjecture for $\mathrm{Rep}_q(\mathrm{F}(4))$

This paper is a part of the series proving the Gaiotto conjecture for basic classical quantum supergroups. The previous part arXiv:2107.02653 [math.RT] , arXiv:2306.09556 [math.RT], proved the Gaiotto conjecture for the general linear quantum supergroups $U_q(\mathfrak{gl}(N|M))$. Here we deal with the exceptional quantum supergroup $U_q(\mathfrak{f}(4))$.

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arXiv2023

Lagrangian subvarieties of hyperspherical varieties

Given a hyperspherical $G$-variety $\mathscr X$ we consider the zero moment level $\Lambda_{\mathscr X}\subset{\mathscr X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that $\Lambda_{\mathscr X}$ is Lagrangian. For the dual $G^\vee$-variety ${\mathscr X}^\vee$, we conjecture that that there is a bijection between the sets of irreducible components $\mathrm{Irr}\Lambda_{\mathscr X}$ and $\mathrm{Irr}\Lambda_{{\mathscr X}^\vee}$. We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.

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arXiv2023

A fusion construction of local L-factors

We propose a new conjectural way to calculate the local $L$-factor $L=L_\chi(\pi,\rho,s)$ where $\pi$ is a representation of a $p$-adic group $G$, $\rho$ is an algebraic representation of the dual group $G^{\vee}$ and $\chi$ is an algebraic character of $G$ satisfying a positivity condition. A method going back to Godement and Jacquet yields a description of $L$ using as an input a certain space ${\mathcal S}_\rho$ of functions on $G$ depending on $\rho$. A (partly conjectural) description of ${\mathcal S}_\rho$ involving trace of Frobenius functions associated to perverse sheaves on the loop space of a semigroup containing $G$ was developed %by Bouthier, Ngo and Sakellaridis, partly based on an earlier work of Braverman and Kazhdan. Here we propose a different, more general conjectural description of ${\mathcal S}_\rho$: it also refers to trace of Frobenius functions but instead of the loop space of a semi-group we work with the ramified global Grassmannian fibering over the configuration space of points on a global curve defined by Beilinson-Drinfeld and Gaitsgory (a relation between two approaches is discussed in the appendix). Our main result asserts validity of our conjectures where $\pi$ is generated by an Iwahori fixed vector: we show that in this case it is compatible with the standard formula for $L$ involving local Langlands correspondence which is known for such representations $\pi$. The proof is based on properties of the coherent realization of the affine Hecke category.

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arXiv2023

Elliptic zastava

We study the elliptic zastava spaces, their versions (twisted, Coulomb, Mirkovic local spaces, reduced) and relations with monowalls moduli spaces and Feigin-Odesskii moduli spaces of $G$-bundles with parabolic structure on an elliptic curve.

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arXiv2022

Orthosymplectic Satake equivalence

This is a companion paper of arXiv:1909.11492. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of $SO(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $SO_N$. We explain how this equivalence fits into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

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arXiv2022

Orthosymplectic Satake equivalence, II

This is a companion paper of arXiv:1909.11492 and arXiv:1912.01930. We prove an equivalence relating representations of a degenerate orthosymplectic supergroup with the category of twisted $Sp(2n,{\mathbb C}[\![t]\!])$-equivariant $D$-modules on the so called mirabolic affine Grassmannian of $Sp(2n)$. We also discuss (conjectural) extension of this equivalence to the case of quantum supergroups and to some exceptional supergroups.

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arXiv2022

Microlocal approach to Lusztig's symmetries

We reformulate the De Concini -- Toledano Laredo conjecture about the monodromy of the Casimir connection in terms of a relation between Lusztig's symmetries of quantum group modules and the monodromy in the vanishing cycles of factorizable sheaves.

math.AG↗
arXiv2022

Fourier transform on hyperplane arrangements

We consider the category of perverse sheaves on a complex vector space smooth with respect to a stratification given by an arrangement of hyperplanes with real equations. As shown in an earlier wotk of two of the authors, this category can be described in terms of certain diagrams of vector spaces labelled by all the faces of the real arrangement (we call such diagrams hyperbolic sheaves). In this paper we calculate, in these terms, several fundamental operations of sheaf theory such as forming the space of vanishing cycles, specialization and the Fourier-Sato transform.

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arXiv2022

A quasi-coherent description of the category D-mod(Gr$_{GL(n)}$)

In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr$_{GL(n)}$) of (all) D-modules on the affine Grassmannian of the group $GL(n)$ as the category of quasi-coherent sheaves on a certain stack (it is explained in loc. cit. that this conjecture "follows" naturally from some heuristic arguments involving 3-dimensional quantum field theory). In this paper we prove a weaker version of this conjecture for the case $n=2$.

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arXiv2022

Kazhdan-Lusztig conjecture via Zastava spaces

We deduce the Kazhdan-Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O from the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and W-algebras.

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arXiv2022

Kostka numbers and Fourier duality

We relate the Fourier transform of perverse sheaves smooth along the coordinate hyperplane configuration in a complex vector space to the Deligne-Lusztig duality of unipotent representations of a general linear group over a finite field. A similar relation is established for arbitrary finite Coxeter groups.

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arXiv2022

Coulomb branches of noncotangent type (with appendices by Gurbir Dhillon and Theo Johnson-Freyd)

We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$ gauge theory corresponding to a choice of a connected reductive group $G$ and a symplectic finite-dimensional reprsentation $\mathbf M$ of $G$, satisfying certain anomaly cancellation condition. This extends the construction of arXiv:1601.03586 (where it was assumed that ${\mathbf M}={\mathbf N}\oplus{\mathbf N}^*$ for some representation $\mathbf N$ of $G$). Our construction goes through certain "universal" ring object in the twisted derived Satake category of the symplectic group $Sp(2n)$. The construction of this object uses a categorical version of the Weil representation; we also compute the image of this object under the (twisted) derived Satake equivalence and show that it can be obtained from the theta-sheaf introduced by S.Lysenko on $\operatorname{Bun}_{Sp(2n)}({\mathbb P}^1)$ via certain Radon transform. We also discuss applications of our construction to a potential mathematical construction of $S$-duality for super-symmetric boundary conditions in 4-dimensional gauge theory and to (some extension of) the conjectures of D.Ben-Zvi, Y.Sakellaridis and A.Venkatesh.

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arXiv2021

Cherednik algebras and Hilbert schemes in characteristic p (with an appendix by Pavel Etingof)

We prove a localization theorem for the type A rational Cherednik algebra H_c=H_{1,c} over an algebraic closure of the finite field F_p. In the most interesting special case where the parameter c takes values in F_p, we construct an Azumaya algebra A_c on Hilb^n, the Hilbert scheme of n points in the plane, such that the algebra of global sections of A_c is isomorphic to H_c. Our localisation theorem provides an equivalence between the bounded derived categories of H_c-modules and sheaves of coherent A_c-modules on the Hilbert scheme, respectively. Furthermore, we show that the Azumaya algebra splits on the formal completion of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland-King-Reid and Haiman.

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arXiv2021

Mirabolic Satake equivalence and supergroups

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of $GL(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $GL_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

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