Related papers: Spectral decomposition of genuine cusp forms over …

Let H be a split reductive group over a local non-archimedean field, and let H^ denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight…

We establish an explicit global spectral decomposition of shifted convolution sums and the second moment of automorphic $L$-functions for Maass forms with explicit integral transforms as well as explicit inversion formulae over every local…

We incorporate covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. We work with all covers that arise from extensions of quasisplit reductive groups by $\mathbf{K}_2$ -- the…

We prove a generalization of the twisted geometric Satake equivalence of Finkelberg--Lysenko in the context of the factorizable grassmannian of a reductive group G relative to a smooth curve X, similar to Gaitsgory's generalization in "On…

We refine the geometric Satake equivalence due to Ginzburg, Beilinson-Drinfeld, and Mirkovi\'c-Vilonen to an equivalence between mixed Tate motives on the double quotient $L^+ G \backslash LG / L^+ G$ and representations of Deligne's…

For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $\Gamma$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the…

We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven…

We introduce a twisted relative trace formula which simultaneously generalizes the twisted trace formula of Langlands et.al. (in the quadratic case) and the relative trace formula of Jacquet and Lai. Certain matching statements relating…

We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension. As an application, we give a description of the nearby cycles on certain Shimura varieties via the…

I extend the ramified geometric Satake equivalence of Zhu from tamely ramified groups to include the case of general connected reductive groups. As a prerequisite I prove basic results on the geometry of affine flag varieties.

In this paper we consider scattering theory on manifolds with special cusp-like metric singularities of warped product type g=dx^2 + x^(-2a)h, where a>0. These metrics form a natural subset in the class of metrics with warped product…

Ben-Zvi--Sakellaridis--Venkatesh described a conjectural extension of the geometric Satake equivalence to spherical varieties, whose spectral decomposition is described by Hamiltonian varieties. The goal of this article is to study their…

We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the…

Fargues-Scholze developed a framework for the geometric Langlands program on the Fargues-Fontaine curve. In particular, they proved the geometric Satake equivalence on the moduli space of closed Cartier divisors on the curve. We prove the…

This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of reductive groups over arbitrary rings. We…

We construct the local Langlands correspondence of essentially unipotent supercuspidal representations under the framework of rigid inner forms and prove a certaion functoriality and compatibilities. This result is stronger than the…

For reductive symmetric spaces G/H of split rank one we identify a class of minimal parabolic subgroups for which certain cuspidal integrals of Harish-Chandra - Schwartz functions are absolutely convergent. Using these integrals we…

We show that Lusztig's theories of two-sided cells and non-unipotent representations of a reductive group over a finite field are compatible with the V. Lafforgue's automorphic-to-galois direction of the Langlands correspondence. To do…

We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the…

Genestier--Lafforgue and Fargues--Scholze have constructed a semisimple local Langlands paramterization for reductive groups over equicharacteristic local fields. Assuming a version of the stable twisted trace formula for function fields,…