arXiv subjects

Anna Romanov

Publications and source records attributed to Anna Romanov.

13 recordsLinked to original sources

arXiv2026

Lusztig-Vogan categories of equal rank 2

Lusztig-Vogan categories are categorifications of the principal block of the Lusztig-Vogan module over the Hecke algebra, which captures information about characters of irreducible admissible representations of a real reductive group. Lusztig-Vogan categories can be constructed as module categories over Soergel bimodules. In this paper, we describe the structure of the rank 2 Lusztig-Vogan categories corresponding to equal rank real groups. More precisely, we classify indecomposable objects and describe the action of generating Soergel bimodules, recovering the $W$-graph of the underlying Lusztig-Vogan module. We also provide an algorithm which completes this procedure for arbitrary finite rank Lusztig-Vogan categories, including those which do not correspond to a real reductive group.

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arXiv2025

On the geometric approach to the discrete series

Harish-Chandra classified discrete series representations of real semisimple Lie groups by describing their characters as tempered distributions with an explicit formula on the elliptic set. His approach was inspired by Weyl's proof of the character formula for irreducible representations of compact Lie groups. Hecht, Mili\v{c}i\'{c}, Schmid and Wolf gave an alternative construction using the localization theory of Beilinson and Bernstein: the discrete series are the global sections of standard Harish-Chandra sheaves on the flag variety attached to the closed orbits of the complexification of a maximal compact subgroup. Their approach was inspired by the Borel-Weil theorem. In this paper, we give an explicit correspondence between these parametrizations. First, for a nilpotent radical $\mathfrak{n}$ of any Borel subalgebra, we establish a geometric formula for the $\mathfrak{n}$-homology of a module over the universal enveloping algebra of a complex semisimple Lie algebra $\mathfrak{g}$, in terms of its localization on the flag variety of $\mathfrak{g}$. Then we give applications of the formula to the discrete series. First, we give a geometric proof of Schmid's result describing the $\mathfrak{n}$-homology of discrete series representations for the nilpotent radical $\mathfrak{n}$ attached to a Borel subalgebra containing the Lie algebra of a maximal torus. Using Osborne's formula, this gives the formula for the character on the elliptic set, and leads to the matching of Harish-Chandra's parameters for discrete series representations with the geometric parameters. This gives an alternative approach to the study of the discrete series. As an example, we deduce Blattner's conjecture on the multiplicities of K-types of the discrete series from the Borel-Weil-Bott theorem for the maximal compact subgroup.

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arXiv2025

A remarkable functor on $G$-modules

We introduce a new functor on categories of modular representations of reductive algebraic groups. Our functor has remarkable properties. For example it is a tensor functor and sends every standard and costandard object in the principal block to a one-dimensional object. We connect our functor to recent work of Gruber and conjecture that our functor is equivalent to hypercohomology under the equivalence of the Finkelberg-Mirkovic conjecture.

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arXiv2024

Two proofs of a Jantzen Conjecture for Whittaker Modules

We define a filtration of a standard Whittaker module over a complex semisimple Lie algebra and and establish its fundamental properties. Our filtration specialises to the Jantzen filtration of a Verma module for a certain choice of parameter. We prove that embeddings of standard Whittaker modules are strict with respect to our filtration, and that the filtration layers are semisimple. This provides a generalisation of the Jantzen conjectures to Whittaker modules. We prove these statements in two ways. First, we give an algebraic proof which compares Whittaker modules to Verma modules using a functor introduced by Backelin. Second, we give a geometric proof using mixed twistor $\mathcal{D}$-modules.

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arXiv2024

An example of the Jantzen filtration of a D-module

We compute the Jantzen filtration of a D-module on the flag variety of $\mathrm{SL}_2(\mathbb{C})$. At each step in the computation, we illustrate the $\mathfrak{sl}_2(\mathbb{C})$-module structure on global sections to give an algebraic picture of this geometric computation. We conclude by showing that the Jantzen filtration on the D-module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson--Bernstein.

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arXiv2022

Contravariant forms on Whittaker modules

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker $\mathfrak{g}$-modules $Y(χ, η)$ introduced by Kostant. We prove that the set of all contravariant forms on $Y(χ, η)$ forms a vector space whose dimension is given by the cardinality of the Weyl group of $\mathfrak{g}$. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules $M(χ, η)$ introduced by McDowell.

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arXiv2022

Contravariant pairings between standard Whittaker modules and Verma modules

We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category $\mathcal{O}$ to the Miličić--Soergel category $\mathcal{N}$. We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of $\mathcal{D}$-modules. We prove that with these costandard modules, blocks of category $\mathcal{N}$ have the structure of highest weight categories and we establish a BGG reciprocity theorem for $\mathcal{N}$.

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arXiv2022

A categorification of the Lusztig--Vogan module

We construct two categorifications of the Lusztig--Vogan module associated to a real reductive algebraic group. The first categorification is given by semisimple complexes in an equivariant derived category, and the second is constructed as a module category over Soergel bimodules. Our categorifications are related by taking equivariant hypercohomology.

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arXiv2021

Langlands correspondence and Bezrukavnikov's equivalence

These are lecture notes (by the first author) from a course (by the second author) given over two extended semesters at the University of Sydney. The first part provides an introduction to the Langlands correspondence from an arithmetical point of view. The second part gives enough background in geometric representation theory to understand Bezrukavnikov's equivalence, which is a categorification of Kazhdan and Lusztig's two realizations of the affine Hecke algebra.

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arXiv2020

Four examples of Beilinson-Bernstein localization

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. The Beilinson-Bernstein localization theorem establishes an equivalence of the category of $\mathfrak{g}$-modules of a fixed infinitesimal character and a category of modules over a twisted sheaf of differential operators on the flag variety of $\mathfrak{g}$. In this expository paper, we give four detailed examples of this theorem when $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$. Specifically, we describe the $\mathcal{D}$-modules associated to finite-dimensional irreducible $\mathfrak{g}$-modules, Verma modules, Whittaker modules, discrete series representations of $SL(2,\mathbb{R})$, and principal series representations of $SL(2,\mathbb{R})$.

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arXiv2019

A Kazhdan-Lusztig algorithm for Whittaker modules

We study a category of Whittaker modules over a complex semisimple Lie algebra by realizing it as a category of twisted D-modules on the associated flag variety using Beilinson-Bernstein localization. The main result of this paper is the development of a geometric algorithm for computing the composition multiplicities of standard Whittaker modules. This algorithm establishes that these multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan-Lusztig polynomials. In the case of trivial nilpotent character, this algorithm specializes to the usual algorithm for computing multiplicities of composition factors of Verma modules using Kazhdan-Lusztig polynomials.

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arXiv2019

Finite Gelfand pairs and cracking points of the symmetric groups

Let $Γ$ be a finite group. Consider the wreath product $G_n := Γ^n \rtimes S_n$ and the subgroup $K_n := Δ_n \times S_n\subseteq G_n$, where $S_n$ is the symmetric group and $Δ_n$ is the diagonal subgroup of $Γ^n$. For certain values of $n$ (which depend on the group $Γ$), the pair $(G_n, K_n)$ is a Gelfand pair. It is not known for all finite groups which values of $n$ result in Gelfand pairs. Building off the work of Benson--Ratcliff, we obtain a result which simplifies the computation of multiplicities of irreducible representations in certain tensor product representations, then apply this result to show that for $Γ= S_k, \ k \geq 5$, $(G_n,K_n)$ is a Gelfand pair exactly when $n = 1,2$.

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arXiv2019

An orbit model for the spectra of nilpotent Gelfand pairs

Let $N$ be a connected and simply connected nilpotent Lie group, and let $K$ be a subgroup of the automorphism group of $N$. We say that the pair $(K,N)$ is a nilpotent Gelfand pair if $L^1_K(N)$ is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs $(K,N)$ where the $K$-orbits in the center of $N$ have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set $Δ(K,N)$ of bounded $K$-spherical functions on $N$ and the set $\mathcal{A}(K,N)$ of $K$-orbits in the dual $\mathfrak{n}^*$ of the Lie algebra for $N$ established by Benson and Ratcliff is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for $N$ a free group and $N$ a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs.

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