arXiv subjects

Calder Morton-Ferguson

Publications and source records attributed to Calder Morton-Ferguson.

8 recordsLinked to original sources

arXiv2025

Markov chains on Weyl groups from the geometry of the flag variety

This paper studies a basic Markov chain, the Burnside process, on the space of flags $G/B$ with $G = GL_n(\mathbb{F}_q)$ and $B$ its upper triangular matrices. This gives rise to a shuffling: a Markov chain on the symmetric group realized via the Bruhat decomposition. Actually running and describing this Markov chain requires understanding Springer fibers and the Steinberg variety. The main results give a practical algorithm for all n and q and determine the limiting behavior of the chain when q is large. In describing this behavior, we find interesting connections to the combinatorics of the Robinson-Schensted correspondence and to the geometry of orbital varieties. The construction and description is then carried over to finite Chevalley groups of arbitrary type, describing a new class of Markov chains on Weyl groups.

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arXiv2025

Weight modules and gluing of sheaves on the flag variety

We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric realization of this category as unipotently monodromic sheaves on the flag variety satisfying a singular support condition. We then explain that a derived version $\mathcal{D}$ of this category is Koszul dual to the Kazhdan-Laumon Category O, a different enlargement of the BGG Category O obtained from a gluing construction for sheaves on the flag variety. This characterizes $\mathcal{D}$ as the category of algebras over a natural monad on a direct sum of copies of the derived Category O. These results give new interpretations of classical algebraic constructions of weight modules over semisimple Lie algebras due to Fernando and Mathieu. We also conjecture that our results fit naturally into a proposed Koszul duality relating the small quantum group and the semi-infinite flag variety to the geometry of affine Springer fibers.

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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.

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arXiv2024

Perverse sheaves and t-structures on the thin and thick affine flag varieties

We study the categories $\mathrm{Perv}_{\mathrm{thin}}$ and $\mathrm{Perv}_{\mathrm{thick}}$ of Iwahori-equivariant perverse sheaves on the thin and thick affine flag varieties associated to a split reductive group $G$. An earlier work of the first author describes $\mathrm{Perv}_{\mathrm{thin}}$ in terms of bimodules over the so-called non-commutative Springer resolution. We partly extend this result to $\mathrm{Perv}_{\mathrm{thick}}$, providing a similar description for its anti-spherical quotient. The long intertwining functor realizes $\mathrm{Perv}_{\mathrm{thick}}$ as the Ringel dual of $\mathrm{Perv}_{\mathrm{thin}}$; we point out that it shares some exactness properties with the similar functor acting on perverse sheaves on the finite-dimensional flag variety. We use this result to resolve a conjecture of Arkhipov and the first author, proving that the image in the Iwahori-Whittaker category of any convolution-exact perverse sheaf on the affine flag variety is tilting.

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arXiv2024

Symplectic Fourier-Deligne transforms on G/U and the algebra of braids and ties

We explicitly identify the algebra generated by symplectic Fourier-Deligne transforms (i.e. convolution with Kazhdan-Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type $A$ by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya's generators of the Yokonuma-Hecke algebra (previously proved case-by-case in types $A, D, E$ by Juyumaya and separately for types $B, C, F_4, G_2$ by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan-Lusztig basis, and finally an explicit formula for the dimension of Marin's algebra in Type $A_n$ (previously only known for $n \leq 4$).

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arXiv2023

Polishchuk's conjecture and Kazhdan-Laumon representations

In their 1988 paper "Gluing of perverse sheaves and discrete series representations," D. Kazhdan and G. Laumon constructed an abelian category $\mathcal{A}$ associated to a reductive group $G$ over a finite field with the aim of using it to construct discrete series representations of the finite Chevalley group $G(\mathbb{F}_q)$. The well-definedness of their construction depended on their conjecture that this category has finite cohomological dimension. This was disproven by R. Bezrukavnikov and A. Polishchuk in 2001, who found a counterexample in the case $G = SL_3$. In the same paper, Polishchuk then made an alternative conjecture: though this counterexample shows that the Grothendieck group $K_0(\mathcal{A})$ is not spanned by objects of finite projective dimension, he noted that a graded version of $K_0(\mathcal{A})$ can be thought of as a module over Laurent polynomials and conjectured that a certain localization of this module is generated by objects of finite projective dimension, and suggested that this conjecture could lead toward a proof that Kazhdan and Laumon's construction is well-defined. He proved this conjecture in Types $A_1, A_2, A_3$, and $B_2$. In the present paper, we prove Polishchuk's conjecture in full generality, and go on to prove that Kazhdan and Laumon's construction is indeed well-defined, giving a new geometric construction of discrete series representations of $G(\mathbb{F}_q)$.

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arXiv2022

Kazhdan-Laumon Category O, Braverman-Kazhdan Schwartz space, and the semi-infinite flag variety

We define and study an analogue of Category O in the context of Kazhdan and Laumon's gluing construction for perverse sheaves on the basic affine space. We explicitly describe the simple objects in this category, and we show its linearized Grothendieck group is isomorphic to a natural submodule of Lusztig's periodic Hecke module. We then provide a categorification of these results by showing that the Kazhdan-Laumon Category O is equivalent to a full subcategory of a suitably-defined category of perverse sheaves on the semi-infinite flag variety.

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arXiv2022

Heaps, crystals, and preprojective algebra modules

Fix a simply-laced semisimple Lie algebra. We study the crystal $ B(n\lambda)$, were $\lambda$ is a dominant minuscule weight and $n$ is a natural number. On one hand, $B(n\lambda)$ can be realized combinatorially by height $n$ reverse plane partitions on a heap associated to $\lambda$. On the other hand, we use this heap to define a module over the preprojective algebra of the underlying Dynkin quiver. Using the work of Saito and Savage-Tingley, we realize $B(n\lambda)$ via irreducible components of the quiver Grassmannian of $n$ copies of this module. In this paper, we describe an explicit bijection between these two models for $B(n\lambda)$ and prove that our bijection yields an isomorphism of crystals. Our main geometric tool is Nakajima's tensor product quiver varieties.

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