David Kazhdan’s research while affiliated with Hebrew University of Jerusalem and other places

We study Hecke operators associated with curves over a non-archimedean local field K and over the rings $O/\mathfrak{m}^{N}$, where $O\subset K$ is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over $O/\mathfrak{m}^{N}$, associated with a connected split reductive group G such that [G,G] is simply connected. The proof uses a Hecke algebra associated with $G(K(\!(t)\!))$ and a global argument involving G-bundles on curves.

Let G be an even orthogonal quasi-split group defined over a local non-archimedean field F. We describe the subspace of smooth vectors of the minimal representation of G(F), realized on the space of square-integrable functions on a cone. Our main tool is the Fourier transform on the cone, for which we give an explicit formula.

We extend the Gelfand and Graev construction of generalized Fourier transforms on basic affine space from split groups to quasi-split groups over a local non-archimedean field F.

The goal of this note is to show that the local L-packet of Fargues-Scholze [FS], corresponding to an elliptic L-parameter, has an endoscopic decomposition. Our argument is strongly motivated by a beautiful paper of Chenji Fu [Fu], where the stable case is proven.

Consider a complex affine variety V∼$\tilde{V}$ and a real analytic Zariski‐dense submanifold VV of V∼$\tilde{V}$. We compare modules over the ring O(V∼)$\mathcal {O} (\tilde{V})$ of regular functions on V∼$\tilde{V}$ with modules over the ring C∞(V)$C^\infty (V)$ of smooth complex valued functions on VV. Under a mild condition on the tangent spaces, we prove that C∞(V)$C^\infty (V)$ is flat as a module over O(V∼)$\mathcal {O} (\tilde{V})$. From this, we deduce a comparison theorem for the Hochschild homology of finite‐type algebras over O(V∼)$\mathcal {O} (\tilde{V})$ and the Hochschild homology of similar algebras over C∞(V)$C^\infty (V)$. We also establish versions of these results for functions on V∼$\tilde{V}$ (resp. VV) that are invariant under the action of a finite group GG. As an auxiliary result, we show that C∞(V)$C^\infty (V)$ has finite rank as module over C∞(V)G$C^\infty (V)^G$.

Given a complex representation of a finite group, Brauer and Nesbitt defined in 1941 its reduction mod p, obtaining a representation over the algebraic closure of $\mathbb{F}_p$. In 2021, Lusztig studied the characters obtained by reducing mod p an irreducible unipotent representation of a finite reductive group over $\mathbb{F}_p$. He gave a conjectural formula for this character as a linear combination of terms which had no explicit definition and were only known in some small-rank examples. In this paper we provide an explicit formula for these terms and prove Lusztig's conjecture, giving a formula for the reduction mod p of any unipotent representation of $G(\mathbb{F}_q)$ for q a power of p. We also propose a conjecture linking this construction to the full exceptional collection in the derived category of coherent sheaves on a partial flag variety constructed recently by Samokhin and van der Kallen.

This paper formulates a conjectural description of of the space of weightless functions (see\cite{BK}) and raises a question about a possibility of extending such a description in a more general context.

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.

We study the behavior of the superperiod map near the boundary of the moduli space of stable supercurves and prove that it is similar to the behavior of periods of classical curves. We consider two applications to the geometry of this moduli space in genus 2, denoted as $\bar{\mathcal S}_2$. First, we characterize the canonical projection of $\bar{\mathcal S}_2$ in terms of its behavior near the boundary, proving in particular that $\bar{\mathcal S}_2$ is not projected. Secondly, we combine the information on superperiods with the explicit calculation of genus 2 Mumford isomorphism, due to Witten, to study the expansion of the superstring measure for genus 2 near the boundary. We also present the proof, due to Deligne, of regularity of the superstring measure on $\bar{\mathcal S}_g$ for any genus.

Let C be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank 2 and fixed odd degree determinant over C, coming from the Schwartz space of $\kappa$-densities on the corresponding stack of bundles (earlier we proved that these functions are locally constant on the locus of very stable bundles). We prove the relevant algebro-geometric statements for curves of genus 2 and for non-hyperelliptic curves of genus 3.