# Research

My papers are also listed on arXiv and Google Scholar.

## Research Papers

• On an invariant bilinear form on the space of automorphic forms via asymptotics, Duke Math. J., to appear. (pdf) (arXiv)
We generalize the definition of the bilinear form $$\mathcal B$$ to an arbitrary split reductive group over a function field. The definition of $$\mathcal B$$ relies on the asymptotics maps defined using the geometry of the wonderful compactification of $$G$$. We show that this bilinear form is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin-Karpelevich formula and certain factorization algebras acting on geometric Eisenstein series. We also give an alternate definition of $$\mathcal B$$ using the constant term operator and the standard intertwining operator.

• On the reductive monoid associated to a parabolic subgroup, J. Lie Theory 27 (3), 637-655 (2017). (pdf) (arXiv)
Let $$G$$ be a connected reductive group over a perfect field $$k$$. We study a certain normal reductive monoid $$\overline M$$ associated to a parabolic $$k$$-subgroup $$P$$ of $$G$$. The group of units of $$\overline M$$ is the Levi factor $$M$$ of $$P$$. We show that $$\overline M$$ is a retract of the affine closure of the quasi-affine variety $$G/U(P)$$. Fixing a parabolic $$P^−$$ opposite to $$P$$, we prove that the affine closure of $$G/U(P)$$ is a retract of the affine closure of the boundary degeneration $$(G \times G)/(P \times_M P^−)$$. Using idempotents, we relate $$\overline M$$ to the Vinberg semigroup of $$G$$. The monoid $$\overline M$$ is used implicitly in the study of stratifications of Drinfeld's compactifications of the moduli stacks $$\mathrm{Bun}_P$$ and $$\mathrm{Bun}_G$$.

• On a strange invariant bilinear form on the space of automorphic forms (with V. Drinfeld), Selecta Math. (N.S.) 22 (4), 1825-1880 (2016). (arXiv)
Let $$F$$ be a global field and $$G:=SL(2)$$. We study the bilinear form $$\mathcal B$$ on the space of $$K$$-finite smooth compactly supported functions on $$G(\mathbb A)/G(F)$$ defined by $$\mathcal B (f_1,f_2):=\mathcal B_{naive}(f_1,f_2)-\langle M^{-1}\text{CT} (f_1)\, ,\text{CT} (f_2)\rangle$$, where $$\mathcal B_{naive}$$ is the usual scalar product, CT is the constant term operator, and $$M$$ is the standard intertwiner. This form is natural from the viewpoint of the geometric Langlands program. To justify this claim, we provide a dictionary between the classical and 'geometric' theory of automorphic forms. We also show that the form $$\mathcal B$$ is related to S. Schieder's Picard-Lefschetz oscillators.

• Radon inversion formulas over local fields, Math. Res. Lett. 23(2), 535–561 (2016). (pdf) (arXiv)
Let $$F$$ be a local field and $$n \ge 2$$ an integer. We study the Radon transform as an operator $$M : \mathcal C_+ \to \mathcal C_-$$ from the space of smooth $$K$$-finite functions on $$F^n \setminus \{0\}$$ with bounded support to the space of smooth $$K$$-finite functions on $$F^n \setminus \{0\}$$ supported away from a neighborhood of $$0$$. These spaces naturally arise in the theory of automorphic forms. We prove that $$M$$ is an isomorphism and provide formulas for $$M^{-1}$$. In the real case, we show that when $$K$$-finiteness is dropped from the definitions, the analog of $$M$$ is not surjective.

• A new Fourier transform, Math. Res. Lett. 22(5), 1541–1562 (2015). (pdf) (arXiv)
In order to define a geometric Fourier transform, one usually works with either $$\ell$$-adic sheaves in characteristic $$p>0$$ or with $$\mathcal D$$-modules in characteristic $$0$$. If one considers $$\ell$$-adic sheaves on the stack quotient of a vector bundle $$V$$ by the homothety action of $$\mathbb G_m$$, however, Laumon provides a uniform geometric construction of the Fourier transform in any characteristic. The category of sheaves on $$[V/\mathbb G_m]$$ is closely related to the category of (unipotently) monodromic sheaves on $$V$$. In this article, we introduce a new functor, which is defined on all sheaves on $$V$$ in any characteristic, and we show that it restricts to an equivalence on monodromic sheaves. We also discuss the relation between this new functor and Laumon's homogeneous transform, the Fourier-Deligne transform, and the usual Fourier transform on $$\mathcal D$$-modules (when the latter are defined).