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David Kazhdan

Publications and source records attributed to David Kazhdan.

At least 19 recordsLinked to original sources
arXiv2026

Effective results in representation theory of reductive groups over local fields

We prove effective versions of several classical results in representation theory of reductive groups over non-Archimedean local fields of arbitrary characteristic. More specifically, we deal with the support of matrix coefficients of cuspidal representations and the recursions satisfied by matrix coefficients of representations of finite length. We provide effective generation results for irreducible representations. Finally, we study the effective smoothness of characters on the set of regular semi-simple elements.

math.RT↗
arXiv2026

Push forwards of smooth measures on the unitary group in odd characteristic under the Chevalley map are almost bounded

In this paper we study analytic properties of the Chevalley map $p:\mathfrak{u} \to \mathfrak{c}$ where $\mathfrak{u}$ is the Lie algebra of a unitary group over a local field of odd characteristics. This map essentially sends a matrix to its characteristic polynomial. Our main result is that smooth compactly supported measures on $\mathfrak{u}$ are pushed by the Chevalley map to measures on $\mathfrak{c}$ with density functions in $L^q(\mathfrak{c})$ for all $q < \infty$. We also provide similar results for the Chevalley map attached to the unitary group. An important consequence is an odd characteristic analogue to results of Harish-Chandra stating the local integrability of the orbital integrals on the unitary group and its Lie algebra. Further possible applications to local integrability of characters of admissible representations of unitary groups over local fields of odd characteristic are also discussed.

math.RT↗
arXiv2026

An analogue of $Γ$-function for irreducible cuspidal representations of the group $\mathrm{GL}(2)$ over a two-dimensional local field

Let $\mathbf F$ be a non-archimedian local field of odd residue characteristic, ${\mathcal O}={\mathbf F}[\![t]\!]\subset{\mathcal K}={\mathbf F}(\!(t)\!)$. We consider the subgroup $\widehat{\mathrm{GL}}(2,{\mathcal O})={\mathcal K}^\times\cdot{\mathrm{GL}}(2,{\mathcal O})\subset{\mathrm{GL}}(2,{\mathcal K})$, and the normalizer ${\mathrm{GL}}'(2,{\mathcal O})\subset{\mathrm GL}(2,{\mathcal K})$ of an Iwahori subgroup $I\subset{\mathrm{GL}}(2,{\mathcal K})$. We calculate the $Γ$-functions for irreducible special representations of $\widehat{\mathrm{GL}}(2,{\mathcal O})$ and ${\mathrm{GL}}'(2,{\mathcal O})$.

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arXiv2026

Automorphic functions for nilpotent extensions of curves over finite fields

We define and study the subspace of cuspidal functions for $G$-bundles (and its ramified analogs) on a class of nilpotent extensions of curves over a finite field. In the case of length $2$ extensions and of $G=\mathrm{GL}_2$, we prove that the space of cuspidal functions (for bundles with a fixed determinant) is finite-dimensional and provide bounds on its dimension. In this case we also relate cuspidality to Hecke finiteness, defined in terms of the action of some natural Hecke operators. Finally, we construct some Hecke eigenfunctions using the relation to Higgs bundles over the corresponding reduced curve.

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arXiv2026

Square integrable representations and Hecke eigenvalues over a local non-archimedian field

Let $G$ be a split connected semi-simple group over a local non-archimedean field $F$. In this paper we study the action of Hecke operators on the Schwartz space of half-densities on the moduli stack $\bpo$ of $G$-bundles on ${\mathbb P}^1$ endowed with a trivialization at $0$ and $\infty$. This space is a representation of $G(F)\times G(F)$ which contains the regular representation of $G(F)$. We show that this inclusion is an isomorphism on the cuspidal part. In addition we show that every irreducible square-integrable representation $π$ of $G(F)$ defines a Hecke eigenvalue. As a byproduct we show that every irreducible square-integrable representation of $G(F)$ with an Iwahori fixed vector defines a ${\mathbb C}$-point on the Calogero-Moser space associated to $G$. We also show that for $G=PGL(N)$ the eigenvalues of certain "simplest non-trivial" Hecke operators are essentially equal to the values of the character of $π$ on regular elliptic elements. These results are parallel to some of the results of our previous publication where some analogous statements are proved in the case when $F$ is a finite field.

math.RT↗
arXiv2026

An analogue of irreducible cuspidal representations for the group $PGL(2)$ over a two-dimensional local field

Let $F$ be a local non-archimedian field of odd residue characteristic and let $G=PGL(2)$. In this paper we study an analog of irreducible cuspidal representations of the group $G(F)$ when $F$ is replaced by the field $K=F((t))$. The story turns out to be similar to the classical case, but also with some differences. We present a construction of such representations essentially (up to a small subtlety) starting from a quadratic extension $L$ of $K$ and a character $θ:L^*/K^*\to \mathbb C^*$ which is not Galois invariant. We also show that the restriction of the representations we construct to the group $P(K)$ (here $P$ is a Borel subgroup of $PGL(2)$) is irreducible. However, contrary to the classical case it turns out that these restrictions are not isomorphic to the "standard" irreducible cuspidal representation of $P(K)$. In the Appendix we propose a notion of cuspidality for smooth representations of the group $H(K)$ for an arbitrary split reductive group $H$.

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arXiv2026

A conjecture on the action of Hecke operators

Let $F$ be a local non-archimedian field, $L$ be a central division $F$-algebra of rank $n$ and ${\mathcal A}_n(L)$ be the convolution algebra of smooth compactly supported $Ad$-invariant complex-valued measures on $L^*$. It is known that for different division $F$-algebras $L$ of rank $n$ the algebras ${\mathcal A}_n(L)$ are canonically isomorphic. In this paper I propose a conjecture extending these isomorphisms to the algebras generated by Hecke operators on spaces of $1/2$-measures on the stacks of $L^*$-bundles on smooth complete curves over $F$.

math.AG↗
arXiv2026

A proof of Harish-Chandra's integrability theorem for cuspidal representations of $\mathrm{GL}_n(\mathbb F_\ell((t)))$

Consider the Chevalley map $$ p:\mathfrak{gl} _n(F)\to (\mathfrak{gl}_n//\mathrm{GL}_n)(F), $$ where $F=\mathbb{F}_\ell((t))$. We show that the push forward via $p$ of every smooth compactly supported measure on $\mathfrak{gl}_n(F)$ is a measure whose density belongs to $L^q$ for every finite $q$. As a consequence, using the main result of [AGKSc], we obtain local integrability for Harish--Chandra's characters of irreducible cuspidal representations of $\mathrm{GL}_n(F)$.

math.RT↗
arXiv2026

Rationality properties of complex representations of reductive p-adic groups

For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). We show that an elliptic G-representation (in the sense of Arthur) can be realized over Qbar if and only if its central character takes values in Qbar. That applies in particular to all essentially square-integrable G-representations. We also study the action of the automorphism group of C/Q on complex G-representations. We prove that the sets of essentially square-integrable representations and of elliptic representations are stable under Gal(C/Q).

math.RT↗
arXiv2026

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↗
arXiv2026

The jet schemes of the nilpotent cone of $\mathfrak{gl}_n$ over $\mathbb{F}_\ell$ and analytic properties of the Chevalley map

We prove dimension bounds on the jet schemes of the variety of nilpotent matrices (and of related varieties) in positive characteristic. This result has applications to the analytic properties of the Chevalley map that sends a matrix to its characteristic polynomial. We show that our dimension bound implies, under the assumption of existence of resolution of singularities in positive characteristic, that the Chevalley map pushes a smooth compactly supported measure to a measure whose density function is $L^t$ for any $t<\infty$. We also prove this analytic property of the Chevalley map, unconditionally, when the characteristic of the field exceeds $n/2$. The zero characteristic counterpart of this result is an important step in the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKSb], we show that also in positive characteristic, this analytic statement implies Harish-Chandra's integrability theorem for cuspidal representations of the general linear group.

math.AG↗
arXiv2026

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.

math.RT↗
arXiv2026

On Harish-Chandra's integrability theorem in positive characteristic

The celebrated Harish-Chandra's integrability theorem states that the distributional character of an irreducible smooth representation of a p-adic group $G(F)$ is integrable, that is represented by an $L^1_{loc}(G(F))$ function. Here $F$ is a non-Archimedean local field of characteristic $0$ and $G$ is a reductive algebraic group defined over $F$. In this paper we focus on cuspidal representations of $GL_n(F)$ for a field $F$ of positive characteristic. We show that in this case the integrability holds under the hypothesis of existence of desingularization of (certain) algebraic varieties in positive characteristics. Furthermore, in the case $char(F)>n/2$ we establish the regularity of such characters unconditionally.

math.RT↗
arXiv2026

Orbital integral bounds the character for cuspidal representations of $GL_n(\mathbb{F}_{\ell}((t)))$

We prove that the character of an irreducible cuspidal representation of $GL_n(\mathbb{F}_{\ell}((t)))$ is locally bounded up to a logarithmic factor by the orbital integral of a matrix coefficient of this representation. The characteristic $0$ analog of this result is part of the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKS] we use this result in order to prove a positive characteristic analog of Harish-Chandra's integrability theorem under some additional assumptions.

math.RT↗
arXiv2026

Invertible top form on the Hilbert scheme of a plane in positive characteristic

We prove that the Hilbert scheme of the plane in positive characteristic admits an invertible top differential form. This implies certain integrability properties of the symmetric powers of the plane. This allows to define a function on the collection of monic polynomials over a local field which can be thought of as a variant of the inverse square root of the discriminant. In characteristic 0 it essentially coincides with this inverse square root, however in general it is quite different, and unlike this inverse square root, it is locally summable. In a sequel work [AGKS] we use this local summability in order to prove the positive characteristic analog of Harish-Chandra's local integrability theorem of characters of representations under certain conditions. The main results of this paper are known in characteristic zero. In fact a stronger result is known: there is a symplectic form on the Hilbert scheme of a plane.

math.AG↗
arXiv2026

Effective local differential topology of algebraic varieties over local fields of positive characteristics

In this paper we provide a framework for quantitative statements on distances and measures when studying algebraic varieties and morphisms of algebraic varieties over local fields. We will concentrate on local fields of the type $\mathbb{F}_\ell((t))$ and work uniformly with respect to finite extensions of $\mathbb{F}_\ell$. In this framework we prove analogues of standard results from local differential topology, including the implicit function theorem and study the behavior of smooth measures under push forward with respect to submersions.

math.AG↗
arXiv2026

$L^2$-property for algebraic stacks over local non-archimedean fields

We introduce an $L^2$-norm on the space of Schwartz half-densities over algebraic stacks over local non-archimedean fields. We show that these $L^2$-norms are finite for the stacks of $PGL_2$-bundles on $\mathbb{P}^1$ with parabolic structures at $\ge 3$ points. The latter property was conjectured in the context of the analytic Langlands correspondence of arXiv:2103.01509.

math.AG↗