arXiv subjects

Tanmay Deshpande

Publications and source records attributed to Tanmay Deshpande.

17 recordsLinked to original sources

arXiv2026

Character Sheaves on Tori over Local Fields

Let $\breve{K}$ be a complete discrete valuation field with an algebraically closed residue field ${k}$ and ring of integers $\breve{O}$. Let $T$ be a torus defined over $\breve{K}$. Let $L^+T$ denote the connected commutative pro-algebraic group over ${k}$ obtained by applying the Greenberg functor to the connected Néron model of $T$ over $\breve{O}$. Following the work of Serre for the multiplicative group, we first compute the fundamental group $π_1(L^+T)$. We then study multiplicative local systems (or character sheaves) on $L^+T$ and establish a local Langlands correspondence for them. Namely, we construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on $L^+T$ and inertial local Langlands parameters for $T$. Finally, we relate our results to the classical local Langlands correspondence for tori over local fields due to Langlands, via the sheaf-function correspondence.

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arXiv2025

A Construction of the Symmetric Monoidal Structure of the Geometric Whittaker Model

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$ and let $\ell$ be a prime number different from $p$. Let $U \subseteq G$ be a maximal unipotent subgroup, $T$ a maximal torus normalizing $U$ and $W$ the Weyl group of $G$. Let $\mathcal{L}$ be a non-degenerate multiplicative $\overline{\mathbb{Q}}_{\ell} $-local system on $U$. R. Bezrukavnikov and the second author have proved that the bi-Whittaker category, namely the triangulated monoidal category of $(U, \mathcal{L})$-biequivariant $\overline{\mathbb{Q}}_{\ell}$-complexes on $G$ is monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}_{W}^{\circ}(T) \subseteq \mathscr{D}_{W}(T)$ of "central sheaves" on the torus. In particular it has the structure of a symmetric monoidal category coming from the symmetric monoidal structure on $\mathscr{D}_W(T)$. In this paper, we give another construction of a symmetric monoidal structure on the above category and prove that it agrees with the one coming from the above construction. For this, among other things, we generalize a proof by Gelfand for finite groups to the geometric setup.

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arXiv2024

Vanishing sheaves and the geometric Whittaker model

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$ and let $\ell$ be a prime number different from $p$. Let $U\subset G$ be a maximal unipotent subgroup, and let $T$ be a maximal torus normalizing $U$ with normalizer $N=N_G(T)$. Let $W=N/T$ be the Weyl group of $G$. Let $\mathcal{L}$ be a non-degenerate $\ell$-adic multiplicative local system on $U$. In this paper we prove that the bi-Whittaker category, namely the triangulated monoidal category of $(U,\mathcal{L})$-bi-equivariant complexes on $G$, is monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}^\circ_W(T)\subset \mathscr{D}_W(T)$ of ''$W$-equivariant central sheaves'' on the torus, answering a question raised by Drinfeld. In particular, the bi-Whittaker category has the structure of a symmetric monoidal category. We also study a certain thick triangulated monoidal subcategory $\mathscr{D}^\circ_G(G)\subset \mathscr{D}_G(G)$ of ''vanishing sheaves'' and prove that it is braided monoidally equivalent to an explicit thick triangulated monoidal subcategory $\mathscr{D}^\circ_N(T)\subset \mathscr{D}_N(T)$ of ''$N$-equivariant central sheaves'' on the torus. The above equivalence is given by an enhancement of the parabolic restriction functor restricted to the subcategory $\mathscr{D}^\circ_G(G)$.

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arXiv2022

Crossed modular categories and the Verlinde formula for twisted conformal blocks

In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Γ$ and a positive integral level $\ell$ under the assumption that "$Γ$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Γ$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Γ$-crossed modular functor and show that it is very closely related to the notion of a $Γ$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $Γ$-twisted conformal blocks associated with a twisted affine Lie algebra define a $Γ$-crossed modular functor. Along the way, we prove equivalence between a $Γ$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.

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arXiv2020

On $G$-crossed Frobenius $\star$-algebras and fusion rings associated with braided $G$-actions

For a finite group $G$, Turaev introduced the notion of a braided $G$-crossed fusion category. The classification of braided $G$-crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a $G$-crossed Frobenius $\star$-algebra and give a classification of (strict) $G$-crossed extensions of a commutative Frobenius $\star$-algebra $R$ equipped with a given action of $G$, in terms of the second group cohomology $H^2(G,R^\times)$. Now suppose that $\mathcal{B}$ is a non-degenerate braided fusion category equipped with a braided action of a finite group $G$. We will see that the associated $G$-graded fusion ring is in fact a (strict) $G$-crossed Frobenius $\star$-algebra. We will describe this $G$-crossed fusion ring in terms of the classification of braided $G$-actions by Etingof, Nikshych, Ostrik and derive a Verlinde formula to compute its fusion coefficients.

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arXiv2019

Cohomology of bimultiplicative local systems on unipotent groups

Let $U_1, U_2$ be connected commutative unipotent algebraic groups defined over an algebraically closed field $k$ of characteristic $p>0$ and let $\mathcal{L}$ be a bimultiplicative $\overline{\mathbb{Q}}_\ell$-local system on $U_1\times U_2$. In this paper we will study the $\overline{\mathbb{Q}}_\ell$-cohomology $H^*_c(U_1\times U_2,\mathcal{L})$, which turns out to be supported in only one degree. We will construct a finite Heisenberg group $Γ$ which naturally acts on $H^*_c(U_1\times U_2,\mathcal{L})$ as an irreducible representation. We will give two explicit realizations of this cohomology and describe the relationship between these two realizations as a finite Fourier transform.

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arXiv2018

On twisted Verlinde formulae for modular categories

In this note, we describe two analogues of the Verlinde formula for modular categories in a twisted setting. The classical Verlinde formula for a modular category $\mathscr{C}$ describes the fusion coefficients of $\mathscr{C}$ in terms of the corresponding S-matrix $S(\mathscr{C})$. Now let us suppose that we also have an invertible $\mathscr{C}$-module category $\mathscr{M}$ equipped with a $\mathscr{C}$-module trace. This gives rise to a modular autoequivalence $F:\mathscr{C}\xrightarrow{\cong}\mathscr{C}$. In this setting, we can define a crossed S-matrix $S(\mathscr{C},\mathscr{M})$. As our first twisted analogue of the Verlinde formula, we will describe the fusion coefficients for $\mathscr{M}$ as a $\mathscr{C}$-module category in terms of the S-matrix $S(\mathscr{C})$ and the crossed S-matrix $S(\mathscr{C},\mathscr{M})$. In this twisted setting, we can also define a twisted fusion $\mathbb{Q}^{ab}$-algebra $K_{\mathbb{Q}^{ab}}(\mathscr{C},F)$. As another analogue of the Verlinde formula, we describe the fusion coefficients of the twisted fusion algebra in terms of the crossed S-matrix $S(\mathscr{C},\mathscr{M})$.

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arXiv2017

Character sheaves on neutrally solvable groups

Let $G$ be an algebraic group over an algebraically closed field $\mathtt{k}$ of characteristic $p>0$. In this paper we develop the theory of character sheaves on groups $G$ such that their neutral connected components $G^\circ$ are solvable algebraic groups. For such algebraic groups $G$ (which we call neutrally solvable) we will define the set $\operatorname{CS}(G)$ of character sheaves on $G$ as certain special (isomorphism classes of) objects in the category $\mathscr{D}_G(G)$ of $G$-equivariant $\overline{\mathbb{Q}}_\ell$-complexes (where we fix a prime $\ell\neq p$) on $G$. We will describe a partition of the set $\operatorname{CS}(G)$ into finite sets known as $\mathbb{L}$-packets and we will associate a modular category $\mathscr{M}_L$ with each $\mathbb{L}$-packet $L$ of character sheaves using a truncated version of convolution of character sheaves. In the case where $\mathtt{k}=\overline{\mathbb{F}}_q$ and $G$ is equipped with an $\mathbb{F}_q$-Frobenius $F$ we will study the relationship between $F$-stable character sheaves on $G$ and the irreducible characters of (all pure inner forms of) $G^F$. In particular, we will prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for neutrally solvable groups and that these almost characters coincide with the "trace of Frobenius" functions associated with $F$-stable character sheaves. We will also prove that the matrix relating the irreducible characters and almost characters is block diagonal where the blocks on the diagonal are parametrized by $F$-stable $\mathbb{L}$-packets. Moreover, we will prove that the block in this transition matrix corresponding to any $F$-stable $\mathbb{L}$-packet $L$ can be described as the crossed S-matrix associated with the auto-equivalence of the modular category $\mathscr{M}_L$ induced by $F$.

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arXiv2016

On centers of bimodule categories and induction-restriction functors

In this paper we study a toy categorical version of Lusztig's induction and restriction functors for character sheaves, but in the abstract setting of multifusion categories. Let $\mathscr{C}$ be an indecomposable multifusion category and let $\mathscr{M}$ be an invertible $\mathscr{C}$-bimodule category. Then the center $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ of $\mathscr{M}$ with respect to $\mathscr{C}$ is an invertible module category over the Drinfeld center $\mathscr{Z}(\mathscr{C})$ which is a braided fusion category. Let $ζ_{\mathscr{M}}:\mathscr{Z}_{\mathscr{C}}(\mathscr{M})\longrightarrow\mathscr{M}$ denote the forgetful functor and let $χ_{\mathscr{M}}:\mathscr{M}\longrightarrow\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$ be its right adjoint functor. These functors can be considered as toy analogues of the restriction and induction functors used by Lusztig to define character sheaves on (possibly disconnected) reductive groups. In this paper we look at the relationship between the decomposition of the images of the simple objects under the above functors and the character tables of certain Grothendieck rings. In case $\mathscr{C}$ is equipped with a spherical structure and $\mathscr{M}$ is equipped with a $\mathscr{C}$-bimodule trace, we relate this to the notion of the crossed S-matrix associated with the $\mathscr{Z}(\mathscr{C})$-module category $\mathscr{Z}_{\mathscr{C}}(\mathscr{M})$.

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arXiv2015

Minimal Idempotents on Solvable Groups

In this paper, we begin to develop a theory of character sheaves on an affine algebraic group $G$ defined over an algebraically closed field $k$ of characteristic $p>0$ using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let $l$ be a prime different from $p$. Following Boyarchenko and Drinfeld, we define the notion of an admissible pair on $G$ and the corresponding idempotent in the $\overline{\mathbb{Q}_l}$-linear triangulated braided monoidal category $\mathscr{D}_G(G)$ of conjugation equivariant $\overline{\mathbb{Q}_l}$-complexes (under convolution with compact support) and study their properties. We aim to break up the braided monoidal category $\mathscr{D}_G(G)$ into smaller and more manageable pieces corresponding to these idempotents in $\mathscr{D}_G(G)$. Drinfeld has conjectured that the idempotent in $\mathscr{D}_G(G)$ obtained from an admissible pair is in fact a minimal idempotent and that any minimal idempotent in $\mathscr{D}_G(G)$ can be obtained from some admissible pair on $G$. We will prove this conjecture in the case when the neutral connected component $G^\circ \subset G$ is a solvable group. For general groups, we prove that this conjecture is in fact equivalent to an a priori weaker conjecture. Using these results, we reduce the problem of defining character sheaves on general algebraic groups to a special case which we call the "Heisenberg case".

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arXiv2015

Crossed S-matrices and Character Sheaves on Unipotent Groups

Let $\mathtt{k}$ be an algebraic closure of a finite field $\mathbb{F}_{q}$ of characteristic $p$. Let $G$ be a connected unipotent group over $\mathtt{k}$ equipped with an $\mathbb{F}_q$-structure given by a Frobenius map $F:G\to G$. We will denote the corresponding algebraic group defined over $\mathbb{F}_q$ by $G_0$. Character sheaves on $G$ are certain objects in the triangulated braided monoidal category $\mathscr{D}_G(G)$ of bounded conjugation equivariant $\bar{\mathbb{Q}}_l$-complexes (where $l\neq p$ is a prime number) on $G$. Boyarchenko has proved that the "trace of Frobenius" functions associated with $F$-stable character sheaves on $G$ form an orthonormal basis of the space of class functions on $G_0(\mathbb{F}_q)$ and that the matrix relating this basis to the basis formed by the irreducible characters of $G_0(\mathbb{F}_q)$ is block diagonal with "small" blocks. In this paper we describe these block matrices and interpret them as certain "crossed $S$-matrices". We also derive a formula for the dimensions of the irreducible representations of $G_0(\mathbb{F}_q)$ that correspond to one such block in terms of certain modular categorical data associated with that block. In fact we will formulate and prove more general results which hold for possibly disconnected groups $G$ such that $G^\circ$ is unipotent. To prove our results, we will establish a formula (which holds for any algebraic group $G$) which expresses the inner product of the "trace of Frobenius" function of any $F$-stable object of $\mathscr{D}_G(G)$ with any character of $G_0(\mathbb{F}_q)$ (or of any of its pure inner forms) in terms of certain categorical operations.

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arXiv2015

Modular categories, crossed S-matrices and Shintani descent

Let $\mathscr{C}$ be a modular tensor category over an algebraically closed field $k$ of characteristic 0. Then there is the ubiquitous notion of the S-matrix $S(\mathscr{C})$ associated with the modular category. The matrix $S(\mathscr{C})$ is a symmetric matrix, its entries are cyclotomic integers and the matrix $(\dim \mathscr{C})^{-\frac{1}{2}}\cdot S(\mathscr{C})$ is a unitary matrix. Here $\dim \mathscr{C}\in k$ denotes the categorical dimension of $\mathscr{C}$ and it is a totally positive cyclotomic integer. Now suppose that we also have a modular autoequivalence $F:\mathscr{C}\to \mathscr{C}$. In this paper, we will define and study the notion of a crossed S-matrix associated with the modular autoequivalence $F$. We will see that the crossed S-matrix occurs as a submatrix of the usual S-matrix of some "bigger" modular category and hence the entries of a crossed S-matrix are also cyclotomic integers. We will prove that the crossed S-matrix (normalized by the factor $(\dim \mathscr{C})^{-\frac{1}{2}}$) associated with any modular autoequivalence is a unitary matrix. We will also prove that the crossed S-matrix is essentially the "character table" of a certain semisimple commutative Frobenius $k$-algebra associated with the modular autoequivalence $F$. The motivation for most of our results comes from the theory of character sheaves on algebraic groups, where we expect that the transition matrices between irreducible characters and character sheaves can be obtained as certain crossed S-matrices. In the character theory of algebraic groups defined over finite fields, there is the notion of Shintani descent of Frobenius stable characters. We will define and study a categorical analogue of this notion of Shintani descent in the setting of modular categories.

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arXiv2015

Shintani descent for algebraic groups and almost characters of unipotent groups

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_q$. For this, it is essential to treat all the pure inner $\mathbb{F}_q$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the "trace of Frobenius" functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko-Drinfeld's theory of character sheaves on neutrally unipotent are in fact positive integral confirming a conjecture due to Drinfeld.

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arXiv2014

On Hilbert bases of cuts

A Hilbert basis is a set of vectors X such that the integer cone (semigroup) generated by X is the intersection of the lattice generated by X with the cone generated by X. Define a graph to be (cut) Hilbert if its set of cuts forms a Hilbert basis. We show that the Hilbert property is not closed under edge deletions, subdivisions, nor 2-sums. Furthermore, no graph having K_6-e as a minor is Hilbert. This corrects an error in [M. Laurent. Hilbert bases of cuts. Discrete Math., 150(1-3):257-279 (1996)]. For positive results, we give conditions under which the 2-sum of two graphs produces a Hilbert graph. Using these conditions we show that all H-minor-free graphs are Hilbert , where H is the unique 3-connected graph obtained by uncontracting an edge of K_5. We also establish a relationship between edge deletion and subdivision. Namely, if G' is obtained from a Hilbert graph G by subdividing an edge e two or more times, then G-e is Hilbert if and only if G' is Hilbert.

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arXiv2013

Modular Categories Associated to Unipotent Groups

Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0 and let l be a prime different from p. Let e be a minimal idempotent in D_G(G), the braided monoidal category of G-equivariant (under conjugation action) \bar{Q_l}-complexes on G. We can associate to G and e a modular category M_{G,e}. In this article, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class C_p^{\pm}.

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arXiv2009

Heisenberg Idempotents on Unipotent Groups

Let G be an algebraic group over an algebraically closed field of positive characteristic such that its neutral connected component is a unipotent group. We consider a certain class of closed idempotents in the braided monoidal category (under convolution of complexes) D_G(G) known as Heisenberg idempotents. For such an idempotent e, we will prove certain results about the Hecke subcategory eD_G(G) conjectured by V. Drinfeld. In particular, we will see that it is the bounded derived category of a modular category.

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arXiv2008

An Exceptional Representation of Sp(4,F_q)

We describe a folklore construction of an exceptional representation of Sp(4,F_q). This representation has the following remarkable combination of properties, namely it is cuspidal, degenerate and unipotent.

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